Optimal Selection of Dominant Measurements and Manipulated

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Ind. Eng. Chem. Res. 2010, 49, 7832–7842

Optimal Selection of Dominant Measurements and Manipulated Variables for Production Control Wuendy Abi Assali and Thomas McAvoy* Department of Chemical and Biomolecular Engineering UniVersity of Maryland, College Park Maryland 20740

The main objective in operating a chemical plant is to improve profit while assuring products meet required specifications and plant operation satisfies environmental and operational constraints. A subobjective that directly affects profit is to improve the control performance of key economic variables in the plant, particularly the production rate and quality. An optimal control-based approach is proposed to determine a set of dominant measurements and manipulated variables and to structure them to improve control of the production rate and product quality. After the dominant variables are identified, a decentralized control structure is designed to control these variables. Then, a model predictive controller (MPC) is built on top of the decentralized control structure. The approach is applied to the Tennessee Eastman Process to demonstrate its effectiveness. 1. Introduction Plantwide control design is complex because (1) the size of the problem is significantly larger than for single units; (2) the variables to be controlled by a plantwide system are not as clearly and easily defined as for single units (Stephanopoulos et al.1); (3) the characteristics of the process are more complicated, such as several recycle streams and energy integration; and (4) there is a large cost involved in developing a detailed dynamic model of the process. To overcome these difficulties, the plantwide control problem is often decomposed into smaller subproblems. In addition, when a plantwide dynamic model is not available, an alternative is to use heuristic rules, based on experience and process insight. Plantwide control design methodologies can be divided into mathematically oriented, process-oriented approaches, and/or a combination of both. The mathematically oriented approaches are based on using models and quantitative methods to determine control structures. The following researchers have presented mathematically oriented approaches: Moore,2 Georgiou and Floudas,3 Narraway and Perkins,4,5 Mohideen et al.,6 Bansal et al.,7 Kookos and Perkins,8 and Kookos9 among others. The process-oriented approaches are based on qualitative methods where heuristics, logic, and experience are used to determine control structures for the process. Buckley,10 McAvoy and Ye,11 Luyben,12 and Skogestad13,14 have proposed process-oriented approaches. Larsson and Skogestad15 presented a review of both processoriented and mathematically oriented approaches. However, there is no systematic procedure that has been adopted by the control community as a general procedure to solve the plantwide problem. Plantwide control design is very much open-ended, and there is not a unique correct solution. In fact, a control structure that is good for a specific objective might not be good for another objective. Therefore, the success of plantwide control design is measured by the extent to which it can achieve the desired control, operating, and economic objectives. The main goal in a chemical plant is to maximize profit while satisfying product specifications, and environmental and operational constraints. However, it is not simple to identify a direct relationship between profitability and how controllers are designed and operated. The key questions to answer are the following: (1) which variables should be controlled; (2) which * To whom correspondence should be addressed. Tel.: +011 410 480 0016. E-mail: [email protected].

measurements and manipulated variables should be used; and (3) what control structure should be used for this purpose? Van de Wal et al.,16 presented a review of methods for input/output selection. Chen17 and Assali18 summarized different methods for control structure selection and measurement/manipulated variable selection. However, the majority of the current plantwide control design methodologies do not focus directly on improving plant profitability. The reasons for this are the following: (1) these methodologies put more emphasis on the control and operation of the plant, and (2) it is not an easy task to quantify profitability when control strategies for the plant are being determined. Among the limited number of researchers who consider economics for control structure design are Nishidaet al.,19 Narraway and Perkins,5 Bansal et al.,7 and Kookos.9 Their approaches are rigorous and produce control structures that are optimal within the limitations imposed by the model and problem definition. However, they involve a detailed evaluation which can involve significant engineering effort and computational time. In general, the most common economic objective in any chemical process is to maximize profit, while considering the environmental and safety regulations. This objective is closely related to the control performance of key economic variables in the plant, particularly production rate, product quality, and purge losses. Often in complex processes, to achieve the desired economic and control objectives, a few dominant variables should be controlled. This approach is called partial control, which has been studied by Arbel et al.20-24 and Tyreus,25,26 among others. Tyreus used thermodynamic principles to identify the dominant variables that affect the production rate and product quality. Then, he used a partial control scheme on these dominant variables to (1) increase and hold the product rate, and (2) improve the control performance of the process for disturbance rejection. The main objective of this work is to use optimal control to determine a set of dominant measurements and manipulated variables in a plant. By improving the control of the dominant variables, the plant can be operated closer to operational constraints. A subobjective is to improve the control performance of product rate and product quality. A methodology based on optimal control theory that uses the idea of partial control is presented. This approach uses a linear dynamic model of the process and optimal control theory to identify the dominant

