Development of Extended Generic Model Control for High-Purity Heat

Article pubs.acs.org/IECR

Development of Extended Generic Model Control for High-Purity Heat Integrated Distillation Column Using Online Concentration Estimation Lin Cong,*,† Liang Chang,‡ and Xinggao Liu*,‡ †

College of Information and Control Engineering, China University of Petroleum (East China), Qingdao, 266580, China Department of Control Science and Engineering, Institute of Industrial Process Control, Zhejiang University, Hangzhou, 310027, China

‡

ABSTRACT: A hybrid control scheme consisting of an online concentration estimator and an extended generic model controller (EGMC) is proposed for the high-purity control of the heat integrated distillation column (HIDiC). The estimator is designed based on a principal component analysis (PCA) based method and the nonlinear wave theory of HIDiC. The PCAbased method is introduced to determine the locations of a limited number of measurements, while the nonlinear wave theory is presented to estimate the stage concentration according to the wave profile parameters estimated from the selected measurements. Then the EGMC controller coupling with the online estimation is established. A benzene−toluene system is taken as an illustration, and a comparative simulation study is carried out among the proposed EGMC, a nonlinear-wave-based classical generic model controller (WGMC), proportional−integral−derivative control (PID), and a model predictive control scheme (MPC). Finally, EGMC shows a superior performance over the other control schemes, which confirms the validity of the proposed control scheme.

1. INTRODUCTION Heat integrated distillation is a new kind of distillation technology which has attracted researchers’ attention in recent years because of its superior energy efficiency over conventional distillation. Figure 1 gives the schematic diagram of a heat

The heat integration during the two parts of HIDiC greatly enhances the energy efficiency of the distillation process, which is quite energy inefficient in a conventional but widely used structure. However, the heat integration, which is also called thermal coupling, makes the dynamics of HIDiC more complex than a conventional distillation column.7,16−21 The highly nonlinear dynamic behaviors make the control of HIDiC more difficult especially when the purity of the product is high. Linear controllers are difficult to regulate the HIDiC process even for moderate-purity production, and therefore several advanced control schemes have been investigated for HIDiC, including generic model control (GMC),22,23 generalized predictive control (GPC),24 and dynamic matrix control (DMC).25 Classical GMC26 is only applicable to handle the model with a relative order of 1, which limits the application of the scheme. For a relative order larger than 1, GMC cannot be used directly and a simplified model is needed. For example, an autoregressive exogenous (ARX) model for HIDiC was introduced in our previous work.22 But the simplification of the model would make a severe model/plant mismatch and also result in worse control effects. Recently a novel control scheme, extended generic model control (EGMC), was proposed based on the classical GMC control scheme.27,28 The newly developed EGMC is applicable to processes with any number of relative order, and the illustration of a conventional distillation column with a relative order of 1 proved the validity of the control scheme. The capability

Figure 1. Schematic diagram of an ideal HIDiC.

integrated distillation column (HIDiC), in which the rectifying and stripping sections are separated into two individual columns. By means of the compressor and the throttle valve, the rectifying column can work at a much higher pressure than the stripping column, and hence heat transfers smoothly from the rectifying column to the stripping column since the higher pressure leads to a higher temperature. Under a deliberate design, the duty of the condenser and the reboiler can reduce to zero during the operation, and this kind of HIDiC is called an ideal HIDiC. More information about HIDiC can be found in the open literature.1−15 © 2015 American Chemical Society

Received: Revised: Accepted: Published: 12897

September 5, 2015 November 6, 2015 December 7, 2015 December 7, 2015 DOI: 10.1021/acs.iecr.5b03300 Ind. Eng. Chem. Res. 2015, 54, 12897−12907

