Ind. Eng. Chem. Res. 1999, 38, 999-1006
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A Quantitative Controllability Index Alex Zheng* and Rama V. Mahajanam Department of Chemical Engineering, University of MassachusettssAmherst, Amherst, Massachusetts 01003-3110
A controllability index, which quantifies the cost associated with dynamic controllability, is proposed. The index depends quantitatively on the process alternative and control system alternative for each process alternative, and for each of these, the process dynamics, process constraints, product variability, disturbance characteristics, and controller/control structures. Its properties are discussed and its dependence on each of these factors is illustrated for the simplest possible case of a single unit (i.e., a distillation column). 1. Introduction The effect of process design on process controllability has been recognized and discussed in the literature over the last 6 decades (see excellent recent reviews6,14,15 and book by Luyben and Luyben13). For example, Ziegler and Nichols24 point out the relationship between process design and process control in 1943: “In the application of automatic controllers, it is important to realize that controller and process form a unit; credit or discredit for results obtained are attributable to one as much as the other. A poor controller is often able to perform acceptably on a process which is easily controlled. The finest controller made, when applied to a miserably designed process, may not deliver the desired performance. True, on badly designed processes, advanced controllers are able to eke out better results than older models, but on these processes there is a definite end point which can be approached by instrumentation and it falls short of perfection.” More recently, several workshops17,23 have been devoted to this topic (see also refs 9, 10, 22, etc.). One of the earliest industrial applications that illustrate the effect of process design on controllability is by Anderson.1 The system is depicted in Figure 1. The author claims that a poor design of the feed-effluent heat exchanger prevents the process from operating at high production rates, regardless of control system design. While the process may be poorly designed, close examination reveals that the original control system is also poorly designed. Let us consider the steady-state behavior of the overall system. The inlet temperature for the cold stream is constant. Control of the vaporizer pressure implies that the outlet temperature for the cold stream is approximately constant. The inlet temperature for the hot stream (i.e., reactor effluent) is also constant since the reactor inlet temperature and pressure are controlled. Therefore, we have a situation where three of the four temperatures in a heat exchanger are constant. The only manipulated variable is the heat-transfer area which may vary at most from 50 to 100%, assuming that the two heat exchangers are identical, since the first heat exchange is operated fully flooded. It is straightforward to show that, with the control system shown in Figure 1, steady state is only * To whom correspondence should be addressed. E-mail:
[email protected]. Voice: (413) 545-2916. Fax: (413) 5452916.
Figure 1. Original reactor/heat-exchange system.1
feasible for some limited range of production rate. Production rate outside this range would cause severe operational difficulties, precisely what was reported by Anderson. However, removing the pressure control loop on the vaporizer should greatly alleviate the observed operational difficulties. As this example clearly illustrates, poor process design and/or poor control system design can cause operational difficulties. The question is: How should we modify the process design and/or control system design so that the “total” cost is minimized? The total cost equals the steady-state operating cost plus the cost associated with dynamic controllability. Before we can address this question, however, we must be able to quantify the cost associated with dynamic controllability. While many qualitative controllability measures (e.g., relative gain array, closed-loop disturbance gain, conditional number, etc.) exist in the literature, few methods for quantifying the cost associated with dynamic controllability exist. Luyben and Elliot11 propose quantifying the cost of dynamic controllability based on the fraction of time that the plant is producing an onspecification product. Luyben and Floudas12 propose a multiobjective optimization approach in which some factor related to dynamic controllability (e.g., integral square error) is added to the traditional steady-state economic factors. The difficulties are that determining suitable weighting factors is nontrivial and that solving the optimization problem for a reasonable large process is computationally intensive. The importance of developing simple quantitative controllability measures is well-recognized among experts. For example, Morari14 points out “... better, simpler controllability criteria are needed before algo-