10.1021/ie901879e  2010 American Chemical Society Published on Web 05/26/2010

Ind. Eng. Chem. Res., Vol. 49, No. 17, 2010

variables that affect production rate and product quality. The original idea of using a linear dynamic model and optimal control theory for control structure design was presented by Schnelle.27 In his work, Schnelle used a linear state space model, a linear quadratic regulator (LQR) design, and two sensitivity matrices, as well as process knowledge to identify (1) if any manipulated variable could be eliminated due to a weak influence on the process measurements that need to be controlled, (2) feasible multiloop control structures with feedforward channels, and (3) interaction between identified control structures. He found two sensitivity matrices that contain information for both feedback paths and feed-forward paths. Then, he used these two matrices and engineering judgment to identify possible control structures. Additional research has been presented by Chen et al.28,29 and Chen.17 The optimal control measurement selection methodology uses a linear quadratic regulator (LQR) design method to generate an optimal static output feedback controller (OSOFC). The basic formulation of the LQR OSOFC can be found in Lewis’s work.30 The OSOFC gain, K, is similar to a process gain matrix in that it can provide dynamic information about the process interaction, while the gain matrix contains static information about process interaction. The K matrix is used to determine the important measurements and manipulated variables that affect product rate and product quality. Then, SISO control structures for these important variables are designed using an approach discussed by Chen and McAvoy.17,28 After closing these loops, the set points of the important measurements become manipulated variables. Finally, the idea of using partial control for controlling product rate and product quality is implemented by using a model predictive controller to manipulate the set points of the dominant variables. 2. The Basic Optimal Static Output Feedback Linear Quadratic Regulator (OSOF LQR) Design

{

x˙ ) Ax + Bu y ) Cx x(0) ) x0

elements describe the system dynamics. Output feedback control is given as u ) -Ky

(1)

This is a linearized state space model where x is the vector of the system states, u is the vector of control inputs, and y is the vector of measured outputs. A, B, and C are matrices whose

(2)

where K is an mxp matrix of constant feedback coefficients. In the approach presented here, elements of K are compared to one another. To make a meaningful comparison, it is necessary that the measurements, manipulated variables, and states be scaled, which results in K being dimensionless. The problem is to find the K that minimizes a quadratic time domain performance index function given by min J ) K

1 2

∫

∞

0

1 2

(yTQy + uTRu) dt +

∑ ∑g k

2

ij ij

i

(3)

j

where Q and R are weighting matrices for y and u, respectively, while gij is a weight on element kij in K. For multivariable control the gij’s are zero. When a single input, single output (SISO) structure is used, the gij’s are used to force the off-diagonal elements of K to be zero. To make the kij elements small, large values of the corresponding gij elements should be used. The design equations needed to calculate K that minimize the performance index (3) are ACT P + PAC + CTKTRKC + CTQC ) 0

(4)

ACS + SACT + X ) 0

(5)

RKCSCT - BTPSCT + g × K ) 0

(6)

AC ) A - BKC

For the approach discussed here, the following data should be available: (1) A linear dynamic state space model. The model can be obtained from a first principle nonlinear model by numerically calculating the first order Taylor expansion coefficients around the operating point. Alternatively, the linear model can be obtained from model identification using process data. (2) Steady state process data for state variables, manipulated variables, and measurements. (3) Operating ranges of the measurements and manipulated variables. (4) Defined control objectives for product rate and quality. (5) Process constraints used to define the safety variables. (6) Information about possible disturbances and/or set point changes. Given a linear time invariant (LTI) state space model, an OSOFC is designed to stabilize the system and bring the states from initial values to zero following a trajectory that minimizes a linear quadratic objective function (LQR; Chen and McAvoy28). The basic formulation of the OSOF LQR design problem is presented by Lewis:30 The LTI process model is given as

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X ) x(0) x(0)T

(7)

where g × K is a matrix with elements gij × kij. These equations result from the first-order necessary condition for optimality given by Lewis.30 Because there is no explicit analytical solution for K, numerical optimization is used to solve eqs 4 to 7 simultaneously. In solving these equations, the following conditions are required: (1) R should be positive definite, and Q should be positive semidefinite to ensure CQC is positive semidefinite. (2) P is positive definite or positive semidefinite as long as AC is stable and (CKRKC + CQC) is positive definite or positive semidefinite. (3) S is positive definite or positive semidefinite as long as AC is stable and X is positive definite or positive semidefinite. Numerical considerations for solving the OSOF problem can be found in Chen and McAvoy.28 Chen17 presented OSOF LQR design methods for specific set point tracking and/or disturbance rejection. The initial condition for eq 7 is given by X)

{

n

∑x x i)1

T di di

}

T + xspixsp i

(8)

where xd represents the disturbance states, n represents the total number of disturbances, and xsp is a vector related to set point changes in the product rate and product quality. The calculation for xd and xsp is given by xdi ) -A-1 × W × di

(9)

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xspi ) C × m

(10)

where W is the matrix whose elements describe the dynamics of the disturbances, d is the vector of disturbances, and m is the vector that specifies the measurements that are considered for control. All the elements of m are zero, except for the elements that correspond to production rate and product quality. 3. Determining Dominant Variables for Product Rate and Quality Control The main problem to be solved in this stage is to find a set of measurements and manipulated variables that affect the product rate and quality, without directly using these two measurements in the control law. These two measurements are used to define the control objective, but they are not considered as available measurements at this stage. By not considering them as available measurements, the idea is to find other dominant variables that can be used to control the production rate and product quality. Two methods have been studied to find the set of measured and manipulated variables that affect product rate and product quality. Both methods are based on OSOFC. The first method uses the weight matrix Q. By changing the elements of this matrix, the control objectives can be defined. Two sets of measurements are generated: a first set (Cs), which includes all the measurements, except the product rate and quality, and a second set (Cc), which includes all measurements. Then, eqs 4, 6, and 7 are replaced by the following equations: ACT P + PAC + CsTKTRKCs + CcTQCc ) 0

(11)

RKCsSCsT - BTPSCsT + g × K ) 0

(12)

AC ) A - BKCs

X ) x(0) x(0)T

(13)