Article

Industrial & Engineering Chemistry Research of EGMC makes it possible to apply in a more complex process, which is not involved in the previous literature. Nonlinear wave theory is an effective method to characterize the dynamics of the separation process,29,30 and recently was extended from conventional distillation column to HIDiC in our previous work.31 The wave theory provides a new way to deal with the complex dynamics of HIDiC. The wave parameters in the wave theory cannot be measured directly, and normally were estimated based on the concentrations of all the stages in HIDiC. The concentration can be inferred from the corresponding temperature in each stage, but it is not practical to place a temperature measuring instrument at each stage in HIDiC. In conventional distillation columns, several online estimations of wave parameters have been developed according to a limited number of stage concentrations,32−34 among which Henson provided a principal component analysis (PCA) based measurement selection method, using several stage measurements to estimate wave parameters and showing a satisfactory result.34 In order to make good use of the wave theory, the estimation of wave parameters in HIDiC is also an important issue that must be solved, which perhaps has not been applied so far on HIDiC. Compared to conventional distillation columns, HIDiC presents relatively high nonlinearity, which makes the control design more difficult.16 The current work focuses on synthesizing the extended GMC coupled with online concentration estimation based on nonlinear wave theory with the application to HIDiC. First, an online estimator using a PCA-based measurement selection method34 for wave parameters in HIDiC is proposed and accordingly the stage concentrations can be estimated based on the known wave parameters. Since the situations of conventional distillation column and HIDiC are different, the estimation method is adjusted to be more suitable for HIDiC. Then, the extended GMC is presented for HIDiC based on the estimation mentioned above, and also a nonlinear wave based classical GMC controller (WGMC) proposed in our previous work,23 proportional−integral−derivative control (PID), and a model predictive control scheme (MPC) presented in our earlier work21 are introduced here as a comparison to EGMC. The benzene−toluene HIDiC separation system is used as the illustration and the proposed EGMC shows high quality performance over other controllers in the simulations both with and without plant−model mismatches.

Table 1. Operating Conditions of HIDiC stage number (n) feed stage ( f) feed composition (benzene) (Zf) feed composition (toluene) (Zf) heat transfer rate in each stage without mismatch (UA) [W/K] Antoine equation constant (a) Antoine equation constant (b) Antoine equation constant (c)

X r_max − X r_min 1 + exp(− k r(j − S1))

,

Xs_max − Xs_min 1 + exp(− ks(j − S2))

,

0.501 100 1.5 30001.1 9313 2.317 0.3387 0.1013

3. ONLINE ESTIMATION DESIGN Since the wave profile parameters, namely, Xr_max, Xr_min, kr, Xs_max, Xs_min, and ks, cannot be measured directly, a estimator design is needed for the application of profile functions. If the concentration at each stage is known, the parameters can be obtained easily by the regression method. However, placing a measurement at each stage is impractical and only a limited number of measurements can be used. Therefore, the locations of the measurements are very critical since they must provide enough information under different operating conditions. In this paper, an improved PCA-based method is presented to better fit HIDiC for the location selection of the measurements. Since the behaviors of the rectifying and stripping columns are different in HIDiC, the profiles of the rectifying and stripping sections are considered separately in HIDiC. Besides, the concentration is usually difficult to get directly and immediately, so the temperature measurement which can also infer the concentration is used here.

j = 1, 2, ..., f − 1 (1)

X̂j = Xs_min +

feed thermal condition (q) feed flow rate (F) [kmol/h] liquid holdup in each stage (H) [kmol] 0.5 latent heat of vaporization [J/kmol] 9803 heat transfer rate in each stage with mismatch (UA) [W/K] 15.9008 relative volatility without mismatch (α) 2788.51 pressure of rectifying section (Pr) [MPa] −52.36 pressure of stripping section (Ps) [MPa]

where Xr_max, Xr_min, kr, Xs_max, Xs_min, and ks are the six wave profile parameters. Xr_min and Xr_max denote the asymptotic limits of the rectifying section when the profile extends to an infinite distance; Xs_min and Xs_max denote the asymptotic limits in the stripping section; Xr_max and Xs_min approximate the concentration (liquid mole fraction) at the top stage and the bottom stage, respectively. kr and ks characterize the tangent of the inflection points S1 and S2. S1 and S2 are the inflection points of profiles and the representation of the profile positions in the rectifying section and the stripping section, respectively. X̂ j denotes the estimation of the concentration at stage j. If the wave profile parameters are determined, the stage concentration can be easily estimated by the above profile functions. Figure 2 gives the profiles reconstructed by the stage concentrations and the profile functions respectively when the feed thermal condition q = 0.48 and 0.501. From Figure 2, the two curves are very close to each other under both conditions, which means that the profile function is quite accurate to describe the stage concentrations as long as an appropriate set of profile parameters is defined. Figure 3 shows the errors between the two curves for clarity. The largest deviation is less than 0.003. By adjusting constraints, the errors near the top and bottom of the column (stage 1 and stage 40) can be close to zero, which means that the profile function can reach higher accuracy at the dual end, so the profile function is very conducive to cooperating with the control scheme, since we concentrate more attention on the top and bottom concentrations which represent the product quality and are the controlled variables.

