A Novel Hierarchical Control Structure with Controlled Variable

Article pubs.acs.org/IECR

A Novel Hierarchical Control Structure with Controlled Variable Adaptation Lingjian Ye,*,† Yi Cao,§ Xiushui Ma,† and Zhihuan Song‡ †

Ningbo Institute of Technology, Zhejiang University, 315100, Ningbo, Zhejiang, China Department of Control Science and Engineering, Zhejiang University, 310027, Hangzhou, Zhejiang, China § School of Engineering, Cranfield University, Cranfield, Bedford MK43 0AL, United Kingdom ‡

ABSTRACT: For control and optimization of chemical processes, the traditional hierarchical control structure (HCS), where an optimizer in the real-time optimization (RTO) layer updates the set-points of controlled variables (CVs) in the lower control layer, has been well-acknowledged and widely adopted in industrial applications. However, a common drawback of such an HCS is that the speed for a plant to converge to an optimal operation is slow because the optimizer has to wait for the process to settle from one steady-state to another to get an accurate disturbance estimation before making any changes to the set-points. In this Article, a novel HCS based on the concepts of controlled variable adaptation (CVA) and nonoptimality detection is proposed. In the CVA strategy, the CVs are determined and adapted on the basis of a so-called just-in-time regression algorithm to approximate the necessary conditions of optimality (NCO), which makes the self-optimizing performance adaptive to operating condition changes. For nonoptimality detection, we apply the theory of statistical process monitoring to monitor the optimality of process operation, where the nonoptimal statuses are treated as a special kind of process faults. The detection results are used as a prerequisite to activate the CVA. With these techniques, the proposed CVA-based HCS exhibits the following distinct features: (1) The regulatory control layer has an ability to approach a near optimal operation automatically through selfoptimizing control, thus accomplishing the majority of the optimization task, and the speed of the process converging to an optimal status is fast. (2) The RTO layer extends the self-optimizing operation range via adapting CVs in the lower control layer, rather than their set-points as in a traditional HCS. (3) The activation of CVA is neither regular nor periodic, but only evoked when it is necessary. Two case studies are provided to demonstrate the basic characteristics and advantages of the proposed CVA-based HCS.

1. INTRODUCTION Both in academic and in industrial fields, the effectiveness of hierarchical control structure (HCS) for regulatory control and real-time optimization (RTO) of chemical processes is wellknown, and its successful applications in real world are frequently reported.1−4 In such an HCS, two distinctly separated layers, the control layer and the RTO layer, are assigned with different tasks. In the control layer, stability and disturbance rejection are achieved through maintaining controlled variables (CVs) at their specified set-points. In some cases, this layer is further divided into a regulatory control layer and a supervisory control layer working in a time scale of seconds and minutes, respectively.2 The former is usually configured with proportionalintegral-derivative (PID) controllers, while the latter more often works with multivariable model predictive controllers (MPC). The RTO layer is normally based on rigorous steady-state process models aiming to optimize daily operation under various disturbances. A commonly used approach for RTO is the so-called two-step approach, sometimes also referred to as the repeated identification and reoptimization approach.5 In this approach, two steps are sequentially carried out. In the first step, unmeasured disturbances are estimated using available measurements, and then the process model is updated with the estimated disturbances. In the latter step, the RTO problem is solved numerically using the updated model and estimated disturbances. The solution of optimal set-points for CVs is passed down to the control layer to drive the plant © 2014 American Chemical Society

toward the optimal operation point. Also, the RTO layer is sometimes divided into local optimization layer where optimization updates in a time scale of hours and plantwide optimization layer, which usually includes planning and scheduling for marketing conditions and updates plant operation in a time scale of days or weeks.2,6 A major drawback of the traditional HCS is that the speed for a RTO layer to update set-points is slow. This is because the disturbance estimation has to be carried out in a steady-state and the settling time for most process plants is generally very large. To better understand this issue, consider a process that is initially operated at an optimal point xopt(d0), where d0 denotes the initial disturbances and x is the system states associated with an operating point. When the disturbances change from d0 to d1, the perturbed system will eventually settle down to a new stationary point x(d1) due to the regulation of the control layer. Note x(d1) is not necessarily optimal. The RTO layer first has to ensure that the steady state x(d1) has been attained, before the remaining RTO procedure can continue. The estimated value d̂1 for d1 is then used to reoptimize the set-points for CVs, which are afterward sent to the lower layer to drive the whole system to the new optimum xopt(d1) (assuming d1 is Received: Revised: Accepted: Published: 14695