Next, all the elements in the Q matrix are set equal to zero, except for the elements that correspond to the product rate and product quality in the Cc matrix. The second method to solve this problem uses the gij elements in eq 3. The key idea is to make all the elements in the rows of the K matrix corresponding to the product rate and product quality measurements very small. To do so, a large value for the corresponding weighting gij elements is used. Several simulations were performed using both methods. The results obtained using both methods were the same; however, the computation speed of the second method was much slower. Because both methods give the same results, the first method is chosen due to the speed of the calculation. Using the first method, gij are equal to 0, and R is chosen to be the identity matrix, so that all manipulated variables are treated equally. After K is calculated, the next question is how to extract information about the best set of measurements and manipulated variables to control the product rate and product quality. Since the process model is scaled, K is dimensionless. Therefore, the absolute value of the elements in K can be compared to one another. In K, the rows represent manipulated variables, while the columns represent measurements. Generally, an element with an absolute value close to zero indicates a weak relationship between the manipulated variable and the measurement. Here, the L1-norm of a vector, defined as follows ||x||1 ) |x1 | + |x2 | + |x3 | + ..... + |xn |

(14)

is used as a measure of the degree of importance for the measurements and manipulated variables. In order to determine which measurements and manipulated variables should be used to control the product rate and product quality, the following rules of thumb are used: • The L1-norm for each row of K is calculated as the sum of the absolute values of all the elements in each row. This sum is called ∑rowi, and it represents the total contribution of each manipulated variable. The manipulated variables that have the strongest effect on the product rate and product quality are those that have the largest values of ∑rowi. • The L1-norm for each column of K is calculated as the sum of the absolute values of the elements in each column. This sum is called ∑coli, and it represents the total contribution of each measurement. The measurements that have the strongest effect on the product rate and product quality are those that have the largest values of ∑coli. • If a row of K contains only small elements, the corresponding manipulated variable should not be considered in the control structure. • If a column of K contains only small elements, the corresponding measurement should not be considered in the control structure. In solving eq 3, it was found that convergence problems occurred when highly correlated variables were included in the C matrix. This is an important issue for overall plantwide control because, when two or more variables are highly correlated, trying to control each of them results in severe interactions. A number of simulations showed that, when highly correlated measurements are considered simultaneously, the algorithm used to calculate K did not converge, or the calculation was very slow. The reason for this is that the condition number of the C matrix increases significantly whenever two or more highly correlated variables are considered. Because the algorithm used to calculate K involves the inversion of the C×S×C′ matrix (eq 6), it is recommended not to work with ill-conditioned C matrices to avoid convergence problems. To assess convergence problems, an analysis is used to determine how the condition number of C is affected when highly correlated measurements are considered simultaneously. To do so, the condition number (CN) of C is calculated, eliminating one variable at a time from each correlated group until there is no significant change in the CN. Application of the correlation analysis to the Tennessee Eastman process is given in the Appendix. 4. Case Study: Tennessee Eastman Plant a. Plant Description. This section presents the application of the optimal control-based approach for determining dominant variables to the well-known Tennessee Eastman (TE) process to demonstrate the effectiveness of the method. This process consists of five operating units that involve the production of two products, G and H, from four reactants, A, D, E, and C. In addition, there is an inert B that enters with one of the feed streams, and two side reactions occur. The exothermic irreversible reactions are A(g) + C(g) + D(g) G(l) Product 1

(15)

A(g) + C(g) + E(g) H(l) Product 2

(16)

A(g) + E(g) F(l) Byproduct

(17)

3D(g) 2F(l) Byproduct

(18)

The model of the process has 50 states, 12 manipulated variables, and 41 measurements. A schematic diagram of the


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Figure 1. Tennessee Eastman plant. Table 1. Operational Modes for TE Process mode

G/H mass ratio

production rate

1 2 3 4 5 6

50/50 10/90 90/10 50/50 10/90 90/10

7038 kg h-1 G and 7038 kg h-1 H 1408 kg h-1 G and 12669 kg h-1 H 10000 kg h-1 G and 1111 kg h-1 H maximum production rate maximum production rate maximum production rate

plant is shown in Figure 1. Downs and Vogel31 provided six operating modes at three different G/H (product 1/product 2) mass ratios that are listed in Table 1. In 1995, Ricker32 calculated the optimal steady state process values for each of the six operating modes. Ricker’s results showed that the base case provided by Downs and Vogel is far from optimal. A detailed description of the TE process, model formulations, physical property data, and steady state process values for each operating mode can be found in Downs and Vogel,31 Ricker,32 and Ricker and Lee.33 Also, Downs and Vogel31 stated that the A, C, and D feed streams have frequency constraints. For this reason, it is not desirable to use these manipulated variables in control loops that require fast responses. Before applying the optimal control methodology to the Tennessee Eastman process, several preliminary steps need to be carried out. First, flow and temperature inner cascade loops are closed. Second, the model is scaled. Third, highly correlated variables are eliminated. Fourth, loops related to plant safety are closed. Finally, a Down’s Drill to ensure component control is carried out. After these steps are completed, the issue of how to best control product rate and composition is addressed. b. Preliminary Considerations. 1. Closing Inner Cascade Loops. McAvoy and Ye11 discuss the issue of closing inner cascade loops for the Tennessee Eastman process. Closing the inner cascade loops eliminates 10 measurements, and the set points of the inner cascade loops become manipulated variables. Table 2 presents the measurements and manipulated variables available after closing the cascade loops. Ten proportional-only controllers are used for the inner cascade loops. 2. Scaling the Tennessee Eastman Model. The measurements are scaled using the following scaling factors: 200 KPa for pressures; 30 °C for temperatures; 50% for levels; and steady state values for mass flow rates, volumetric flow rates, molar compositions, and compressor work. Since the manipulated

