A Profit Index for Assessing the Benefits of Process Control

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Ind. Eng. Chem. Res. 2007, 46, 5614-5623

A Profit Index for Assessing the Benefits of Process Control Margret Bauer,† Ian K. Craig,*,† Elsa Tolsma,‡ and Hannes de Beer‡ Department of Electrical, Electronic and Computer Engineering, UniVersity of Pretoria, 0002 Pretoria, South Africa, and Sasol Synfuels, Instrumentation and Control Engineering, AdVanced Solutions, 1 Synfuels Road, Secunda, South Africa

In industrial processes, the potential benefit that arises from improved control is assessed before the installation of a new control strategy. Estimating the economic benefit motivates the investment in a new control system and minimizes the risk of the new technology not living up to its promises. In this paper, a method for a priori estimation of the benefit from improved process control is presented. The method builds on conventional estimation techniques and evaluates the performance function associated with the controlled variable by assuming a reduction in variance. Three frequently used performance functions, namely, a quadratic performance function, a linear performance function with constraints, and the so-called clifftent performance function, are investigated. The profit index as a function of the variance is computed, and the implications of a reduction in variance for each performance function are assessed. A further contribution of this article is the analytical derivation of the optimal operating point of the controlled variable for a given performance function and a given variance. The estimated benefit from process control originates from a reduction in variance. Several approaches have been proposed to estimate and assess the variance reduction. Here, an index for the assessment of the current economic benefit incorporates the minimum achievable variance to estimate the reduction in variance. The economic benefit resulting from process control and the online monitoring index are demonstrated on an industrial refining process. Introduction Weighing cost against economic benefits is crucial when justifying the investment of a new process control system or system upgrade, since most such decisions in the chemical industry are based on profitability.1 An overview of estimation techniques for the economic benefit of process control has recently been presented.8 Common business drivers for process control are the minimization of product and quality variability. While the cost can be obtained from control system vendors, past experience, or other sources, the economic benefit has to be estimated for each plant and process individually. Estimation approaches that have been pursued in the chemical and petrochemical industry for quantifying the benefit are often based on the reduction of variability.2-5 In many cases, an upper or lower limit for a controlled variable is assumed, which must not be violated excessively. A reduction in variance allows shifting the mean of the controlled variable closer to the constraint and thus ensures better performance.6 Often, important controlled variables can also be directly linked to a performance function.4,5,7 If a performance function is available, then the improvement of the economic benefit is derived from both the shift in the operating point and the reduced variance. In this work, a profit index is derived that uses the performance function and thus links the benefit improvement directly to the reduction in variance. The profit index allows the direct deduction of the economic benefit resulting from a reduction in variance. Benefits from Process Control Economic assessment of process control systems has been a concern of process control engineers and advanced process * To whom correspondence should be addressed. Phone: +27(12)420-2172. Fax: +27(12)362-5000. E-mail: [email protected]. † University of Pretoria ‡ Sasol Synfuels.

control (APC) system vendors since the first implementation of such systems. Many of the applications cited in this work have therefore been reported by industrial authors for whom the issue is part of their day-to-day business. Although highly relevant and of practical value, reported methods often lack a theoretical basis and rely heavily on simplified assumptions. Papers written by APC vendors are sometimes met with a degree of skepticism as it is in their interest to report economically successful applications. Furthermore, there is some tendency to beautify the results of APC projects, as found in a recently conducted Web-based survey on the economic performance assessment of APC.8 It is however in the best long-term interest of APC users and vendors alike to get to the truth, a fact that is appreciated by the industry at large. A conventional approach to estimate the benefits from process control is to reduce the variability in a process or quality variable first and then to shift the operating point close to the specification limit. Several guidelines for the shift in operating points have recently been summarized by Martin2 and analyzed in their statistical sense by Muske and Finegan.9 The most commonly used technique is the same percentage rule or equal operation at the limit, which is illustrated in Figure 1 using the probability density function, also called frequency distribution, of a controlled variable. The distribution is assumed to be Gaussian, and it will be shown later that this is a reasonable assumption for most industrial processes. The same percentage rule is recommended when a reasonable percentage of the process operation data violates the specification limit xL. The shift in the operating point from the base case mean xjb to the improved control mean xji, indicated in Figure 1 by ∆x, results from allowing the same percentage of process data to violate the limit xL. Because of the reduction in variance, the operating point can be shifted closer to the limit and still show the same percentage of violation. The shift ∆x is therefore sometimes referred to as “the hidden benefit from process control”.14

10.1021/ie0614273 CCC: $37.00 © 2007 American Chemical Society Published on Web 07/21/2007

Ind. Eng. Chem. Res., Vol. 46, No. 17, 2007 5615

Figure 1. Graph showing that the same percentage rule estimates the economic benefit by allowing the same violation of the limit before and after the introduction of a new control system.