February 9, 2014 August 21, 2014 August 21, 2014 August 21, 2014 dx.doi.org/10.1021/ie500562t | Ind. Eng. Chem. Res. 2014, 53, 14695−14711

Industrial & Engineering Chemistry Research

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of CVs to improve the economic performance. However, the set-points of CVs should be persistently perturbed to evaluate current gradients, which may unnecessarily upset the process. In this Article, a different type of HCS is proposed on the basis of a concept of controlled variable adaptation (CVA). Precisely, the RTO layer adapts the CVs based on a so-called just-in-time regression algorithm to track the NCO so as to enable the SOC layer to reduce the economic loss of a nonoptimal operation to an acceptable level such that the “acceptable loss” operation range is effectively enlarged. According to the theory of SOC, the self-optimizing performance is mainly determined by CVs selected.16 The new CVA strategy moves CV selection from off-line to online to make CVs adaptive to operating conditions. Therefore, the economic performance can be further improved. Because the regulatory control loops configured with CVs for the aim of self-optimizing control still play a major role for optimizing control, the activation of CVA is not always required but depends on how well current CVs behave. In this Article, a novel nonoptimality detection technique is proposed on the basis of the theory of statistical process monitoring, where the nonoptimality statuses are treated as a special kind of process faults. The T2 and squared prediction error (SPE) statistical criteria are used to judge whether the CVs need to be adapted. With the aforementioned techniques developed, a more general framework of the CVA-based HCS is constructed. Different from the traditional HCS, the control layer not only stabilizes the whole process, but also accomplishes most of the optimizing task, and thus the “slowness” drawback is overcome; the RTO layer neither estimates the disturbances and resolves the mathematical optimization problem, nor passes down the set-points of CVs, but adapts the CVs if a nonoptimality status is identified. The main contributions of this work can be summarized as follows: (1) proposed the concept of CV adaptation for optimizing control together with a novel solution of the CVA; (2) proposed the concept of nonoptimality detection as a prerequisite for CVA activation together with a novel solution based on the theory of statistical process monitoring; and (3) a novel HCS framework with two constitutive blocks, CVA and nonoptimality detection, which makes the RTO performance superior to the traditional HCS. To the best of our knowledge, the aforementioned concepts and solutions are all first-time proposed in the literature. The rest of this Article is organized as follows: In section 2, after the general ideas of SOC and regression-based CV selection method are briefly reviewed, a CVA strategy is proposed. In section 3, a nonoptimality detection technique is developed to provide a prerequisite for adapting CVs. Moreover, in section 4, a new HCS framework is proposed on the basis of these two contributions. Section 5 provides two case studies to investigate the effectiveness of various techniques proposed here. Finally, the work is concluded in section 6.

Figure 1. RTO procedure of a two-step approach.