Table 2. Measurements and Manipulated Variables after Closing Cascades Loops # measur.

measurements

manipulated variables

1 2 3 4 5 6 7 8 9 10 11 12 13-18 19-26 27-31

recycle flow reactor feed reactor pressure reactor level reactor temperature separator temp separator level separator pressure stripper level stripper pressure stripper temperature compressor work reactor feed composition purge compositions product composition

D feed set point E feed set point A feed set point C feed set point purge set point separator exit flow set point stipper exit flow set point product flow set point reactor cooling water temp SP condenser cooling water temp SP

Table 3. Condition Number of C Matrix measurements included BM BM BM BM BM BM BM BM BM

+ + + + + + + + +

reactor pressure reactor and separator pressure reactor and stripper pressure reactor, separator, and stripper pressure composition of G in the purge composition of H in the purge compositions of G and H in the purge reactor feed reactor feed and recycle flow

CN of the C matrix 982.0030 1.6637e+004 1.3718e+004 2.3442e+012 9.0452e+004 5.9808e+004 7.5591e+016 1.2037e+004 2.5037e+007

variables are all set points of inner cascade loops, these set points are scaled in the same manner as the measurements are scaled. 3. Correlation Analysis. Steady state correlation and condition number analysis of the C matrix is carried out. The measurements considered for this analysis are presented in Table 2. These measurements are divided into two groups: (1) Highly correlated measurements: reactor pressure, separator pressure, stripper pressure, composition of G in the purge, composition of H in the purge, reactor feed, and recycle flow. (2) Basic measurements (BM): all the measurements in Table 2 except for the highly correlated measurements. Table 3 shows the values of the condition number of the C matrix. The results from Table 3 demonstrate that the condition number of the C matrix has a significant increase when two or more highly correlated variables are considered simultaneously.

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Table 4. Measured and Manipulated Variables after Eliminating the Correlated Variables # measur

measurements


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22

reactor feed rate reactor pressure reactor level reactor temperature separator temp separator level stripper temperature stripper level compressor work comp of A in feed comp of B in feed comp of C in feed comp of D in feed comp of E in feed comp of F in feed comp of A in purge comp of B in purge comp of C in purge comp of D in purge comp of E in purge comp of F in purge comp of G in purge

D feed set point E feed set point A feed set point C feed set point purge set point separator exit flow set point separator exit flow set point product flow set point reactor cooling water temp SP condenser cooling water temp SP

Table 5. Control Structures for Controlling the Safety Variables candidate

reactor P

reactor L

reactor T

separator L

stripper L

1 2 3 4 5 6 7 8 9

purge purge CON CW purge purge CON CW purge purge CON CW

E RCT CW Sep.Bottom Str.Bottom CON CW RCT CW Sep.Bottom Str.Bottom E RCT CW Sep.Bottom Str.Bottom E RCT CWT Sep.Bottom Str.Bottom CON CW RCT CWT Sep.Bottom Str.Bottom E RCT CWT Sep.Bottom Str.Bottom E Sep.Bottom Str.Bottom CON CW Sep.Bottom Str.Bottom E Sep.Bottom Str.Bottom

Table 4 shows the available measurements and manipulated variables after eliminating the correlated measurements. 4. Control of Safety Variables. Safety variables are identified, and decentralized control structure candidates for them are generated. These variables are those that have operational limits and can cause the shutdown of the plant if they exceed their limit. Chen17 used eigenvalue analysis, the process gain matrix, and engineering judgment to determine the safety variables in the plant. Moreover, Chen17 used an OSOFC methodology to generate a controller for the safety variables. Details on the identification of the safety variables and the generation of control structure candidates can be found in Chen.17 In this stage, the control structure candidates for controlling the safety variables proposed by Chen17 for the operational modes of the Tennessee Eastman process are used. Table 5 shows the control structures recommended by Chen.17 Here, only candidates 4, 5, 6, 7, 8, and 9 are considered for the following reason: candidates 1, 2, and 3 are almost the same as candidates 4, 5, and 6, respectively. The difference between them is that, in candidates 1, 2, and 3, the reactor temperature is controlled using the reactor cooling water valve, while in candidates 4, 5, and 6, a cascade configuration is used. In this paper, candidate 4 is considered in detail. The same procedure needs to be applied to the remaining candidates. Five proportional-only controllers are automatically tuned for candidate 4 by calculating an optimal K that contains only diagonal terms, using the decentralized approach discussed in section 2 above. Averaging level control is used for the two integrating levels (separator and stripper levels). The gains for the averaging level controls are +1 or -1 (%/%), depending on the sign of the process gain. After the safety loops are closed, the