Statistical estimation approaches thus consider the situation before the introduction of a new control scheme. This situation is often referred to as the base case4 and gives the current performance. The economic benefit improvement is also estimated for the improVed control case, quantifying and assessing the performance of the new system. Zhou and Forbes10 introduced two further performance estimates: the performance after retuning the existing control scheme and the maximum achieVable performance. The reason for considering the performance after retuning is that otherwise the economic benefit is attributed to the control system upgrade while retuning of the existing system could potentially improve the situation considerably with little additional cost. The maximum achievable performance on the other hand is considered to ensure that the control system upgrade results in a significant improvement. The measure of significance is, in this case, the minimum achievable variance. The economic benefit resulting from minimum achievable variance was also discussed by Muske.6 To make the decision of investing in a new process control strategy, the performance of the base case should be carefully compared against the achievable, retuned, and improved control situation. Some conventional approaches derive the economic benefit directly from the shift of the operating point ∆x. This often assumes a linear relationship between the economic benefit and the shift ∆x. Martin and co-workers,11 for example, summarize methods that are frequently used in industry and state that if the operating point is shifted closer to the constraint then “the profit increases with x”, where x is the process or quality

variable. That the relationship is not necessarily linear will be shown for the examples below. Instead of deriving the benefit from the shift, it can also be given by a performance function ϑ(x), which is a function of the controlled variable x. If a performance function as well as the frequency distribution f(x) of the controlled variable is available, then the aVerage profit can be calculated by integrating the product of ϑ(x) and f(x) over x. This profit integral plays a significant role when estimating the economic benefit from control improvement. The proposed profit index, which will be introduced in the next section, for example, is the maximized profit integral. The maximization is achieved by adjusting the operating point to its optimum value. Three commonly used types of performance functions will be discussed in the following and are shown in Figure 2: a quadratic performance function, a linear performance function with constraints, and the clifftent performance function. The performance functions used in this paper are the ones most commonly referred to in the literature, albeit often without much theoretical justification. The methodology presented here is however valid for essentially any performance function imaginable. Performance functions are difficult to obtain. They differ from process to process, and comprehensive knowledge of the process, the economic environment, and sales agreements are required. There is a need for a systematic method to determine performance functions, which appears not to be available in the literature. The development of such a method is considered outside the scope of this paper. Good process engineers will, however, at least have some knowledge of the economic impact of important controlled variables. This expert knowledge can be converted to a performance function through carefully considered questioning.12 Performance functions can also be obtained experimentally,25 or from an analysis of the product value versus production costs.10 Quadratic Performance Function. For many process variables, the maximum benefit ϑm is achieved when the process variable is at an optimal operating point x0. The benefit decreases if the variable deviates from x0. In the case of the quadratic performance function as described in eq 1, the optimum operating point x0 is equal to the point xm at which the performance function is at its maximum ϑm. The economic benefit can often be approximated by a quadratic performance function that uses a second-order polynomial fit as shown in Figure 2i. Martin et al.11 refer to the quadratic performance

Figure 2. Commonly used performance functions: (i) quadratic function,13 (ii) linear function with constraints,14 and (iii) clifftent performance function.15

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Ind. Eng. Chem. Res., Vol. 46, No. 17, 2007

function as the curvature rule. An example of a quadratic performance function was given by Stout and Cline13 for an ammonia production process in which the profit has a linear relationship with throughput, and throughput is a quadratic function of a composition ratio. When using a quadratic performance function, a reduction in variance will center the process data closer to x0. Unlike the operating point for constraint performance functions, the optimal operating point x0 determined by the quadratic performance function is unaffected by an increase or decrease of the variance. The quadratic performance function can be expressed by the following equation:

ϑquad(x) )

{[

ϑm 1 0

( )] x - xm x1

2

xm - x1 e x e xm + x1 x > xm + x1, x < xm - x1 (1)

where ϑm is the maximum achievable profit and xm ( x1 represents the thresholds above which the profit is zero. The zero constraint is introduced because if the performance function should continue to fall beyond the threshold xm ( x1 the profit would become negative. A negative performance function will contribute to the cost rather than the benefit and is therefore excluded in this description of the performance function ϑ(x). Linear Performance Function with Constraints. A common behavior of the profit is a linear relationship with constraints as shown in Figure 2ii. The performance increases linearly from a point x1 until a threshold x2 at which, if exceeded, the product turns out as waste and the profit is zero. This relationship has been described by Latour14 for the example of a fuel oil blending process. In this example, the profit increases linearly with the amount of sulfur content, which is the quality variable. If the sulfur content exceeds a limit, then environmental regulations will be violated and the product is considered worthless. Latour14 extends this example also to consider cases where overspecification material encounters a sliding penalty on the selling price, leading to perhaps a more realistic piecewise linear performance function. What is important, however, is that eq 2 can be utilized to construct more complex piecewise linear performance functions. The clifftent function described below is one such function, as is the multivariate performance function described by Zhou and Forbes10 for a bleach plant in a pulp mill. The linear performance function with constraints can be described as follows:

{

x2 - x x e x e x2 ϑm x 2 - x1 1 ϑlin(x) ) x < x1, x > x2 0

(2)

where x2 is the constraint limit and x1 is the limit below which the performance function is equal to zero. The maximum economic benefit ϑm can only be achieved if the variance is zero and the operating point is at x2. For a nonzero variance, an optimal operating point has to be determined that will lie in the interval between x1 and x2, close to x2. Clifftent Performance Function. The clifftent performance function was introduced and named by Latour15 and has been discussed for the case of multivariate optimization by Zhou et al.16 The name clifftent is derived from the shape of the function, as illustrated in Figure 2iii. The function increases and decreases linearly, resembling a tent with a cliff at the top. The clifftent performance function can be expressed as the sum of the two linear constraint variables, ϑcliff(x) ) ϑ1(x) + ϑ2(x) with the

following formulations for ϑ1 and ϑ2:

{ {

x - x1 x e x e x2 ϑm x 2 - x1 1 ϑ1(x) ) x < x1, x > x2 0 x3 - x x e x e x3 cϑm x3 - x2 2 ϑ2(x) ) x < x2, x > x3 0

(3)

where x1 and x3 are the zero profit thresholds and x2 is the point at which the maximum economic benefit ϑm is achieved for zero variance. The maximum value of the second “tent” can be expressed in terms of ϑm, that is, ϑc ) cϑm where c is a constant 0 e c e 1. For a nonzero variance, an optimal operating point has to be determined that will lie, similar to the linear performance function with constraints, in the interval between x1 and x2 close to x2. Profit Index The objective of the profit index is to express the economic benefit estimate in terms of variance, or standard deviation σx, rather than in terms of the controlled variable x as given in eqs 1-3. The reason for this is that the performance after retuning, that is, the improved control performance and the minimum achievable performance, are all based on the variance estimate. A profit index as a function of variance or standard deviation relates the reduction of variance directly to the profit to be gained from that reduction. The profit integral, also called the time profit by Latour,15 will be referred to as the profit index in the following. It is given by the following expression:

Ψ(σx,µx) )

∫-∞+∞ ϑ(x) f(x,σx,µx) dx

(4)

where ϑ is the performance function, f is the frequency distribution or probability density function of the time series of the controlled variable x, and µx is the average or mean of the controlled variable x. The function f is either estimated or determined from process data. Authors in the control loop performance assessment literature often assume that the time series of control loop outputs are stationary and ergodic, as is done in this paper (see for example Choudhury et al.18). This is a reasonable assumption provided the control loop is functioning properly. Furthermore, eq 4 is valid for just about any performance function as ϑ is bounded for industrial processes. The contribution of this paper is the derivation of the profit integral as a function of only the standard deviation σx for the commonly introduced performance functions described in eqs 1-3. To achieve this, the optimum operating point x0 is determined for a given standard deviation σx such that x0 ) g(σx) where g(‚) is an invertible function. The mean of the controlled variable is then shifted to the optimum operating point such that x0 ) g(σx). The profit index Ψ is then only a function of σx, that is, Ψ[σx,g(σx)] ) Ψ(σx). For the most common performance functions, the profit index decreases monotonically with σx and the maximum of Ψ occurs for zero standard deviation. Time series are usually assumed to be Gaussian if generated by a linear source. As most process control loops are approximately linear in a particular operating region, the time series of a controlled variable is often assumed to be Gaussian (see, e.g., Zhou and Forbes10). In fact, the non-Gaussianity of a control loop time series is used as an indication that the loop


Figure 3. Profit index Ψ of a quadratic performance function as a function of the standard deviation σx (i) and first derivative of the performance function over σx (ii).

might be performing poorly.17,18 The controlled variable time series x used in this paper is therefore assumed to be Gaussian. A Gaussian distribution is fully determined if the standard deviation σx and the mean µx are given. If the mean µx is assumed to be equal to the optimal operating point, that is, µx ) x0, the frequency distribution can be described by

f(x) )

1

x2πσx

[

exp -

]

(x - x0)2 2σx2

(5)