accurately estimated). As indicated in Figure 1, the system unnecessarily undergoes a nonoptimal status x(d1); hence the transiting time spent in stage (I) is wasted. This defect of the two-step approach significantly degrades the RTO performance as discussed in ref 3. Various efforts have been made to overcome the “slowness” of traditional HCS. For example, shorter RTO intervals were used to accelerate the optimization frequency,7,8 hence improving the RTO performance. However, inappropriate RTO intervals may cause potential stability problems.3 Another method is the so-called two-stage cascade MPC,9 where the upper layer MPC solves a static LP or QP problem associated with the economic performance of the plant to update set-points for the lower layer MPC. The set-points are updated faster than a conventional RTO scheme because the upper layer MPC uses the disturbances estimated by the lower layer MPC, and hence can work in the same sampling rate as the lower layer MPC. Nevertheless, the LP or QP problem is only an approximation of the true economic optimization problem. Therefore, a RTO layer is still necessary to work with the two-stage MPC. Among various possible solutions for enhancing RTO performance, self-optimizing control (SOC)6 is demonstrated to be very promising. In SOC, a set of appropriately chosen CVs are maintained at constant set-points. Normally, such control scheme will incur some economic losses. However, with carefully selected CVs, such losses can be kept within an “acceptable” level, hence achieving “self-optimizing” despite various disturbances. The idea of SOC is indeed very attractive because here optimizing is integrated with feedback regulation, hence significantly enhancing the speed converging to optimum. In SOC, an RTO layer is not necessary when the economic loss is “acceptable”. In the past decade, various methods for finding CVs pursuing good self-optimizing performance have been proposed.10−16 Among various SOC methods, the regression-based approach proposed in ref 16 provided a different way to determine CVs approximating the necessary conditions of optimality (NCO). Because the approximation is within the whole operating region, the selfoptimizing performance with the selected CVs is global, while other existing SOC methods are only locally effective because of the linearization round the nominal point involved. The “global” nature of the regression-based approach significantly enlarges the operation region where the economic loss under SOC is acceptable as compared to other local approaches. SOC aims to maintain economic losses at an acceptable level without repeated optimization. However, this can only be achieved within a limited operation range for a single SOC layer. The economic loss may, in some circumstances, not be negligible due to changes of operation conditions. Recently, an HCS has been proposed17 where the control layer is configured with CVs selected through the null space method11 and the RTO layer adopts the NCO tracking.18 In their HCS, the NCO tracking scheme periodically perturbs and updates the set-points

2. A CVA STRATEGY FOR SELF-OPTIMIZING CONTROL In this section, we first review the basic idea of SOC6 and, in particular, an NCO approximation approach16 for CV selection, where the CVs are found by approximating the NCO with measurements through regression. The regression-based CV selection approach forms some preliminary basis of the CVA approach to be proposed in this section. More specifically, the just-in-time (JIT) regression approach19−21 is adopted for CVA such that the operation region with an acceptable economic 14696

dx.doi.org/10.1021/ie500562t | Ind. Eng. Chem. Res. 2014, 53, 14695−14711


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Assume ηj, j = 1,...,nu − na are unmeasurable NCO elements, which in most cases are, hence are assumed in this work as, the reduced gradients. Besides, it is assumed that the active constraints remain unchanged in all expected disturbance scenarios. To facilitate the regression, data samples of yi and ηji are generated by evaluating f(u, d) and ηj at N sampling points, ui and di for i = 1,...,N. A regression model, Z(y,θ j), with nj adjustable parameters, θ j ∈ nj , to approximate ηj can then be obtained by solving the following least-squares problem:

loss can be adaptively enlarged to cover the entire operation space. Description of Self-Optimizing Control. SOC is termed as a control strategy “when we can achieve an acceptable loss with constant set point values for the controlled variables (without the need to reoptimize when disturbances occur).”6 The key step for achieving SOC is choosing the appropriate CVs, such that when tracking these CVs with constant setpoints, the whole system will be automatically steered to optimum or near optimum. Because tracking CV set-points can be accomplished along with the function of regulatory controllers, SOC is more advantageous to optimal operation than a traditional RTO method in the sense that the SOC avoids the requirements for a system to settle to a nonoptimal steady-state before the optimality can be restored via a traditional RTO method. For various SOC methods developed in the literature, the self-optimizing control with an “acceptable” loss can only be achieved with respect to a limited operating region. Therefore, one still needs a solution to extend or shift the “acceptable loss” range, when disturbances make process operation outside of the current “acceptable loss” range. NCO Approximation Approach for CV Selection. Consider a static optimization problem for continuous processes:

N

min ∑ (εi j)2 for j = 1, ..., nu − na j θ

εji

γi =

with available measurements: (2)

where J is a scalar cost function to be minimized; u ∈ nu , d ∈ nd , y̅ ∈ ny , and y ∈ ny are the manipulated, disturbance, true, and measured measurement variables, respectively; n ∈ ny are the measurement noises; g : nu × nd → ng and f: nu × nd → ny are the operational constraints and measurement equations, respectively. It is assumed that disturbances are observable through measurement equations f. Moreover, the process is assumed to be open-loop stable in this Article. Solving eq 1 gives the following NCO that should be satisfied at the optimal point:16,22,23 ga = 0,

∇r J =

∂J V2 = 0 ∂u

ηji.