measurements corresponding to these variables are no longer available as measurements. Instead, the set points of these loops become the new manipulated variables. This procedure drops the number of available measurements from 22 to 17. 5. Down’s Drill for Component Control. It is well-known that, to satisfy the material balance in a plant, all the reactants fed into the system must be consumed in the reaction or leave the system as impurities in the product or purge streams. In the case of inerts, they should also be removed from the process through the purge or product streams. In chemical plants, any imbalance in the number of moles of any reactant will cause an accumulation of the reactant that is in excess. For this reason, it is very important to control the inventory of components so that exactly the right amount of the reactants is fed in. There are three ways to avoid component accumulation: (1) limit the feed flow of reactants, (2) control their reaction, or (3) adjust the product in the plant or the purge flow. Jim Downs, of the Eastman Chemical Company, has pointed out the importance of verifying if a control structure satisfies component balances. To do so, it is necessary to check for each component the specific mechanism or control loop that guarantees that there will be no accumulation of that chemical component. This procedure is called Downs Drill Analysis. Luyben34 recommended the use of this analysis for checking component balances in a control scheme. Chen17 used Downs Drill Analysis to identify the components that need to be controlled for the Tennessee Eastman process. Chen analyzed reactants, inerts, and products. Here, only the reactants, inerts, and byproducts are considered for the Downs Drill Analysis since products will be controlled in the next stage. The Downs Drill Analysis for reactants and purge components for candidate 4 is given in Table 6. The second column of Table 6 tells if the component is self-regulating or not. If it is self-regulating, the third column shows which loop makes it self-regulating. For example, component B is self-regulating because the purge is used to control the reactor pressure (RCT P). The fourth and fifth columns indicate the measured and manipulated variables that can be used to control the inventory variables. From the Downs Drill Analysis for candidate 4, components A, C, and D are left uncontrolled, and they need to be controlled. The manipulated variables used for controlling the inventories of A, C, and D are their respective feeds (see Chen17). There are two analyzers in the gas loop that can be used for measuring the inventories of A, C, and D. One is in the reactor feed, and the other is in the purge stream. In this case, the analyzer in the reactor feed is used. It should be pointed out that, even though the A, C, and D feeds have frequency constraints, these feeds can be used to control the compositions of A, C, and D in the reactor feed without affecting upstream processes, provided that no aggressive changes are made to the flows. In the Tennessee Eastman process, loops that involve analyzers are slow because of delays in the measurements. At this point, candidate 4 involves eight loops (five safety variables, and the compositions of A, C, and D). Three proportional-only controllers are automatically tuned using OSOFC to control the component loops (%A in the reactor feed - A feed, %C in the reactor feed - C feed, and %D in the reactor feed - D feed). These loops are incorporated into the model for use in later stages. c. Dominant Variables for Production Rate and Product Quality Control of the TE Plant. In this stage, the optimal static output feedback controller K is calculated and used to determine the dominant measurements and manipulated variables that affect product rate and product quality. The idea is to control the dominant variables to improve the control of the


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Table 6. Downs Drill Analysis candidate

component

self-reg

candidate 4

A (reactant) B (inert) C (reactant) D (reactant) E (reactant) F (byproduct)

no yes no no yes yes

Table 7. Measurements and Manip Variables after Closing the Safety and Inventory Variables # measur.

measurements


1 2 3 4 5 6 7 8 9 10 11

reactor feed separator temp stripper temperature compressor work comp of B in feed comp of E in feed comp of F in feed comp of B in purge comp of E in purge comp of F in purge comp of G in purge

%A in feed set point reactor pressure set point %C in feed set point %D in feed set point reactor temperature set point cond cooling water temp SP reactor level set point separator level set point stripper level set point

product rate and quality. The available measurements and manipulated variables are shown in Table 7. To find dominant variables, the following weights are used in calculating K (Dominant Variable Problem): (1) R ) I; this gives the same importance to all the manipulated variables (2) gij ) 0 Q is the weight matrix that can be used to define the desired control objective. All the elements of this matrix should be zero except for those that correspond to the product rate and product quality. Table 8 shows the resulting K matrix. The numbers that are in bold in Table 8 correspond to the strongest measurements and manipulated variables to control the product rate and product quality. The strongest manipulated variables are reactor temperature, % D in the reactor feed, condenser cooling water temperature SP, % A in the reactor feed, reactor level, and % C in the reactor feed. This result is reasonable considering that throughput changes can be achieved only by altering, either directly or indirectly, conditions in the reactor. In addition, having the % D as a dominant variable is reasonable, since it affects the rate of formation of component G, which is one of the control objectives. The strongest measurements are separator temperature, % G in the purge, and stripper temperature. In this case, only the separator temperature is considered as an important measurement for three reasons: (1) it has the largest value in ∑col, which is more than twice the value of closest important measurement; (2) there is no need to control product G in the purge because it is going to be controlled in the product stream; and (3) the manipulated variable that is most often used to control the stripper temperature, the steam flow, is fixed at its minimum value for all the optimal operating modes.32 The other manipulated variables are located too far from the stripper temperature or do not seem to have a large effect on the stripper temperature. To control the separator temperature, an optimal K is calculated using the separator temperature and all the manipulated variables. The following parameters are used for the calculation of K: (1) R ) I; this gives the same importance to all the manipulated variables (2) gij ) 0 (3) Q All the elements of Q should be zero, except those that correspond to the separator temperature. The resulting K