In the following, the profit index will be calculated and discussed for each performance function individually. Quadratic Performance Function. The analytical expression for the profit integral from eq 4 using the quadratic performance function of eq 1 is as follows after substituting z ) x1/(x2σx):

[

Ψquad(σx) ) ϑm (1 - 2z2) erf{z} +

1 -z2 e xπz

]

(6)

where “erf” is the Gaussian error function. The relationship between Ψquad and σx is displayed in Figure 3i where Ψquad and σx are scaled to the maximum profit ϑm and threshold x1, respectively. For example, a reduction in standard deviation by ∆σx from σx ) 2x1 to σx ) x1 and the resulting increase in economic benefit ∆Ψquad are plotted in Figure 3i. This shows that the increase in economic benefit can be directly obtained from the performance function. The profit decreases monotonically but not linearly with σx, and the maximum profit, ϑm, is achieved for zero standard deviation. For small values of σx, only small improvements in the economic benefit result in a reduction of the profit. For example, if σx is reduced from σx ) 0.2x1 to σx ) 0.1x1, that is, a reduction by 50%, the profit increases from Ψquad ) 0.96ϑm to Ψquad ) 0.99ϑm or by 3%. If, on the other hand, σx is reduced from σx ) 0.5x1 to σx ) 0.4x1, that is, a reduction by 20%, the profit increases from Ψquad ) 0.77ϑm to Ψquad ) 0.84ϑm or by 8%. To illustrate at which values of σx the reduction in standard deviation results in the highest benefit, the change in profit, that is, the first derivative of Ψ(σx), is derived as follows:

[

]

dΨquad(σx) x2 2 -z2 1 ) ϑm e - erf{z} , z ) x1/(x2σx) dσx x1 xπ z (7) The first derivative is shown in Figure 3ii and illustrates that the improvement of economic benefit is small for small values of σx but also for very large values of σx. The maximum achievable profit is at the local minimum of dΨquad(σx)/dσx,

Figure 4. Profit index for a linear performance function Ψlin with constraints as a function of the operating point x0 (x1 ) 0, x2 ) 1, and σx ) 0.1).

≈ which is the inflection point of Ψquad(σx) and is at σinfl x 0.47x1. Thus, if the standard deviation is improved from a base standard deviation that lies around σinfl x , then a reduction in standard deviation will have the largest impact on the profit improvement. Linear Performance Function with Constraints. The linear performance function with constraints is shown in Figure 2ii and described in eq 2. The analytical expression for the profit integral from eq 4 using the linear performance function with constraints of eq 2 and the assumption of a Gaussian distribution from eq 5 is as follows:

Ψlin(σx,x0) ) ϑm 2 1 -z22 [e - e-z1 ] + z1[erf{z2} - erf{z1}] 2(z2 - z1) xπ

{

}

(8)

where z1 ) (x0 - x1)/(x2σx) and z2 ) (x0 - x2)/(x2σx). The optimum operating point x0 can be derived by maximizing the economic benefit. Figure 4 shows the profit index as a function of the operating point for a given standard deviation after setting, without any loss of generalization, x1 in eq 8 to zero. The profit index is at its maximum for x0 values smaller than but close to x2. In order to find the maximum, the derivative of the performance function Ψlin over the operating point x0 is derived as follows:

[

]

ϑm erf{z2} - erf{z1} e-z22 dΨlin(σx,x0) ) dx0 2(z2 - z1) x2σx xπ

(9)

The maximum profit is achieved for dΨlin(σx,x0)/dx0 ) 0. This equation can be solved numerically. The dependency of the

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Figure 5. Optimum operating point x0 of a linear performance function with constraints as a function of the standard deviation (x1 ) 0).

optimum operating point x0 on the standard deviation is shown in Figure 5. For zero standard deviation, the profit is at its maximum ϑm if x0 ) x2. For large values of σx, the optimum operating point approaches a final value of x0(σx f ∞) ) 2/3, which can be assumed for σx > x2. Note that, whereas the optimum operating point approaches a final value as σx increases, the profit index decreases monotonically as shown in Figure 6i. Once the optimum operating point is determined, it can be substituted in the original definition of the performance function in eq 8. The resulting performance function Ψlin, which is now only a function of the standard deviation, is shown in Figure 6i. As for the quadratic performance function, the economic benefit that results from a reduction in variance can be directly deduced and is shown for the example of a reduction from σx ) 2x1 to σx ) x1 in Figure 6i. The performance function decreases monotonically but not linearly with σx. Rather, the main economic improvement results in a reduction of standard deviation below σx < x2 for which the potential improvement is ∆Ψlin ) Ψ(0) - Ψ(x2) ≈ 0.8ϑm, as can be seen in Figure 6. This implies that a reduction of above σx < x2 does not result in a significant improvement of the economic benefit, which can be observed best by obtaining the first derivative of ∆Ψlin over σx as follows:

dΨlin(σx) ) dσx ϑm

{[1 - 2z1z2 + 2z22] e-z2 - e-z1 } (10) 2

2

2xπσx(z2 - z1)