1 T −2 εi ∇r Jεi 2

(5)

where of the reduced Hessian of J and εi = [ε1i ... the more accurate is the regression model, the better the economic performance can be. This regression approach is effective in the sense that the CVs can be conveniently obtained via numerical methods, and, more importantly, the optimizing performance is globally effective in the whole operating range because the data used for regression can be generated without being limited to the vicinity around a nominal point. The control loops for NCO approximation approach are illustrated in Figure 2.

(1)

y̅ = f(u , d)

j

∇−2 r J is the inverse εni u−na]T. Therefore,

u

y = y̅ + n ,

(4)

where = Z(yi,θ ) − In principle, any linear or nonlinear regression models can be adopted for Z(y,θj). Particularly, linear, polynomial, and artificial neural network (ANN) models have been proposed in ref 16. Furthermore, the economic loss γi at a specific sampling point, i, can be estimated using the regression error:16

min J(u , d) s.t. g(u , d) ≤ 0

i=1

Figure 2. Control loop of regression approach for CV selection.

Remark 1: Choosing the NCO as the CVs to achieve optimality is also proposed in another class of feedback optimizing control approaches, the NCO tracking.18,23 As its name indicates, by choosing NCO as CVs with 0 set-points, NCO tracking converts the optimization problem into a control problem, which is also suggested in the NCO approximation method. Although similarly motivated, the NCO approximation method is positioned as a kind of SOC, rather the NCO tracking, as discussed below. NCO tracking does not specify particular ways of computing the gradients.24 For a class of NCO tracking where the gradients are experimentally evaluated using finite differences, the RTO speed is slow because gradient calculation has to wait for a new steady state to bereached. Therefore, NCO tracking with finite differences is not highly effective for continuous processes. Most reported NCO tracking applications are for batch processes, where both the cost function and the gradient can be evaluated from batch to batch.23 On the other hand, for continuous processes, the speed of NCO approximation-based SOC to converge to a near optimal operation is faster than NCO tracking because of measurement combinations as CVs naturally use the transient information for optimization convergence without waiting for

(3)

where the active constraint ga: nu × nd → na is a subset of g and ∇r J: nu × nd → nu − na is the reduced gradient. V2 is defined through the singular value decomposition of the Jacobian matrix ∂ga/∂u = USVT and V = [V1 V2], where V2 are nu − na right singular vectors corresponding to nu − na zero singular values. Therefore, the reduced gradients ∇rJ can be intuitively interpreted as the gradients of cost function J with respect to the remained degrees of freedom of u after ga has been satisfied. The NCO are suggested as the CVs to achieve perfect selfoptimizing control, which is an intuitive observation from eq 3. However, because not all NCO are online measurable, particularly the reduced gradients, an alternative way is to approximate the NCO using available measurements y. Unfortunately, to derive such an approximation analytically is generally difficult. An alternative approach is through regression,16 which is briefly reviewed as follows. 14697



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nonoptimal steady-states to be reached. For another class of NCO tracking, the neighboring-extremal control (NEC),25 control laws are derived to force the first-order variations of NCO to be zero, resulting in a fast optimizing speed. However, because NEC is also developed around the nominal value, the RTO performance is local, the same as other local SOC methods. In a very recent work, it is shown that the NEC and existing local SOC are equivalent through appropriate tuning.26 In the literature, there are also discussions and comparisons for NCO tracking and SOC (see, e.g., refs 16, 17). Remark 2: Static optimization formulation is considered in the present work. For feedback optimizing control, a nice property is that when using the transient information, the system will ultimately settle at the (approximate) optimal point upon convergence. Therefore, the optimizing performances for such schemes are much quicker than those using successive steady-state information, for example, two-step RTO approach. This subject is explored in more detail in a recent work.27 Basic Idea of CVA. The recently proposed NCO approximation approach16 uses an off-line algorithm to find the CVs, where the data samples are generated in the whole operating range with the available process model. Once determined, the CV models will remain unchanged in online applications, which is also the characteristic of other existing SOC methods. As has been acknowledged, the economic performance highly relates to the accuracy of Z(y) approximating the NCO. Regressing over a large operation range with some simple model structures, for example, linear combinations, may lead to large regression errors so that the corresponding economic average loss over the entire region may not be acceptable. Generally, increasing the complexity of Z(y), such as polynomial or ANN models, will improve the predicting accuracy.16 However, there still exist the following reasons that make restricting complexity of CVs favorable: (1) If the number of available measurements is very limited, then the CV models to make losses acceptable may be too complicated to be adopted. (2) Increasing the complexity of Z(y) may lead to overfitting, which is harmful to the approximation accuracy within the entire operation space. (3) The ultimate aim is not only to get a good approximation, but also to maintain Z(y) = 0 through feedback control. The complexity of Z(y) may increase the difficulty of control design, and may introduce some nonoptimal solutions unnecessarily. For example, if Z(y) is chosen as a second-order polynomial function, there may exist two distinct solutions satisfying Z(y) = 0 so that a carelessly tuned controller may lead the process to stick to the nonoptimal one satisfying Z(y) = 0. (4) A complex Z(y) may not be intuitively understood by field operators, whose operating confidence is crucial for practical applications. To use simple measurement models as CVs but still with satisfactory optimizing performance over the entire operation range, an online CVA strategy is proposed as follows. With this new strategy, the satisfactory SOC performance can be ensured over the entire operation range even with a simple structure of Z(y, θ) by adapting parameters, θ. The basic idea of CVA strategy is illustrated in Figure 3, with the linear form of Z(y, θ) used for an illustrative purpose. The left plot shows the situation of off-line approach for NCO approximation,16 Z(y, θ0) is obtained in a global sense and