why self-reg

man var

measurement

A feed

%A RCT feed, %A purge

C feed D feed

%C RCT feed,%C purge %D RCT feed,%D purge

purge-RCT P E feed-RCT L RCT CW-RCT T

is given in Table 9. From Table 9, the best manipulated variable to control the separator temperature is the condenser cooling water set point. After closing this loop, the separator temperature set point is included as a manipulated variable. Now, the optimal control calculation (Dominant Variable Calculation) is carried out with this additional manipulated set point. The condenser cooling water set point is no longer available to be manipulated. Table 10 shows K for this case. As can be seen by comparing Tables 8 and 10, the order of importance for the manipulated variables and the measurements remains almost the same. However, the main differences are that now the separator level is in the fifth place of importance, and that %A and %C switched in their order of importance. Now the problem of determining which manipulated variables to use for controlling the product rate and product quality is addressed. The strongest manipulated variables according to the ∑row value in Table 10 are added one at a time in descending order of importance. Every time a new manipulated variable is added, a new control system is generated, and each new control system is called a case. Each case starts with the number of the base candidate (candidate for safety variables in Table 5) being evaluated. Each case has the product rate and quality as control variables and a different number of manipulated variables. According to Table 10, the order of importance of the manipulated variables (from the strongest to the weakest) is as follows: separator temperature, composition of D in the reactor feed, separator temperature set point, reactor level, and composition of C and A in the reactor feed. The multivariable K, calculated for each case, and transient responses are used to determine the number of dominant variables that should be employed, and to eliminate manipulated variables with poor performance. All the generated cases are tested for a set point change of a 50% increase in the production rate, and to reject the first two step process disturbances IDV(1) and IDV(2). Transient response characteristics, such as settling time and offset value for critical variables in the plant, as well as the integral of the absolute value for the error (IAE), are used as a measure of the case performance. The critical variables are chosen as the reactor pressure, temperature, and level and product rate and product quality. Indirectly, the purge flow, an important economic variable, is already taken into consideration because the purge flow is used to control the reactor pressure. The offset value is calculated as the difference between the set point of a critical variable and the final steady state value reached by that particular variable after a disturbance or set point change. Then, the summation of the absolute value of the offset for the critical variables is calculated for disturbances IDV(1) and IDV(2) and for product flow changes for each case. Because the offsets of different critical variables will be added together, these offset values are scaled by dividing them by their steady state values. In order to calculate the summation of the offsets, the following weights are given to the critical variables: product rate, 0.25; product quality, 0.25; reactor temperature, 0.20; reactor level, 0.20; and reactor pressure, 0.1. These weights are chosen on the basis of the control objectives of the plant and the operating cost function given by Downs and Vogel.31 The

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Table 8. OSOFC Matrix Control Objective: To Control Product Rate and Quality SP

RF

SpT

StT

CpW

BF

EF

FF

BP

EP

FP

GP

Σ row

%A RP %C %D RT CCW RL SL St L Σ col

0.036 0.054 0.026 0.225 -0.120 -0.238 -0.093 -0.101 -0.011 0.905

0.676 -0.514 0.571 -2.839 4.923 2.102 1.327 0.237 -0.388 13.58

-0.185 0.028 -0.368 0.236 -0.822 -0.353 -0.197 0.070 0.236 2.496

0.043 0.021 -0.011 0.211 0.087 -0.264 -0.126 -0.098 -0.002 0.863

-0.029 0.023 -0.001 0.045 -0.190 0.054 0.005 0.027 0.016 0.390

-0.022 0.011 0.014 -0.145 -0.147 -0.049 0.241 -0.030 -0.024 0.681

-0.013 0.012 -0.000 0.025 -0.073 0.047 0.008 -0.005 -0.002 0.185

-0.019 -0.002 -0.013 0.006 -0.057 -0.186 -0.002 -0.047 -0.014 0.346

0.006 0.013 -0.017 0.048 0.034 0.075 0.002 0.043 0.025 0.263

-0.013 0.001 -0.014 0.033 -0.065 -0.096 -0.025 -0.010 0.008 0.265

-0.211 0.190 -0.176 1.065 -1.579 -0.599 -0.503 -0.183 0.119 4.626

1.252 0.871 1.212 4.876 8.097 4.065 2.530 0.851 0.845

Table 9. Optimal Static Output Feedback Controller for Separator Temperature Man Var

ST

%A RP %C %D RCT T SP CCW SP RCT L SP Sep L SP Stp L SP

-0.022 -0.001 -0.009 0.007 0.129 0.512 0.009 -0.004 0.000

summation of the offset values for each candidate is calculated as follows: 3

5

∑ ∑ R abs(critical variables offset(j,i))

Offset Valuek )

*

j)1 i)1

(19) where R represents weights for the critical variables (0.25, 0.25, 0.20, 0.20, 0.10); i represents critical variables offsets: product rate, product quality, reactor temperature, reactor level, and reactor pressure; j represents disturbances and set point change: J ) 1 IDV(1), J ) 2 IDV(2), and J ) 3 set point change; and k represents the total offset value for each candidate k ) 1 to 7. The IAE is calculated as follows: IAE for critical variablesk )

∫

∞

0

[SP(t) - CV(t)] dt (20)

The IAE for each case is calculated as follows 3

IAE )

5

∑ ∑ IAE for critical variables(j,i)