10.1021/ie980337y CCC: $18.00 © 1999 American Chemical Society Published on Web 02/13/1999
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rithmic synthesis techniques to trade off controllability and economics become meaningful.” The purpose of this paper is to introduce a quantitative controllability measure which allows designers to quantitatively rank, according to economics (environmental impact and safety can be directly related to costs), alternative plantwide control systems, and alternative process designs. A one-product and continuous chemical process is assumed. Furthermore, product specifications are assumed to be one-dimensional (e.g., product purity), rather than multi-dimensional (e.g., molecular-weight distribution). It should be emphasized that such a measure is complementary to many qualitative measures existing in the literature: Qualitative measures are useful in discarding “poor” alternatives from further consideration quickly in early process design/control system design stages while such a quantitative measure can be used to rank a set of attractive alternatives, both for the process alternatives and for control system alternatives for each process alternative, in the later stages. The rest of the paper is organized as follows: In section 2, the proposed controllability index is defined. Its properties are stated in section 3. Computation of the index is discussed in section 4. Section 5 illustrates its quantitative dependence on controller structure, disturbance, product variability specifications, and so forth, for the simplest possible case of a single unit (i.e., a distillation column). Section 6 concludes the paper. 2. A Quantitative Controllability Index (ν) Since many definitions of controllability exist in the literature, we adopt the following definition for process controllability in this paper.20 Definition 1 (Controllability). A process, with a given control/controller structure, is said to be controllable if all the control objectives and constraints are met dynamically for all the expected disturbances; otherwise, it is uncontrollable with respect to the given control/controller structure. We have defined controllability to be dependent on control/controller structures so that their economic impacts can be evaluated. A process is inherently controllable if it is controllable for some control/controller structures and inherently uncontrollable if it is uncontrollable for all control/controller structures. It is clear from this definition that a quantitative controllability measure should depend on process dynamics, process constraints, disturbance characteristics, product variability, and controller/control structures. Process Dynamics Characteristics. The achievable performance, by any control system, is largely limited by the process dynamics. This has been wellunderstood for linear systems for many years (e.g., see refs 16 and 20). Process Constraints. In this paper, process constraints refer to equipment, chemical (e.g., molar ratio), safety, and environmental constraints that cannot be violated (as opposed to product variability). Qualitatively, one would expect that it is much more difficult to control a process operating close to constraints. Disturbance Characteristics. The disturbance magnitude, frequency, and origin affect process controllability. For example, if measurement noise of a key variable is larger than its variability specifications, then it may be impossible to design a control system such that the variable is within the specifications. Similarly, if a feed
composition disturbance is sufficiently large, it might not be possible to satisfy the overall process material balance. Product Variability. Downs and Ogunnaike6 give a good discussion on the effects of product variability specifications on process controllability. Control/Controller Structures. While it is true that the more sophisticated a controller is, the better the performance is, it is less appreciated that better performance does not always translate into more profit. This is true because the additional cost associated with implementing a more sophisticated controller may exceed any benefit that it may generate. It should be noted that the controller design for a given structure also depends on model uncertainty. The first four factors are largely fixed by design and external forces (e.g., EPA, market, etc.). The purpose of this paper is