2. PROFILE FUNCTION IN NONLINEAR WAVE THEORY A benzene−toluene separation system is utilized as an illustration for the study, and the structure of HIDiC has already been given in Figure 1. The initial operating conditions are listed in Table 1, and the research in the current work is based on the initial steady state. In nonlinear wave theory, the distribution of concentrations at different column stages is considered as a whole to study.30,31 Since the behaviors of the rectifying and stripping sections in HIDiC are quite different during the dynamic process, the concentration distributions of these two sections are characterized by two individual equations, named as profile functions, which provides a new method to describe the stage concentration: X̂j = X r_min +

40 21 0.5

j = f , f + 1, ..., n (2) 12898

DOI: 10.1021/acs.iecr.5b03300 Ind. Eng. Chem. Res. 2015, 54, 12897−12907

Article

Industrial & Engineering Chemistry Research

Figure 2. Profiles reconstructed by the stage concentrations and the profile functions.

Figure 3. Errors between the stage concentrations and the profile functions.

3.1. Concentration Estimation from Temperature. The vapor and liquid concentrations of different stages can be estimated from the temperature of the corresponding stages under some simplifying assumptions. The following assumptions are invoked: perfect liquid and vapor mixing on each stage; the temperature and the concentration on each stage being uniform; the relative volatility is constant. Then the concentration in each stage can be derived from the following assumptions. According to the Antoine equation with Raoult’s law and Dalton’s law Tj = b/(a − ln pvp, i ) − c

(3)

pvp, j = p /[Xj + (1 − Xj)/α]

(4)

In this work, the top stage of the rectifying section and the bottom stage of the stripping section must be selected first since the concentrations of the two stages are controlled variables and must be monitored. Since the methods used for both sections are technically the same, we take the rectifying section as the illustration. Suppose that the number of estimated variables is q for the rectifying section, then the first measurement selection is the top stage and the rest of the first q selections can be decided based on the following iterative algorithm: 1. The elements of the sensitivity matrix between the candidate measurements and the estimated variable can be expressed as dxi dvj

where Tj is the temperature of stage j, pvp,i is the vapor saturated pressure of stage j, p is the pressure of rectifying or stripping column, and α is the relative volatility. Therefore, the relationship of concentration and temperature is derived: X j = (αp e

b /(Tj + c) − a

− 1)/(α − 1)

sij =

vj =vj̅

xi̅ / vj̅

(6)

where x is the vector of the candidate measurements, that is, the stage concentration which can be derived from temperature, v is the vector of the estimated variables, and v ̅ and x̅ are the nominal values of the initial steady state. The larger sij is, the more sensitive the ith candidate measurement is to the jth estimated variable and the more useful this measurement is for the estimation. By introducing a 1% step change of the estimated variable vi, sij can be computed numerically according to eq 1 and eq 6. Denote the

(5)

3.2. PCA Method for the First q Selections. The selections of the measurement locations for the rectifying and stripping sections are made respectively in HIDiC. Xr_max, Xr_min, kr, and S1 are estimated variables for the rectifying section, and Xs_max, Xs_min, ks, and S2 are estimated variables for the stripping column. 12899


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Industrial & Engineering Chemistry Research weighted sensitivity matrix as s = {sij} ∈Rn/2×q, n is the total stage number and n/2 is the stage number of the rectifying section. 2. Define the covariance matrix as Xc = s·sT ∈ Rn/2×n/2, and it can be computed from the sensitivity matrix s. Denote λj as the jth nonzero eigenvalue of Xc, and [P1j, P2j, ..., Pij, ..., Pn/2j] as the corresponding unit eigenvectors of λj. The weighed sum of Pij and λj can be expressed as