(21)

j)1 i)1

where i represents IAE for critical variables: product rate, product quality, reactor temperature, level, and pressure; j represents disturbances and set point change: J ) 1 IDV(1), J ) 2 IDV(2), and J ) 3 set point change; and k represents total IAE for the critical variables for each candidate k ) 1 to 7. There are three conditions that a case has to pass in order to be considered: (1) it must reject disturbances IDV(1) and IDV(2), (2) it must increase the production rate by 50% in less than 5 h, and (3) it must have a less than (5% product rate and product quality variability for disturbance rejection. Every time a new manipulated variable is added, the values for the summation of the offset and IAE are compared with the previous case to evaluate whether there is significant improvement or not. Table 11 shows the first four cases generated, and the ability of each case to reject disturbances IDV(1) and IDV(2) and to achieve maximum production rate change in less than 5 h. In addition, it shows the summation of the offset, and the IAE

values for the critical variables in the plant for each case. It also shows the percentage change for the summation of the offset values and the IAE between consecutive candidates. After case 4-4, there is almost no change in either the offset or the IAE summations. Cases 4-1 and 4-2 are not able to achieve the required set point change. Case 4-3 is able to achieve all objectives, and case 4-4 achieves the best performance. This structure is considered for implementation using model predictive control. In this manner, all candidates given in Table 5 need to be evaluated. The evaluation can be done by comparing the final transient responses and the resulting IAE and offset values to determine the best case(s) for further evaluation via nonlinear simulation. Once the best case(s) has been identified, a Model Predictive Controller (MPC) can be implemented on top of the dominant variable loops. The main objective of the MPC is to improve the control of the product rate and product quality. Case 4-4 was the best case that was found for the Tennessee Eastman process. A model predictive controller is built using these four manipulated variables and the function scmpcnl in Matlab. This function designs an MPC controller for constrained problems and simulates closed loop systems with hard constraints. Additional details on the MPC controller are given by Assali.18 The MPC controller was tested using the full nonlinear dynamic simulation. Figures 2 and 3 show the nonlinear simulations for a 50% set point change in the production rate (maximum production rate for case 4-4). Figure 2 shows transients of the eight variables (product flow, product quality, purge flow, reactor pressure, reactor level, reactor temperature, separator level, and stripper level). Figure 3 shows transients for eight other variables (%A feed, %B feed, %C feed, %D feed, A feed, D feed, E feed, and C feed). Figure 2 shows how the large and fast changes in the production rate are handled. It is worth noting that the production rate changes more than 50% in less than 2 h. Although this is a large change, the proposed control structure can handle it without valve saturation and/or plant shutdown. An important consideration from an economic point of view is the amount of purge used. In this case, the purge flow is less than what Tyreus25 reported for his control scheme. Figure 2 shows the purge flow. Figures 4 and 5 show the response of the proposed control scheme for IDV(1), a step change in the composition of A and C in the C feed stream. This figure shows that the control scheme can easily reject IDV(1). When the amount of A in the system decreases, the A feed flow increases to counteract this fact. Figures 6 and 7 show the response of the proposed control scheme for IDV(2), a change in the composition of B in the C feed stream. This figure shows how the system increases the purge flow to control the amount of B (inert). The final control system that results from the methodology described here is shown in Figure 8. The same control system architecture worked for the other two optimal operating modes, and details are presented by Assali.18


7839

Table 10. OSOFC Matrix Control Objective: to Control Product Rate and Quality SP

RF

StT

CpW

BF

EF

FF

BP

EP

FP

GP

∑ row

%A RP %C %D RT SepT RL SL St L ∑ col

0.071 0.018 0.048 0.016 0.193 -0.171 0.025 -0.092 -0.038 0.671

-0.080 -0.065 -0.263 -0.301 -0.017 -0.110 0.077 0.127 0.205 1.246

0.072 0.008 0.044 0.047 0.321 0.093 -0.018 -0.079 -0.024 0.707

-0.050 0.024 -0.047 0.083 -0.302 -0.054 -0.021 0.016 0.024 0.622

0.014 0.007 -0.004 -0.202 0.025 -0.092 0.281 -0.040 -0.031 0.695

-0.010 0.014 -0.008 0.022 -0.089 -0.003 0.014 -0.009 -0.001 0.171

0.028 -0.021 0.044 -0.136 0.255 0.006 0.083 -0.037 -0.029 0.638

0.022 0.010 0.007 -0.028 0.051 0.066 0.066 0.058 0.015 0.324

0.001 -0.011 0.003 -0.032 0.066 -0.018 0.003 -0.003 0.001 0.137

-0.070 0.053 -0.084 0.375 -0.389 -0.297 -0.197 -0.135 0.020 1.620

0.418 0.231 0.552 1.243 1.709 0.909 0.784 0.597 0.388

Table 11. Cases for Candidate 4

case number

controlled variables


case 4-1

production rate product quality

reactor temperature SP

case 4-2


reactor temperature SP %D in reactor feed SP

case 4-3


reactor temperature SP %D in reactor feed SP sep temp SP-CCWT

case 4-4


reactor temperature SP %D in reactor feed SP sep temp SP-CCWT reactor level SP

Comparison with Tyreus’ Control Scheme. During the past decade, several control structures were proposed for the TE process. The main differences among these control schemes are the ways in which the production rate (throughput), the reactor pressure, and the liquid levels are controlled. The majority of authors have focused on controlling the production rate with a single flow rate. A different idea to control the production rate using the partial control idea was introduced by Tyreus.25 He controls the production rate by manipulating the set points of the composition of A in the purge, reactor temperature, and separator temperature. In the control structure discussed here, the production rate is controlled by manipulating the set points of the reactor temperature, the separator temperature, the reactor level, and the composition of D in the reactor feed. Tyreus is the only author who demonstrated the performance of his control strategy by increasing the set point of the production rate by 50%. To do so, Tyreus ramped the set point of the dominant variables that he identified in a 5 h period. The

Figure 2. Maximum production rate for operation mode 1 (candidate 4-4).

can maximize production rate and reject disturbances?