to introduce a controllability measure that quantifies the economic impacts of all these factors for a continuous chemical process. Many qualitative indexes existing in the literature do not consider all of the above factors. For example, the relative gain array (RGA) does not consider disturbance characteristics, product variability, and process constraints. Definition 2 (the Controllability Index (ν)). Consider a continuous plant. Given a set of disturbances, a set of constraints, a set of control objectives, and a control system, the dynamic controllability index (ν) is defined to be the smallest additional total surge volume (The surge volume may correspond to more than one surge tank or to increasing the surge capacity of more than one process unit, or some combination) required to meet all the control objectives and constraints dynamically for all of the disturbances. The basic idea behind this definition is “what is the minimum surge capacity necessary to meet all the objectives?” Implicit is the assumption that variances of product quality measures can be reduced by mixing which we will assume throughout this paper. (This is always true if the product quality measure is onedimensional (e.g., product impurity) but may fail if the product quality measure is two- or higher-dimensional (e.g., particle-size distribution or molecular-weight distribution, as well as purity). We have intentionally defined ν to be dependent on the control system. The reason is that it allows a direct and quantitative comparison of different controller/control structures. An inherent process controllability index can be thought of as the smallest value of ν over all possible control/ controller structures. This controllability index is motivated by Buckley’s dynamic process control concept.2 Buckley proposed distinguishing between material balance controllers and product quality controllers. Material balance controllers act as low-pass filters (i.e., regulation) and product quality controllers act as high-pass filters (i.e., servo systems). Surge capacities can be chosen such that the break frequency of the material balance control system is an order of magnitude lower than that of the quality control system to avoid significant interaction between the two control systems. Thus, poor quality control can be overcome by installing sufficiently large surge capacities in the process. 3. Properties of ν All the properties stated below assume a continuous one-product process whose product specifications are
Ind. Eng. Chem. Res., Vol. 38, No. 3, 1999 1001
fixed and one-dimensional. From the definition, the following property is obvious. Property 1. A process is controllable if and only if ν e 0. If 0 < ν < ∞, then the process can be made controllable by installing additional surge volume of ν. If ν < 0, then a surge volume of -ν can be removed from the design without affecting controllability. For an existing plant, normally ν g 0 since it may not be possible to remove any surge volume; in this case, such a plant is controllable if and only if ν ) 0. The following property should greatly facilitate the analysis. Property 2. The transfer function for a surge tank is 1/(τs + 1) where the time constant τ is proportional to its volume V. The actual proportional constant depends on the process variable, whose dynamic variation needs to be minimized. For example, if the process variable is component composition (or temperature), then V ) Fτ where F is the flow associated with the stream. If the process variable is the flow, then V ) 2∆F∫∞0 e-t/τ dt ) 2∆Fτ, where ∆F is the change in flow rate, assuming that the flow increases (or decreases) from F to F + ∆F (or F - ∆F) via a first-order transfer function. [Other level control strategies can also be used (see, for example, ref 3).] The factor 2 is needed since the tank is assumed to be 50% full. Additional surge capacities only affect the dynamic behavior of the process, not the steady-state behavior. Thus, ν is unbounded if the problem is not feasible at steady state. Property 3. 1. ν is bounded only if the steady-state control problem is feasible. 2. ν is bounded if the steady-state control problem is strictly feasible and the closed-loop system is asymptotically stable. Strict feasibility means that the steadystate input and output strictly satisfy their respective constraints for all the disturbances. Proof. Since part 1 is obvious, we focus only on part 2. Suppose that the relationship between the output (y) and the disturbance (d) is represented by the following mapping:
y(t) )
∫0tf(t-τ,d(t-τ)) d(τ) dτ
Without loss of generality, we would want |y(t)|∞ e 1 ∀ |d(t)|∞ e 1. The closed-loop stability and strict steadystate feasibility imply that there exists a finite T such that