Considering the worst case over all the combinations, that is, Ij = minr Ir,j. Calculate Ij for each candidate measurement (n/2 − v candidate stages), and select the candidate measurement with the largest Ij as the next measurement. Then the number of the measurement becomes v + 1. Carry out the above algorithm iteratively until all the measurements are selected. In conclusion, the measurement selection can be determined based on the above method. Table 2 gives the measurement

q

Ci =

∑ j = 1 |λjPij| q

∑ j = 1 |λ j |

Table 2. Measurement Ranking Results for PCA Method

(7)

ranking of stage

where Ci ∈ [0 1], characterizing a synthetic evaluation of the ith stage measurement to the total q estimated variables. 3. Suppose that m measurements have been selected, and 1 ≤ m < q. For all l ∈ [1 m], the sensitivity vectors corresponding to the lth selection are denoted as ski = [skl1 skl2 ··· sklq], where kl ∈ [1 n/2] is the stage index. The elements of skl can be directly got from the sensitivity matrix s. Any vector in the space determined by sk1, sk2, ..., skm can be expressed as

∑ alskl l=1

fifth

sixth

rectifying column stripping column

1 40

20 31

19 26

16 37

8 23

13 38

X r_max , X r_min , k r , s1

∑ (X ̂ k

v

− X kv)2

(12)

i=1

subject to X̂ kv = X r_min +

(9)

X r_max − X r_min 1 + exp( −k r(kv − S1))

X r_min − X r_max < 0

0 < X r_max < 1 0 < X r_min < 1

where kv is the stage number of the selected measurement, Xkv is the stage concentration obtained by the measurements, and X̂ kv is the estimated concentration. Then the estimated variables Xr_max, Xr_min, kr, and S1 can be known, and therefore all the stage concentrations can be estimated according to the profile function eq 1 whether there exists a measurement at the corresponding stage or not. Figure 4 and Figure 5 show the estimation performance under different steady states, the initial steady state list in Table 1, and the feed thermal condition q + 10%, respectively. The profiles by the process and the profile functions are both given. The profile parameters are estimated first, and then the concentration of each stage is estimated by the profile function and shown in Figures 4 and 5. From the figures, the PCA-based method can handle a large operating range.

(10)

which makes a good trade-off between the magnitude and direction of the sensitivity. Calculate the ranking indices of all the unselected stages, and choose the ith stage with the largest Ii as the next measurement. 5. Let m = m + 1 and use the method above iteratively, then another measurement can be decided. Since the rank of the matrix s is q, there exist at most q linearly independent sensitivities sk1, sk2, ..., skq. Therefore the calculation will terminate when m = q, and the first q selections are available. 3.3. Selection for the Rest of the Measurements. The method above can select q measurements, the same number of estimated variables. For more selections, the method is invalid and another selection principle is needed. Suppose that v measurements have been selected, then v ≥ q and any choice of q − 1 measurements from the v measurements selected form a combination. The number of all the possible combinations is r = Cq−1 v . Considering any of the unselected candidate stages, for example the jth stage, calculate the closest vector to sj in the space determined by any of the r combinations using the method mentioned above, and then the angle between sj and ŝj,min can be calculated and denoted as θr,j. r different angles can be obtained from all the r combinations, and accordingly r different ranking indices are determined as Ir , j = Cj sin θr , j

fourth

v

and the closest vector is expressed as ŝi,min = ∑l m= 1 ãlskl. 4. Denote the angle between si and ŝi,min as θi, reflecting the collinearity degree between si and ŝi,min. The greater θi is, the more unique information si can offer. Define a ranking index as Ii = Ci sin θi

third

min

T

al

second

(8)

For any of the rest (n/2 − m) unselected stages, the corresponding sensitivity vector is denoted as si, and the closest vector to si in the space determined by sk1, sk2, ..., skm can be calculated by solving the following minimization problem min(si − s )̂ (si − s )̂

first

ranking results for both rectifying and stripping sections, and six measurements are utilized for each section. 3.4. Concentration Estimation for HIDiC. Suppose that v measurements are utilized, then the profile parameters in the rectifying section can be derived by solving the following minimum problem:

m

ŝ =

section

4. CONTROL SCHEME FOR HIDIC 4.1. Review of Wave-Based GMC. Recently, a nonlinear wave based classical GMC scheme (WGMC) was proposed in our previous work,23 and is used in the current work as comparison. Also, the estimation proposed above is coupled with WGMC. In nonlinear wave theory, S1 and S2 are the representations of the profile positions in the rectifying section and stripping section, respectively. The concentrations of the top and bottom products, Y1 and Xn, are controlled, with the pressure of the rectifying section (Pr) and feed thermal condition (q) as the manipulated variables. The movements of S1 and S2 can reflect

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Figure 4. Estimation performance under initial steady state.