Figure 6ii shows the first derivative of the profit index Ψlin. Unlike the quadratic performance function, the first derivative

of the linear performance function with constraints has no local minimum other than for zero variance. Instead, the first derivative increases monotonically. Thus, the smaller the standard deviation, the higher the resulting profit improvement will be from reducing the standard deviation. The conventional approach of computing the shift of the operating point due to the improved variance can now be examined more closely. Figure 7 shows the profit index Ψlin as a function of the optimum operating point x0 where both parameters are a function of standard deviation σx. If σx ) 0, then Ψlin is at its maximum and x0 is equal to the constraint limit x2. If σx approaches infinity, the profit index goes to zero and the optimal operating point approaches the limit value of 2/3. The relationship between the shift in mean and the economic benefit improvement, which was assumed to be linear, is in fact almost linear for small values of σx. For values larger than σx > 0.1x2, however, the relationship is clearly nonlinear. Assuming a linear relationship between the shift of the operating point and the economic benefit improvement is therefore not always valid. Clifftent Performance Function. The clifftent performance function can be expressed as the sum of two “tents”, as illustrated in Figure 2iii and described in eq 3. The first tent is the linear performance function with constraints as given in eq 8, and the second tent can be viewed as a linear performance function with a constraint as the lower limit. The clifftent performance function can therefore be expressed as

Ψcliff(σx,x0) ) Ψlin(σx,x0) + Ψlin2(σx,x0)

(11)

with

Ψlin(σx,x0) ) ϑm 1

{

2

Ψlin2(σx,x0) ) cϑm 1

{

}

(12)

}

(13)

[e-z2 - e-z1 ] + z1[erf{z2} - erf{z1}] 2

2(z2 - z1) xπ

2(z2 - z3) xπ

[e-z3 - e-z2 ] + z3[erf{z3} - erf{z2}] 2

2

where z1 ) (x0 - x1)/(x2σx), z2 ) (x0 - x2)/(x2σx), and z3 ) (x0 - x3)/(x2σx). The profit index is therefore always larger than the profit index of a linear performance function with constraints as given in eq 8. Also, the optimal operating point of the clifftent performance function will lie closer to the constraint x2 since the profit is not zero if x is larger than x2. Both the larger profit index and the shifted optimal operating

Figure 6. Profit index Ψ of a linear performance function with constraints as a function of the standard deviation σx (i) and first derivative of the performance function over σx (ii).


Figure 7. Profit index Ψlin as a function of the operating point x0. The resulting benefit from a shift of the operating point ∆x can be seen directly in the graph (∆Ψlin).

Figure 8. Profit index for the clifftent performance function Ψcliff as a function of the operating point x0 (x1 ) 0, x2 ) 1, x3 ) 1.2x2, c ) 0.6, and σx ) 0.1). The dashed line indicates the profit index of a linear performance function with constraints using the same parameters for x1 and x2.

point can be seen in Figure 8 for a set of parameters for x2, x3, and c. These parameters are set to match the process requirements. Expert knowledge is required to establish the connection between the process variable and a financial value, which will depend on the process under investigation. An example of the parameters settings for c, x2, and x3 for a refinery process is given by Latour.15 Again, without any loss of generality, x1 is set to zero. In order to find the maximum, the derivative of the performance function Ψcliff over the operating point x0 is computed as follows:

dΨcliff(σx) dΨlin(σx) dΨlin2(σx) ) + dx0 dx0 dx0

[ [

(14)

] ]

ϑm erf{z2} - erf{z1} e-z22 dΨlin(σx) ) dx0 2(z2 - z1) x2σx xπ

(15)

cϑm erf{z3} - erf{z2} e-z22 dΨlin2(σx) ) + dx0 2(z2 - z3) x2σx xπ

(16)

The maximum profit is achieved for dΨcliff(σx,x0)/dx0 ) 0. Again, the solution can be found numerically, and the next steps are similar to the linear performance function with constraints. The dependency of the optimum operating point x0 on the variance is shown in Figure 9. For zero standard deviation, the profit is at its maximum ϑm if x0 ) x2. For large values of σx, the optimum operating point approaches a final value, which

Figure 9. Optimum operating point x0 of the clifftent performance function with constraints as a function of the standard deviation (x1 ) 0, x3 ) 1.2x2, and c ) 0.6). The dashed line indicates the optimum operating point for the linear performance function with constraints.