∫T |f(t,d(t))|1 dt e , ∞
∫0
T
|f(t,d(t))|1 dt e γ,
|∫ f(t,d(t)) dt| T
0
1
e 1 - δ,
with these surge tanks installed, the new output |yˆ (t)|∞ e 1 ∀ t for all the disturbances. By property 2, the dynamics of a surge tank can be represented by a first-order transfer function. Suppose surge tanks are installed to minimize the dynamic variations of all the disturbances. Thus, it is without loss of generality to assume that the new disturbance entering the process with the installed surge tanks is
dˆ (s) )
where R is the time constant and can be made arbitrarily large. Clearly, |dˆ (t)|∞ e 1 ∀ t g 0. Furthermore, for any arbitrarily small positive constant 1, there exists a finite R such that [We have assumed that the plant is initially at steady state (i.e., d(t) ) 0, t < 0)]
|dˆ (t)|∞ e 1, |dˆ (t) - dˆ (t1)|∞ e 1,
yˆ (t) )
t1 e t e t1 + T, t1 g 0
∫0tf(t-τ,dˆ (t-τ))dˆ (τ) dτ
We want to show that there exists a finite R such that |yˆ (t)|∞ e 1 ∀ |d(t)|∞ e 1, t g 0. Below, we will consider the two cases 0 e t e T and t > T. For 0 e t e T, we have
|∫ f(t-τ,d(t-τ))dˆ (τ) dτ| e ∫ |f(t-τ,dˆ (t-τ))| |dˆ (τ)| dτ e ∫ |f(t-τ,dˆ (t-τ))| dτ e γ
|yˆ (t)|∞)
t
∞
0
t
∞
1
0
t 1 0
1
1
Therefore, |yˆ (t)|∞ can be made less than 1 by making 1 sufficiently small (i.e., choosing R sufficiently large but finite). For t > T, we have
|∫ f(t-τ,dˆ (t-τ)) dˆ (τ) dτ| ) |∫ f(t-τ,dˆ (t-τ)) dˆ (τ) dτ +
|yˆ (t)|∞)
t
∞
0
t-T
0
e
|∫
t f(t-τ,dˆ (t-τ)) dˆ (τ) dτ|∞ ∫t-T
t-T
0
e
)
|
f(t-τ,dˆ (t-τ)) dˆ (τ) dτ ∞ +
|∫
t
t-T
|d(t)|∞ e 1, 0 e t e T
hold for any arbitrarily small constant > 0, a finite γ > 0, and some constant 0 < δ < 1. Notice that since ∫∞0 |f(t,d(t))|1 dt is not necessarily less than 1, |y(t)|∞ may be greater than 1 for some disturbance sequence. We want to prove, however, that there exist surge tanks with finite volumes such that,
0eteT
The new output becomes
|d(t)|∞ e 1, t g T |d(t)|∞ e 1, 0 e t e T
1 d(s) Rs + 1
|
f(t-τ,dˆ (t-τ)) dˆ (τ) dτ ∞
∫0t-T|f(t-τ,dˆ (t-τ))|1 dτ + t f(t-τ,dˆ (t-τ)) dˆ (τ) dτ|∞ |∫t-T t f(t-τ,dˆ (t-τ)) dˆ (τ) dτ|∞ ∫Tt|f(τ,dˆ (τ))|1 dτ + |∫t-T
e+
|∫
t
t-T
|
f(t-τ,dˆ (t-τ)) dˆ (τ) dτ ∞
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)+
)+
|∫
t
t-T
|∫
t
t-T
f(t-τ,dˆ (t-τ)) (dˆ (τ)-dˆ (t-T)) dτ + t f(t-τ,dˆ (t-τ)) dˆ (t-T) dτ|∞ ∫t-T
|∫
t
t-T
e+
|
f(t-τ,dˆ (t-τ)) (dˆ (τ) - dˆ (t-T)) dτ ∞ +
|
f(t-τ,dˆ (t-τ)) dˆ (t-T) dτ ∞
t |f(t-τ,dˆ (t-τ))|1 dτ max ∫t-T τ∈[t-T,t]
|dˆ (τ) - dˆ (t-T)|∞ +
|∫
t
t-T
e + 1
∫
Figure 2. Dynamic degrees of freedom.
|
f(t-τ,dˆ (t-τ)) dτ dˆ (t-T) ∞
t
|f(t-τ,dˆ (t-τ))|1 dτ + t-T
|∫
t
t-T
|
f(t-τ,dˆ (t-τ)) dτ 1
e + 1γ + (1 - δ) Since δ > 0, |yˆ (t)|∞, t g T can be made less than 1 by choosing and 1 sufficiently small (i.e., T and R sufficiently large, respectively). Therefore, we conclude that there exist surge tanks with finite volumes such that |yˆ (t)|∞ e 1 ∀ t. 9 Remark 1. Notice in the proof that we have installed surge tanks to minimize the dynamic variation of the disturbances. This is not the only way as one can prove the property by installing surge tanks to minimize the dynamic variation of the output instead. This property reinforces the importance of proper inventory control:5 ν would be unbounded if the material balance is not maintained for some component (which implies an unstable, i.e., integrating, closed-loop system). Additional surge capacities can be thought of as increasing dynamic degrees of freedom. This can be explained by referring to Figure 2. It is obvious that the steady-state degree of freedom is 1 as Fout must equal Fin. However, dynamically, Fout can differ from Fin for a “short” time period. How independently we can manipulate both flows depends on V, the volume of the tank. In the extreme case where V ) ∞, we can manipulate both flows independently, resulting in 2 degrees of freedom for the system. Thus, it is intuitively clear that any surge tank of finite volume increases the dynamic degrees of freedom. Similarly, the vapor rate in a distillation column can exceed the flooding velocity of the column for a short time, and the length of this time period depends on the column design (e.g., holdup). Now we illustrate the utility of this property on the system shown in Figure 1. Example 1. Consider the system shown in Figure 1. At steady state, the total amount of heat being transferred for the two heaters is
Q ) Qcoldout - Qcoldin ) hA∆T where h is the heat-transfer coefficient, A is the total heat-transfer area which is assumed to vary between A0 and 2A0, and ∆T is some average temperature difference between the cold stream and the hot stream. Controlling the vaporizer pressure implies Tcoldout g Tsat, where Tsat is the saturation temperature corresponding to the pressure setpoint. Thus, Q g F(Cp(Tsat - Tcoldin) + λ) } RF, where Cp is the heat capacity and