Figure 5. Estimation performance under q + 10%.

⎡ dS 2 1 = ⎢⎢ (− Vf Y f + Lf − 1X f − 1 − FZf − LnX n) dt H ⎣

the values of Y1 and Xn by nonlinear wave theory. Therefore the control of the concentrations can be replaced by control of the positions of S1 and S2. When S1 and S2 deviate from the set points S*1 and S*2 , the rate of change of S1 and S2 should be controlled to return toward the desired steady states and has zero offset, and then WGMC formulation has the following forms: dS1 = K11(S1* − S1) + K12 dt

∫0

dS 2 = K 21(S2* − S2) + K 22 dt

t

∫0

(S1* − S1) dt t

−

−

∑

X r_max X r_min

j=1 f −1

+

∑ j=1

⎡ ×⎢− ⎢⎣

−

×

∑ j=1

∑ j=f

(S2* − S2) dt

⎡ ×⎢− ⎢⎣

(14)

f − 1 dX r_min (X r_max dt

∑ j=1

− X j)

X r_max X r_min

k r(X r_max − Xj)(Xj − X r_min) ⎤ ⎥ ⎥⎦ X r_max − X r_min

(Xj − Xs_min)

X max − Xs_min

n

−

dX s_min

∑ j=f

dt

(Xs_max − Xj)

Xs_max − Xs_min

⎤ (Xs_max − Xj)(Xj − Xs_min) dks (S2 − j)⎥⎥ Xs_max − Xs_min dt ⎦ n

∑ j=f

ks(Xs_max − Xj)(Xj − Xs_min) ⎤ ⎥ ⎥⎦ Xs_max − Xs_min

−1

(16)

The profile parameters Xr_max, Xr_min, kr, Xs_max, Xs_min, and ks and the stage concentrations Xj can be estimated by the estimator designed above. The derivative terms of the profile parameters can be replaced by a difference approximation. The meanings of the other parameters can be seen in the Nomenclature, and these parameters can be calculated based on the variables estimated above. 4.2. Development of the Extended GMC. Considering the following state space representation of a nonlinear multi-input/multi-output (MIMO) system:

⎤ (X r_max − Xj)(Xj − X r_min) dk r (S1 − j)⎥⎥ X r_max X r_min dt ⎦ f −1

∑

dt

n

(13)

⎡ dS1 ⎢ 1 = ⎢ ( − V1Y1 + Vf Y f − Lf − 1X f − 1) dt H ⎣ − X r_min)

dX s_max

j=f

where

f − 1 dX r_max (Xj dt

n

−1

(15) 12901

x(̇ t ) = f (x , d , θ ) + g (x , d , θ ) ·u(t )

(17)

y(t ) = h(x)

(18) DOI: 10.1021/acs.iecr.5b03300 Ind. Eng. Chem. Res. 2015, 54, 12897−12907

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Industrial & Engineering Chemistry Research Here, the state vector x(t) ∈ Rn, the input vector u(t) ∈ Rn, the output vector y(t) ∈ Rm, d is the disturbance vector, and θ is the process parameter vector. There are three nonlinear functions involved, namely, f, g, and h. If the output yi has a relative order of ri, then we have the following relationships: yi(α) = Lαf hi(x)

α = 0, 1, ..., ri − 1

Therefore, for the convenience of discussion, X1 and Xn are regarded as controlled variables. According to the extended GMC control law, the EGMC formulation for HIDiC can be expressed as X1″ + α12X1′ + α11X1 = k11(X1* − X1) + k12

(29)

(19)

X n″ + αn2X n′ + αn1X n = kn1(X n* − X n) + kn2

yi(ri) = Lrfihi(x) + Lg Lrfi − 1hi(x) ·u

L0f hi(x) = hi(x)

Lαf hi(x) =

(30)

where α11, α12, k11, k12, αn1, αn2, kn1, and kn2 are the control parameters. For the benzene−toluene system, the component mass balances in stages 1 and n of HIDiC are as follows: 1 X1′ = (V2Y2 − V1Y1 − L1X1) (31) H

(21)

∂hi(x) f (x ) ∂x 1 ∂Lα− hi(x) f

∂x

(22)

f (x )