can be approximated by x0(σx f ∞) ) 1/x2 for the given parameters of x3 and c. Once the optimum operating point is determined, it can be substituted in the original definition of the performance function in eq 11. The resulting performance function Ψcliff, which is now only a function of the standard deviation, is shown in Figure 10i. As previously shown for the quadratic and linear performance function with constraints, the economic benefit that results from a reduction in variance can be directly deduced and is shown for the example of a reduction from σx ) 2x1 to σx ) x1 in Figure 10i. The performance function decreases monotonically but not linearly with σx. The maximum improvement can be achieved for small values of σx. This can also be observed when obtaining the first derivative of ∆Ψcliff over σx that can be derived as follows:

dΨcliff(σx) dΨlin(σx) dΨlin2(σx) ) + dσx dσx dσx

(17)

dΨlin(σx) ) dσx ϑm

{[1 - 2z1z2 + 2z22] e-z2 - e-z1 } (18) 2

2

2xπσx(z2 - z1) dΨlin2(σx) ) dσx cϑm

{[1 - 2z2z3 + 2z22] e-z2 - e-z3 } (19) 2

2

2xπσx(z3 - z2)

and is plotted in Figure 10ii. As for the linear performance function with constraints, the smaller the standard deviation, the higher the resulting profit improvement will be from reducing the standard deviation. The conventional approach of computing the shift of the operating point due to the improved variance can also be examined more closely. Figure 11 shows the profit index Ψcliff as a function of the optimum operating point x0 where both parameters are a function of standard deviation σx. As in the linear performance function, if σx ) 0, then Ψcliff is at its maximum and x0 equals the constraint limit x2. If σx approaches infinity, the profit index goes to zero and the optimal operating point approaches the limit value of x0 ) 1/x2. Variance Reduction Estimate The first step when estimating the reduction of variance is the determination of the base case. The variance of the base

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Figure 10. Profit index Ψ of the clifftent function (x1 ) 0, x3 ) 1.2x2, and c ) 0.6) as a function of the standard deviation σx (i) and first derivative of the performance function over σx (ii). The dashed lines indicate the profit index and derivative of the profit index for the linear performance function with constraints.

Figure 11. Profit index Ψcliff as a function of the operating point x0.

case can be approximated as follows if historical data of the process or quality variable x are available:

σB2 ≈

1

N

∑

N - 1k)1

(xk - x0)2

(20)

where N is the number of samples, xk is the value of x at sample k, and x0 is the average value of x. The variance σC of the proposed control system is then estimated in a second step. At this point, it should be noted that both the variance and standard deviation are used for estimating the economic benefit. The tendency is that authors with an industrial background use the standard deviation, which gives the average size of deviation from the operating point,3,4 while academic papers refer to the variance.6,10 Many reported industrial applications assume a reduction in standard deviation by a fixed percentage derived from a rule of thumb. The most commonly adopted reduction2,7,11,14 in the standard deviation is 50% of the base case, that is, σC ) 50%σB. Other publications assume even larger reductions of 50-66%,13 85%,19 or even up to 90%.3,4 In this sense, Bozenhardt and Dybeck5 are the exception by proposing a reduction of only 35%. The reduction of variance is therefore a result of control improvement rather than comparing the no-control against the controlled case. The implications that the reduction of σx has for the economic benefit are shown in Figure 12 for a linear performance function with constraints. The economic benefit depends on the base case and is given by

∆Ψ ) Ψ(σC) - Ψ(σB)

(21)

The economic benefit ∆Ψ is positive because Ψ is a monotonically decreasing function and σC is known to be smaller than

Figure 12. Profit index of a linear performance function with constraints: two cases of reduction of the standard deviation by 50% result in significantly different improvements of the economic benefit ∆Ψlin.

σB. A significantly smaller economic benefit results for a base case of σB ) 4x2 versus that for σB ) x2 as illustrated in Figure 12. A fixed percentage reduction in standard deviation, however, has to be taken with caution. Most of the reduction estimates were made more than 20 years ago. Advances have been made in the implementation of distributed control systems, and previously exceptional proportional-integral-derivative control has become a standard control scheme. In a recent publication, Zhou and Forbes10 therefore propose to implement and tune a controller using a dynamic simulation model. σC for the proposed control system is then determined from simulation data. Furthermore, a more objective estimate of the reduction of variance, namely, the minimum variance estimate, has been suggested for assessing the economic benefit. Minimum variance as a lower limit for variance reduction is discussed in the following. Minimum Variance Estimate. The concept of minimum variance has been proposed as a measure to estimate the potential economic benefit6,10 by establishing a lower bound. The maximum achievable profit improvement can be obtained from a minimum achievable variance estimate or rather the minimum achievable standard deviation σMV.