λ is the heat of vaporization. Thus, the following inequality must be satisfied at steady state:
hA∆T g RF However, this equation is only feasible for some range of production rate F since A is only allowed to vary between A0 and 2A0 and ∆T decreases with increasing F. Steady state would not be feasible if the expected production rate is outside this range; in this case, ν would be unbounded, regardless of how the control system is designed. This steady-state infeasibility problem is caused by improper design of the control structure. Many alternatives exist for resolving this difficulty. For example, one can simply remove the pressure control loop on the vaporizer and control the level on the vaporizer. In this case, ν would be bounded if the furnace and the reactor have adequate capacities to ensure steady-state feasibility for all of the expected production rates. The next property allows us to quantify the cost associated with controllability. Property 4. If ν > 0, then the cost associated with achieving controllability equals the cost associated with installing surge tanks with a total volume of ν (although some surge capacity might also be removed). It should be emphasized that determining ν is one method to quantify the cost associated with controllability (or, more precisely, provide an upper bound on the cost). It is by no means the only method, nor is it necessary to actually install surge tanks with a total volume of ν. For example, one can think of more sophisticated ways of quantifying the cost (e.g., through design modifications to each individual unit). The following property illustrates some qualitative relationships between ν and disturbances and between ν and allowable product variability. Property 5. Consider a fixed flowsheet, a fixed controller, and a fixed set of process constraints. Let ν be the controllability index for a set of disturbances D and a set of allowable product variability Y. 1. Let ν1 be the controllability index for a set of disturbances D1 and a set of allowable product variability Y. If D ⊆ D1, then ν e ν1. 2. Let ν2 be the controllability index for a set of disturbances D and a set of allowable product variability Y1. If Y ⊆ Y1, then ν g ν2. 4. Computation of ν For a continuous process with Ns streams, a surge tank of volume νi is installed for stream i. This is illustrated for a simplified butane alkylation process (Figure 3). Then, the controllability index equals the sum of the optimal values of νi, that is,
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Figure 3. Determination of process controllability cost for a butane alkylation process. N
ν)
νopt ∑ i i)1
where νopt is the solution to the following optimization i problem:
subject to
{
min Φ(νi) νi
y(t) ∈ Y ∀ t d(t) ∈ D ∀ t u(t) ∈ U ∀ t process dynamics controller dynamics
Figure 4. Binary distillation column: (a) schematic diagram; (b) surge tanks locations for determining ν.
(1)
where y is a vector of the controlled variables, u a vector of the manipulated variables, and d a vector of the disturbances. The objective function Φ(νi) depends on N νi the particular application. For example, Φ(νi) ) ∑i)1 N minimizes the total volume while Φ(νi) ) ∑i)1fi(νi), where fi(νi) is the annualized cost associated with installing and maintaining the surge tank of volume νi, minimizes the total cost. The process dynamics can be represented by a process model and its associated uncertainty. Also, for existing processes where surge volumes may not be removed, we have to impose additional constraints (i.e., νi g 0). If the steady-state problem is feasible and the controller stabilizes the process, then property 3 implies that a solution always exists. In general, solving the optimization problem (1) may be computationally intensive. Here, we consider the simplest case where the controller is linear and the process dynamics is described by a linear model, that is,
y ) G(s)u + Gd(s)d u ) C(s)(r - y) Adding surge capacities would introduce first-order lags (i.e., 1/(τs+1)) to the variables d and y. Suppose
yˆ ) Ty(s)y, dˆ ) Td(s)d where Ty(s) and Td(s) are diagonal transfer matrices whose diagonal elements equal 1/(τs+1) with various values of τ. In the special case where all values of τ are zeros, we recover the original problem. Thus, we have, for the regulation problem (i.e., r ) 0),
yˆ ) Ty(s)(I + G(s)C(s))-1Gd(s)Td(s)d Here, we have assumed that the original output y was used for control. Depending on the characteristics of d,
various norms can be minimized. For example, if we want |yˆ (t)|∞ e 1 ∀ t and |d(t)|∞ e 1 ∀ t, then the optimization problem becomes