1 ( −VnYn + Ln − 1X n − 1 − LnX n) H Then time derivatives of (31) and (32) are X n′ =

(23)

Assume the reference trajectory for the ith process output, yi is as follows: yiri

+

αiriyi(ri − 1)

+ ki2

∫0 u(t )

t

(33)

∫ (yi* − yi ) dt

X n″ = ( −Vn′Yn − VnY n′ + Ln′− 1X n − 1 + Ln − 1X n′− 1 − Ln′X n

(24)

− LnX n′)/H

[H(x , u , d)]T WH(x , u , d)] dt

⎛ 1 Q j = UA ·b⎜⎜ ⎝ a − ln{pr /[Xj + (1 − Xj)/α]}

(25)

−

x ̇ = f (x , d , θ ) + g (x , d , θ ) ·u(t ) y = h(x)

L1 = Q 1/λ

hi = yiri + αiriyi(ri − 1) + ... + αi2yi ̇ + αi1yi − ki1(yi* − yi ) (26)

Ln − 1 =

yiri + αiriyi(ri − 1) + ... + αi2yi ̇ + αi1yi − ki1(yi* − yi ) i = 1, ..., m

j − 1, 2, ..., n

(38)

Ln = F − V1

(39)

V2 = V1 + L1

(40)

Vn = V1 +

f −2

∑ Q k /λ − F(1 − q) − ∑ Q k /λ k=1

Substituting (19) and (20) into the above equations, u can be solved out. 4.3. Extended GMC for HIDiC. For the HIDiC system, the controlled variables (Y1 and Xn) and the manipulated variables (Pr and q) are the same as WGMC discussed above. The mole fraction of vapor Y1 can be easily derived from the mole fraction of liquid X1 by vapor−liquid equilibrium relationships

(37)

k=1

V1 = F(1 − q)

f −1

(27)

f −2

∑ Q k /λ + Fq − ∑ Q k /λ k=1

If no constraints are involved and at least one element of u appears in each of the m equations represented by (26), the solution of the optimal control problem is equivalent to the solution of the following equations:

Yj = αXj /[(α − 1)Xj + 1],

(36)

f −1

∫ (yi* − yi ) dt

(35)

Vapor and liquid molar flow rates are

where W is a weighting matrix, H(x,u,d) = [h1, h2, ..., hm] , and

∫ (yi* − yi ) dt = 0,

⎞ 1 ⎟, a − ln{ps /[Xj + f − 1 + (1 − Xj + f − 1)/α]} ⎟⎠

j = 1, 2, ...f − 1 T

− ki2

(34)

The other relations are listed as follows: The heat transfer rate between stage j in the rectifying section and stage j + f − 1 in the stripping column is

subject to

− ki2

(32)

X1″ = (V2′Y2 + V2Y 2′ − V1′Y1 − V1Y1′ − L1′X1 − L1X1′)/H

+ ... + αi2yi ̇ + αi1yi = ki1(yi* − yi )

where αiri, ..., αi1, ki1, and ki2 are control parameters and y* is the set point. The control problem can be transformed as the following optimal control problem: min

∫ (Xn* − Xn) dt

(20)

with

Lf hi(x) =

∫ (X1* − X1) dt

k=1

(41)

All the stage concentrations Xj can be estimated by the estimator proposed in this work, and the variables involved on the right sides of (31)−(41) can be divided into several types: (1) Xj obtained by the estimator; (2) known variables, such as the operating conditions or the constants; (3) variables which can be calculated from the known variables and the estimated variables, such as mole fraction of vapor Yj; (4) manipulated variables (Pr and q).

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Figure 6. Servo performance of WGMC and EGMC with step changes in both set points.