∆Ψmax ) Ψ(σMV) - Ψ(σB)

(22)

Minimum variance is the variance that is caused by an external disturbance which cannot be eliminated by the controller. The concept of minimum variance and a controller that achieves minimum variance was proposed in the 1960s by Åstro¨m.20 Harris21 used the concept to derive a performance measure of


- 1 impulse response coefficients as follows: d-1

σMV2 ) (1 +

Figure 13. Profit index Ψlin of a linear performance function with constraints: the minimum variance determines the maximum achievable profit improvement ∆Ψlinmax.

control loops, which is now a standard tool for assessing control loop performance. Recent tutorials by Qin22 and Jelali23 describe the mechanism and implementation schemes of control loop performance using minimum variance. The underlying assumption is that an additive disturbance affects the process variable. The disturbance is modeled as a Gaussian noise sequence wk with zero mean and variance σw2 filtered by an ARIMA transfer function Gw. The closed loop transfer function with process variable xk as the output and noise input sequence wk as the input can be represented by an impulse response model as follows:

xk ) )

Gw(z-1)

wk

1 + Gp(z-1) Gc(z-1) B(z-1) wk A(z-1) ∞

)

φiwk-i ∑ i)0

(23)

where Gp is the process model and Gc is the controller transfer function. The process model is assumed to be of ARMA form with a process time delay d. If sufficient and representative process data are available, the disturbance transfer function B(z-1)/A(z-1), also assumed to be in ARMA form, can be derived through system identification24 together with the noise variance σw2. The closed loop transfer function B(z-1)/A(z-1) is then transformed to an impulse response model, represented by coefficients φi, through long division. The process time delay d is either known or has to be estimated from process data.24 Because of the process time delay, the controller will have no influence on the first d - 1 impulse response coefficients φi, where i ) 0 ... d - 1. Therefore, no matter which controller is used, the variance will always be larger than the minimum variance, which can be derived from the first d

φi2)σw2 ∑ i)0

(24)

The consequences for the profit index improvement as given in eq 22 are illustrated in Figure 13 for the example of a linear performance function with constraints. The maximum achievable profit will always be smaller than the maximum profit ϑm of the performance function ϑ(x). Therefore, instead of comparing the profit improvement ∆Ψ to the maximum profit, the profit improvement of eq 21 should rather be compared to the maximum achievable profit improvement ∆Ψmax by the following index:

β)

∆Ψ ∆Ψmax

(25)

If index β is close to one and the profit improvement ∆Ψ is sufficiently large to cover the cost of the investment, then the new control system is favorable. If, however, β is close to zero, a different control strategy might achieve better results for the standard deviation σC, and in this case, the control system design should be reconsidered. The profit improvement ∆Ψ can be derived through simulation of the standard deviation σC as described by Zhou and Forbes10 rather than by a fixed percentage of reduction in standard deviation. Index β also gives a useful assessment when using a fixed percentage reduction of standard deviation. In this case, if β is larger than one, then the fixed percentage reduction is too high and the profit performance is not achievable, that is, σC < σMV. It is important to note that process re-engineering, not improved control, might be the preferred option for realizing additional benefits when ∆Ψmax is small and when σMV is large. Industrial Example The industrial example for which the control benefit is studied is an acid reactor within a refining unit. The acid reactor has three inlets, and the critical parameter during operation is the acid specification quantified by an acid number x. The acid number depends on the temperature of the three inlets and is therefore calculated by an inferential from the three temperatures: the lower the temperature, the higher the acid number. The acid number is biased on a regular basis by making use of laboratory samples. The profit ϑ(x) of this process increases linearly with the acid number because less hydrogen is consumed at lower temperatures. However, if a threshold of x2 ) 1 is exceeded, then the acid specification is violated and the product becomes waste. In the current control setup, the acid number x is controlled. Figure 14 shows an example of a time trend of the acid number as well as the performance function ϑ(x). The acid number is sampled every 5 s and occasionally

Figure 14. Time trend of the controlled variable and performance function ϑ(x) (linear function with constraint) of the industrial case study.