min Φ(νi) νi subject to |Ty(s)(I + G(s)C(s))-1Gd(s)Td(s)|1 e 1 5. An Illustrating Example Now, let us illustrate how the major factors mentioned earlier affect ν quantitatively for a binary distillation column separating methanol and ethanol, denoted by A and B, respectively (relative volatility is assumed to be 1.5) (Figure 4a). Notice that ν for the system is the sum of the three surge tanks volumes (i.e., ν1, ν2, and ν3 shown in Figure 4b). In order to compute ν, we have to specify the process dynamics, process constraints, product variability, disturbance characteristics, and controller/control structures. Below, we will evaluate ν for the base case for two controllers. The impacts of product variability, process constraints, and disturbance characteristics on ν will then be evaluated individually for the two controllers. 5.1. The Base Case. Process Dynamics. The column has 41 stages, including the partial reboiler and total condenser, and the feed stage is 21. The nominal feed flow rate and molar composition are 100 gpm and 0.5, respectively. The tray holdup is 50 gal, reflux drum holdup is 50 gal, and the sump holdup is 50 gal. All the simulations are carried out using the rigorous dynamic model of the column, developed by Skogestad.20 Process Constraints. There are none. Product Variability. The product purity specifications are (99 ( 0.2)% of light component A for the distillate and (99 ( 0.2)% of heavy component B for the bottoms. Disturbance Characteristics. The feed flow can change stepwise by up to (10% (i.e., 90-110 gpm) (gpm ) gallons per minute). Controller/Control Structures. The column pressure is assumed to be perfectly controlled at 1 atm, and
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Figure 5. Closed-loop responses for the two controllers (solid, CMIMO; dotted, CPI; dashed, product variability limits) to a 10% increase in feed flow.
the reflux drum and reboiler levels are controlled by the distillate flow and the bottoms flow, respectively. The distillate and bottoms compositions (yD and xB, respectively) are controlled by manipulating the reflux flow rate (L) and vapor flow rate (V), that is,
[] [
SPy - yD L ) C SP D - x V xB B
]
where C is the controller, SPyD the setpoint for yD, and SPxB the setpoint for xB. Two controllers (one consisting of two single-loop PI controllers, CPI, and one multivariable controller, CMIMO), which are designed for robust performance (i.e., model uncertainty has been incorporated into controller design) by Skogestad and Morari,19 are used to investigate effects of the controller structure on ν. The transfer function matrices for the two controllers are given below.
[
CPI(s) )
Figure 5 shows that the product variability specifications are satisfied with both controllers for the worstcase disturbance. Thus, ν e 0 for both controllers (i.e., no additional surge capacity is needed and some should possibly be removed, resulting in cost savings, if the calculations are made before the column is built). For an existing plant, ν is assumed to be non-negative and thus ν ) 0. 5.2. Effects of Product Variability on ν. Figure 6a shows quantitative effects of product variability specifications, which are assumed to be the same for both products for simplicity, on ν. All other specifications are the same as the base case. Initially, decreasing product variability specifications have no effect on controllability since all the specifications are well within their limits at the nominal operating conditions (Figure 5). When product variability specifications become 0.12% (versus 0.2%), all objectives are barely met. Thus, further decreases would result in an uncontrollable process, and
[
2.4(75s + 1) 1 0 0 -1 s
5870(s+16.1)(s+0.0324)(s2+3.07s+2.35) s(s+0.151)(s+9.03)(s+583) CMIMO(s) ) 5000(s+3.14)(s-0.0211)(s2+1.2s+8.4) s(s+0.151)(s+9.03)(s+583) additional surge volume would be needed to ensure that all objectives are met. For example, for product variability specifications of 0.02%, the additional surge volumes for CPI and CMIMO are 8600 and 1860 gal, respectively. Figures 6b and 6c show that the much tighter specifications are indeed met for the two controllers with respective additional surge volumes. The surge volumes are computed by minimizing the following objective function (i.e., total annualized cost):
Φ ) 150(ν0.65 + ν0.65 + ν0.65 1 2 3 )
]
-3820(s+2.7)(s-0.0544)(s2-3.37s+8) s(s+0.151)(s+9.03)(s+583) -4880(s+19.3)(s+0.0413)(s2+2.79s+2.05) s(s+0.151)(s+9.03)(s+583)
]
The cost correlation 150ν0.65 for computing the annualized cost of installing and maintaining a surge tank of volume ν was obtained from Peters and Timmerhaus.18 Notice that this correlation does not consider the inventory and environmental costs of the material inside the tank, which can be significant in many cases. It turns out that ν ) ν1 and ν2 ) ν3 ) 0 for this example. Without disturbance, the annualized utility cost (i.e., steam and cooling water) is about $2 million (based on an operating time of 8000 h/year and the fuel cost of $4/million BTU for both steam and cooling water4). The