Figure 7. Regulatory performance of WGMC and EGMC under F + 10%.

response than EGMC. Figure 8 depicts the comparative regulatory performance between WGMC and EGMC for the feed mole fraction Zf + 10%. EGMC shows a better disturbance rejection performance than WGMC. Adjust the control parameters to make WGMC have a servo response similar to that of EGMC, and then compare the regulatory performances of the two controller. In this way, a fairer comparison is carried out. The control parameters for WGMC are adjusted as K11 = 30, K12 = 200, K21 = 60, and K22 = 500, and Figure 9 shows the servo control performance of WGMC and EGMC with the same step change of the set points above. However, the adjustment is not satisfactory. As shown in Figure 9, WGMC and EGMC have about the same servo responses by tuning the control parameters. We can see clearly in Figure 9, although the performances of WGMC are improved to some extent, the response of the bottom is seriously oscillatory during about 0.1−0.3 h. Figure 10 shows the changes of the manipulated variables with about the same servo responses, in which the oscillation can be seen more clearly. Besides, if Zf increases 10% or more, WGMC would be fail to work in this set of control parameters. The simulation above proves that the method of adjusting the parameters of WGMC to improve the performance would sacrifice the robustness and stability of the controller. 5.2. Simulation with Plant−Model Mismatch. In a real industrial process, there always exists plant−model mismatch.

Substitute (31)−(34) into (29) and (30), and utilize the relationships of (35)−(41), then the manipulated variables (Pr and q) can be solved out.

5. SIMULATION RESULTS 5.1. Simulation without Plant−Model Mismatch. In the simulation test without plant−model mismatch, both WGMC and EGMC are carried out for the benzene−toluene HIDiC system and the initial operating conditions are listed in Table 1. The control parameters for WGMC are K11 = 15, K12 = 100, K21 = 7, and K22 = 100, and the control parameters for EGMC are α11 = 0.2, α12 = 1.5, αn1 = 0.3, αn2 = 1.2, K11 = 30, K12 = 200, Knl = 60, and Kn2 = 500. Figure 6 shows the servo control performance of WGMC and EGMC, when the set point of the concentration of the top product (Y1) increases from 0.998995 to 0.9992, and meanwhile the set point of the concentration of the bottom product (Xn) decreases from 0.002997 to 0.0028. It is clear that the responses of WGMC are more sluggish in both the rectifying and stripping sections. The feed flow rate F and the feed mole fraction Zf are two main disturbances for HIDiC. Step disturbances from F and Zf are introduced into the system respectively and the responses of the two controllers are investigated. With the feed flow rate F + 10%, the performances shown in Figure 7 demonstrate that WGMC has a more oscillatory 12903


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Figure 8. Regulatory performance of WGMC and EGMC under Zf + 10%.

Figure 9. Servo control of WGMC and EGMC with about the same servo responses.

Figure 10. Changes of the manipulated variables with about the same servo responses.

In order to make our simulation closer to the actual process, plant−model mismatch is introduced in the following simulation. In the ideal situation, the benzene−toluene mixture has a constant relative volatility which is used in the above control scheme. However, in the actual separation system, the relative volatility would be nonconstant and have unknown bias. Therefore a nonideal vapor−liquid equilibrium is used to describe the plant, while the ideal vapor−liquid equilibrium is used in the

control method in which the real nonideal relation is not known. As shown in Figure 11, the solid line represents the ideal vapor− liquid equilibrium used in the model and the control method, determined by (28), while the dashed line indicates the nonideal vapor−liquid equilibrium used in the plant. We assume that there are no mismatches in the vapor−liquid equilibrium relations of the first stage and the nth stage, since the compositions of the vapor leaving the first stage and the liquid leaving the nth stage are controlled variables in the simulation. If the mismatches exist 12904


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Figure 11. Vapor−liquid equilibrium of the model and the plant.

Table 3. Temperature (Absolute Temperature, K) and Composition of Each Stage at the Initial Steady State stage no.

temp

composition

stage no.

temp

composition

stage no.

temp

composition

1 2 3 4 5 6 7 8 9 10 11 12 13 14

398.36 398.37 398.39 398.41 398.44 398.48 398.53 398.61 398.71 398.87 399.10 399.44 399.93 400.65

0.9977 0.9972 0.9965 0.9956 0.9943 0.9926 0.9902 0.9869 0.9821 0.9753 0.9653 0.9507 0.9296 0.8994

15 16 17 18 19 20 21 22 23 24 25 26 27 28

401.68 403.11 405.00 407.32 409.96 412.69 365.99 368.10 370.42 372.88 374.90 376.62 378.27 379.48

0.8572 0.8006 0.7294 0.6466 0.5590 0.4749 0.4011 0.3430 0.2828 0.2247 0.1726 0.1292 0.09496 0.06917