5622


Figure 15. Profit index Ψlin of the industrial case study (linear performance function with constraints). The minimum variance determines the maximum achievable profit improvement ∆Ψlinmax.

exceeds the threshold x2, for example, at around sample 1200. For the performance function, for reasons of simplicity and confidentiality, the maximum profit is specified here as 100% and the acid number is scaled to a threshold x2 ) 1. The base case standard deviation σB can be estimated from the process data shown in Figure 14 after eq 20 as σB ) 0.103. Furthermore, the average operating point can be estimated from the data and is x0B ) 0.856. Using the performance function ϑ(x) together with the base case standard deviation gives the optimum operating point x0 ) 0.828 after eq 9. The optimum operating point x0 is therefore close to the actual operating point x0B, implying that the operating point has been chosen appropriately. As a next step, the minimum variance can be estimated from eq 24 using the process data and assuming a second-order transfer function and a time delay of d ) 3 samples.22 The minimum variance can be estimated as σMV2 ) 0.0092 after eq 24. In this industrial example, it is assumed that an improved controller will reduce the standard deviation by 50%, as most commonly proposed. Thus, the standard deviation is reduced from σB ) 0.103 to σC ) 0.0515. The three values for the standard deviation, σB, σC, and σMV, are indicated in Figure 15 together with the resulting profits for each standard deviation: Ψ(σB) ) 77.9%, Ψ(σC) ) 87.4%, and Ψ(σMV) ) 97.2%, which indicates the maximum achievable profit. Following the definition of eq 21, the economic benefit resulting from improved control is ∆Ψ ) 9.54%, while the maximum achievable benefit after eq 22 is ∆Ψmax ) 19.4%. Thus, the index that compares the improved control profit to the achievable profit is β ) 0.49. This is significant but can be improved by a further reduction in variance. Improved control rather than process re-engineering is the preferred option for realizing additional benefits in this case, as ∆Ψmax is relatively large and σMV is relatively small. Implementation of the Profit Index The implementation of a procedure to estimate the profit index is relatively straightforward and is briefly explained below, using the steps required to calculate it. Most of these steps are covered in detail in the rest of the paper and are only briefly stated here for completeness. Step 1: The first step would be to collect data of the controlled variable of interest. As these data are assumed to be Gaussian, the distribution is fully determined by estimates of the base case average x0B and standard deviation σB, which can be obtained from the process data. Step 2: A performance function ϑ(x) appropriate for the process is then selected.

Step 3: The minimum variance (σMV) is estimated from the process data using a suitable transfer function and time delay. Step 4: The standard deviation improvement possible from improved control (σC) is estimated. Step 5: The optimum operating points (x0B, x0C, and x0MV) for the chosen performance function ϑ(x) and standard deviations (σB, σC, and σMV) are determined. For example, this requires solving eq 9 in the case of a linear performance function with constraints. This equation can be solved numerically to yield the result shown in Figure 5. For implementation purposes, it can be precomputed and implemented as a lookup table. As the improvements are expressed as proportions (e.g., reduce standard deviation by 50%), the limits x1 and x2 can be scaled to accommodate engineering units. Step 6: The maximum achievable profit for each case, Ψ(σB), Ψ(σC), and Ψ(σMV), can be calculated for each pair (x0i, σi) from, for example, eq 8 in the case of a linear performance function with constraints. Step 7: ∆Ψ, ∆Ψmax, and β can then be calculated from eqs 21, 22, and 25, respectively. Conclusions In this paper, a profit index for a priori estimation of the economic benefit resulting from improved process control has been introduced. The index presents a simple method based on conventional approaches to estimate the profit improvement for three commonly used performance functions. The advantage of this index is that previously “hidden” benefits from process control can now be seen directly from this profit index. It shows that the relationship between the shift of the operating point and the economic benefit is clearly not linear for all three performance functions that were investigated. In case of a constraint performance function or, for that matter, any asymmetrical performance function, the derivation of the profit index involves the optimization of the operating point x0. The profit index can also be used as an online monitoring tool for the economic performance of a process that gives the profit as a monetary value if the standard deviation is estimated for a fixed, moving time window. The implementation of the profit index is straightforward. A drawback, however, is that, instead of linear or quadratic functions for the performance function, exponential and error functions have to be implemented for the profit index. This is potentially problematic when implementing the index directly on a control platform, although lookup tables can be generated for commonly used performance functions. A minimum variance estimate provides an upper bound for the achievable economic benefit which results from control improvements. The proposed profit index β assesses the economic performance of a new control system relative to the maximum achievable performance of the process. In the industrial example, it was shown that a reduction of the standard deviation by 50% through improved control would result in a significant profit. Improved control rather than process re-engineering is the preferred option for realizing additional benefits in this case, as the maximum achievable profit is relatively large and the minimum variance estimate is relatively small. Acknowledgment The principal author gratefully acknowledges the financial support of the Claude Leon Foundation and the University of Pretoria Postdoctoral Fellowship program. This work was supported by the National Research Foundation of South Africa, grant number GUN 2069476.


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ReceiVed for reView November 7, 2006 ReVised manuscript receiVed May 23, 2007 Accepted June 1, 2007 IE0614273

A Profit Index for Assessing the Benefits of Process Control

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