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Figure 6. (a) Effects of product purity specifications on ν (solid, CMIMO; dotted, CPI); (b) closed-loop responses for CMIMO with (dotted) and without (solid) an additional surge volume of 1860 gal for the worst-case disturbance (10% increase in feed flow rate) when product purity specifications are (0.02% (dashed); (c) closed-loop responses for CPI with (dotted) and without (solid) an additional surge volume of 8600 gal for the worst-case disturbance (10% increase in feed flow rate) when product purity specifications are (0.02% (dashed). Table 1. Annualized Controllability Costs for the Two Controllers vs Product Variability Specifications ((10% Flow Disturbance and No Constraints on ∆V) product variability specifications
CPI
CMIMO
0.2% 0.1% 0.05% 0.02%
0 $5,000 $18,000 $53,000
0 $2,000 $8,000 $20,000
cost of controllability (i.e., annualized cost for surge tanks of volumes ν1, ν2, and ν3) for various values of product variability specifications is summarized in Table 1. Without the additional surge capacity, product purity specifications of 99 ( 0.02% will be met about 80% of the time for the PI controller (Figure 6c), if the worst-case disturbance occurs once a day. Thus, the utility cost of reworking the off-specification materials, assuming that the column has the required capacity (The cost would be much higher if the column is already operating at its capacity) would be increased by about 25%, or $0.5 million, significantly larger than the cost of installing the required surge volume (i.e., $53,000). While installing excessive surge capacities can be expensive, insufficient surge capacities can be much more costly. The controllability index ν can be used to guide a design with just-enough surge capacity. 5.3. Effects of Process Constraints and Disturbance Characteristics on ν. The effects of the constraint on ∆V (rate of change for vapor flow rate) and flow disturbance magnitude are shown in Figures 7a and 6b, respectively. While the qualitative trends are similar to that for product variability specifications, their quantitative effects on v are much smaller, for this simple example. 6. Conclusions A quantitative controllability index has been proposed and its quantitative dependence on the process dynam-
ics, process constraints, product variability, disturbance characteristics, and controller/control structures has been illustrated for the simplest possible case of a single unit (i.e., a binary distillation column). Such an index allows one to compare the process design alternative and control system alternative quantitatively in terms of economics. Thus, it can be directly used in the synthesis of optimal plantwide control systems, where alternative control systems are compared economically, and in the integration of process design and process control, where alternative process designs and alternative control systems are compared. It is worthwhile to emphasize again that the index is most useful in the later design stages in ranking alternative plantwide control systems and alternative process designs according to economics. To make such an index applicable in early design stages, short-cut methods to quickly evaluate it need to be developed. Acknowledgment The author wishes to thank Professor J. M. Douglas of the University of Massachusetts-Amherst for numerous discussions and valuable comments. The financial support from the National Science Foundation (CTS9713599) is gratefully acknowledged. Nomenclature A, A0 ) heat-transfer area C ) controller CPI ) decentralized controller consisting of two single-loop PI controllers CMIMO ) multivariable controller d ) disturbance vector h ) heat-transfer coefficient L ) reflux flow rate
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Q ) heat SPxB ) bottoms composition setpoint SPyD, ) distillate composition setpoint T ) temperature ∆T ) average temperature difference between hot and cold streams u ) manipulated input vector V ) reboiler vapor flow rate (or tank volume) ∆V ) rate of change for reboiler vapor flow xB ) bottoms composition y ) output vector yD ) distillate composition ν ) controllability index νi ) surge tank volume associated with stream i τ ) time constant in a first-order transfer function
Figure 7. (a) Effects of ∆V constraint on ν; (b) effects of disturbance magnitude on ν. (Solid, CMIMO; dotted, CPI.)
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Received for review May 28, 1998 Revised manuscript received December 4, 1998 Accepted December 14, 1998 IE980337Y