29 30 31 32 33 34 35 36 37 38 39 40

380.58 381.20 381.58 381.97 382.25 382.48 382.62 382.82 382.91 383.01 383.07 383.10

0.05030 0.03673 0.02706 0.02016 0.01522 0.01164 0.009023 0.007082 0.005623 0.004512 0.003658 0.002997

Figure 12. Servo performance with plant−model mismatches.

in the two stages, certain offsets would occur, which could be compensated easily since usually in the real plant the set points do not change frequently and the certain offsets could be easily eliminated by correcting the set points according to the data from the real process. Also, a plant−model mismatch from the heat transfer rate UA is introduced. UA used in the control method retains the original value of 9803, and UA in the plant is adjusted as

9313, 95% of the original value, which is not known in the control method. With the existence of the mismatches, WGMC cannot work properly and the divergence phenomenon occurs. Therefore we choose two conventional control methods as comparisons, i.e., PID and MPC, which are more robust. PID is realized by two single-loop sub PID controllers, and the MPC control scheme of HIDiC was proposed in our previous work.21 Table 1 and 12905


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Figure 13. Regulatory performance of PID, MPC, and EGMC under F + 10%.

Figure 14. Regulatory performance of PID, MPC, and EGMC under Zf + 10%.

Based on the online concentration estimation, the EGMC control scheme for HIDiC is proposed. The benzene−toluene separation system is taken as the illustration, and several control systems are utilized for comparison both with and without plant−model mismatches. The simulation results show that EGMC outperforms the other control schemes in both servo and regulatory performances.

Table 3 show the complete information on the base case conditions of the system with plant−model mismatches. In the servo control, the same as above, the set point of the concentration of the top product increases to 0.9992, and meanwhile the set point of the concentration of the bottom product decreases to 0.0028. The performances of the three controllers, PID, MPC, and EGMC are shown in Figure 12. From Figure 12, PID and MPC are more sluggish than EGMC. In the regulatory control, with the feed flow rate F + 10%, the performances are shown in Figure 13. Figure 14 depicts the comparative regulatory performances for the feed mole fraction Zf + 10%. Also, EGMC is superior to PID and MPC in both the top and bottom responses.

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AUTHOR INFORMATION

Corresponding Authors

*E-mail: [email protected] (L.C.). *E-mail: [email protected] (X.L.). Notes

The authors declare no competing financial interest.

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6. CONCLUSION In this work, an extended GMC control system is developed for HIDiC. First, an online estimator aiming at estimating the state variables required by the EGMC control law is presented. A PCA-based selection method is introduced to select the location of the measurements. Based on the measurements, the estimator first estimates the wave profile parameters which have physical meanings in the nonlinear wave theory, and then the stage concentrations can be online estimated by the profile functions and the online updating profile parameters. The tests under different conditions show that the estimation of the concentrations can maintain the accuracy over a wide range.

ACKNOWLEDGMENTS This work is supported by the Scientific Research Foundation of China University of Petroleum and the Fundamental Research Funds for the Central Universities (15CX02036A), and the support is thereby acknowledged.

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NOMENCLATURE a, b, c = coefficients of the Antoine equation F = feed [kmol/h] f = feed stage DOI: 10.1021/acs.iecr.5b03300 Ind. Eng. Chem. Res. 2015, 54, 12897−12907

Article


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H = stage holdups in each stage [kmol] j = stage number (counted from the top to the bottom) kr, ks = characterize the tangent of the inflection points S1 and S2 L = liquid flow rate [kmol/h] n = stage number Pvp = vapor saturated pressure [MPa] Pr = pressure of rectifying section [MPa] Ps = pressure of stripping section [MPa] P = representation of either Pr or Ps [MPa] q = feed thermal condition Q = heat transferred rate between the coupling stages [kW] S1 = position of wave profile in the rectifying section S2 = position of wave profile in the stripping section t = time [h] T = absolute temperature [K] UA = heat transfer rate in each stage [W/K] V = vapor flow rate [kmol/h] X = mole fraction of liquid X̂ = estimation of liquid mole fraction Xr_min, Xr_max = asymptotic limits of the rectifying section Xs_min, Xs_max = asymptotic limits in the stripping section Y = mole fraction of vapor Zf = mole fraction of feed Greek Symbols

α = relative volatility λ = latent heat [J/mol]

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Development of Extended Generic Model Control for High-Purity Heat

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