summation offset values/% change btw candidates

IAE /% change btw candidates

SP change no IDV(1) yes IDV(2) yes SP change no IDV(1) yes IDV(2) yes SP change yes IDV(1) yes IDV(2) yes SP change yes IDV(1) yes IDV(2) yes

0.0092

330.67

0.0094 2.12

321.89 2.8

0.0083 13.25

291.29 10.51

0.0072 15.27

231.70 25.71

control structure discussed here is faster than Tyreus’ structure. It accomplishes the same production rate change in 2 h. In addition, it uses less purge flow than Tyreus’ control structure, demonstrating its economic benefit. The control scheme discussed here allows working with a larger amount of component B in the purge (21%) than Tyreus’ scheme (15%). Therefore, the amount of purge (0.3 kscmh) in the proposed scheme is less than the amount of purge (0.5 kscmh) in Tyreus’ control scheme. By using the economic numbers given by Downs and Vogel,31 the purge loss is reduced by $68/h, which is more than half a million dollars a year. Some limitations with Tyreus’ control structure for the Tennessee Eastman plant are the following. His approach uses steady state sensitivity information, while the approach discussed here uses dynamic information. His control structure is difficult to apply in a real process since the final product flow controller requires changing four set points (three dominant variables and %B purge) simultaneously. However, it would be easy to implement an MPC control system to manipulate these set points, as was done here. Also, Tyreus’

Figure 3. Maximum production rate (candidate 4-4).

7840


Figure 4. IDV(1) for operation mode 1 (candidate 4-4).



product rate and product quality. The main characteristics of this methodology are as follows: (1) It uses a linear time invariant (LTI) state space model of the plant and optimal control theory to determine key variables in the plant that affect product rate and product quality. To do so, an optimal static output feedback controller is calculated. The control objective is to control production rate and product quality, using other variables in the plant. The information about the interaction and the effect of the variables on production rate and product quality is determined by analyzing and comparing the relative values of the elements of the optimal controller gain. (2) It aims to improve economics in the plant because of the following: (a) It improves the control performance of production rate and product quality. Therefore, the plant can be operated closer to operational limits. (b) The calculation of the OSOFC is done for specific disturbance rejection and set point changes. Therefore, the resulting control structures are best suited for these cases. (3) It is a hierarchical design procedure that divides the plantwide control problem into subproblems which are more manageable and are easy to solve. In the case of the Tennessee Eastman Plant, the key variables obtained are similar to the variables obtained using Tyreus’ thermodynamic-based approach. The resulting control strategies have demonstrated their efficiency, since the production rate can be easily increased and held to more than 50% of the steady state value. The resulting control system is also able to reject disturbances IDV1 and IDV2. Although, this methodology is based on a linear methodology, all of the generated control structures were tested using the nonlinear model of the plant. Appendix: Correlation Analysis


control structure ignores the frequency constraints on the C, D, and A feeds, and negative RGA pairings result from his structure. 7. Summary and Conclusion This paper presents a systematic procedure to determine measurements and manipulated variables that affect a plant’s

In this work, the interaction between measurements is determined by analyzing the correlation of the measurements in each of the left singular vectors of the U matrix, which is obtained from the singular value decomposition of the process gain matrix. For processes with integrators, such as the Tennessee Eastman process, the approach of Arkun and Downs35 can be used to calculate the process gain matrix. Once the singular value decomposition of the process gain matrix is carried out, the model for measurements is calculated as follows: yi )

∑a

i,jµj


7841

Figure 8. Final control scheme.

where ai,j is an element of the left singular vectors by the singular values (U×Σ) from the SVD analysis. If each measurement is assumed to have the same variance random noise added to it, then the correlation coefficient between the measurements can be calculated as ri,j ) ε(yiyk)/(σ(yi)σ(yk)) where ε is the expected value, σ is the variance, and i,k refer to different rows. The correlation matrix has elements

Table A1. Condition Number Analysis of C Matrix measurements included

CN of the C matrix

reactor pressure reactor and separator pressure reactor and stripper pressure reactor, separator and stripper pressure composition of G in the purge composition of H in the purge compositions of G and H in the purge

982.0030 1.6637e+004 1.3718e+004 2.3442e+012 9.0452e+004 5.9808e+004 7.5591e+016

Condition Number Analysis of the C Matrix ri,k )

∑a

i,jak,j /

j

∑ (

j

a2i,j

×

∑

a2k,j)

j

The elements of the correlation matrix are known as coefficients of determination (ri,k2). These elements measure the variation of the measurement i that is explained by measurement k. Therefore, to determine the correlation between two measurements, the elements of the correlation matrix in each column are analyzed. If ri,k2 is close to 1, then yi and yk are correlated. If there is more than one measurement that has a value of ri,k2 close to 1, in the same column, this means that these measurements have interaction among them. After this correlation analysis, we found that the following groups of variables are correlated for the Tennessee Eastman process: Group # 1: reactor pressure, separator pressure, and stripper pressure Group # 2: Separator temperature, stripper temperature, stripper steam flow, recycle flow, and reactor feed Group # 3: composition of E in the reactor feed and composition E in the purge Group # 4: composition of F in the reactor feed and composition of F in the purge Group # 5: composition of G and H in the purge

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ReceiVed for reView November 29, 2009 ReVised manuscript receiVed May 6, 2010 Accepted May 10, 2010 IE901879E

Optimal Selection of Dominant Measurements and Manipulated

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