Figure 3. Comparison of two methods for approximating the NCO: (a) off-line approach; (b) online CVA.

results in an average performance. Therefore, the approximation accuracy would be hampered at a specific operating point. As a comparison, for the CVA as shown in the right plot, when the system is operated in the area of s1, where the off-line obtained CV is not accurate enough to approximate the NCO, the CV is adapted to Z(y, θ1) by updating coefficients of the regression model to θ1 to approximate the NCO more closely. Moreover, when the system operation is further moved from s1 to the area of s2, which is far away from s1, the regression coefficients will be updated again, resulting in Z(y, θ2) to improve approximation accuracy. By doing this, the approximation accuracy can be significantly improved and consistently maintained over the entire operation region so as to ensure the operation optimality. Just-in-Time Regression for CVA. To adapt the CVs so as to maintain an acceptable economic loss within a wide operation range, in this Article, the just-in-time (JIT) technique for online regression is introduced. The JIT, which was previously used for nonlinear process modeling and soft sensor development,19−21 is an online modeling technique where a local regression model is built from the historical database around a new query data sample. Denote the data set for regression as {yi, ηi}i=1,2,N. For JIT regression, the following three main steps are carried out in sequence at each regression instant: (1) For a new coming sample ynew, L relevant input samples are selected from the database according to some similarity criteria. Typical similarity criteria can be determined from the Euclidean distance and angle between samples; the following expression gives a combined similarity criterion ϕi between ynew and a database sample yi:19 ϕi = ρ exp( −Di) + (1 − ρ) cos(ωi)

(6)

where Di and cos(ωi) denote the Euclidean distance and the cosine value of the angle ω between ynew and sample yi, respectively. ρ (0 ≤ ρ ≤ 1) is an index balancing the weight between distance and angle information. Specially, ρ = 1 indicates that only the distance information is taken into consideration when evaluating the similarity, while ρ = 0 indicates that the angle information is solely considered. Di and cos(ωi) can be calculated as Di = ynew − yi

cos(ωi) =

(7)

2

⟨ynew , yi⟩ ynew 2 · yi

2

(8)

where ∥·∥2 is an operator for calculating Euclidean norm and ⟨·,·⟩ for calculating inner product between vectors. Generally, the data samples should be scaled prior to calculating the similarity to eliminate the impact of adopting different units. With the similarity criterion defined in eq 6, L samples with the 14698



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(3) Online adapting CVs using the JIT approach may be in some cases time-consuming, especially when the size of the overall database {yi, ηi}i=1,2,N is large. Because of these reasons, if the process operation is near optimum and the economic loss is negligible, it is not necessary to adapt the CVs. Therefore, instead of periodically adapting the CVs, CVA will be activated only when the system significantly deviates from the optimum over a certain level.

largest similarity values are selected from the database for construction of a local regression model. As shown in Figure 3, when the system is operated in the region of s1, those samples marked with “*” will be selected, while in the region of s2, those samples marked with “o” will be selected. (2) When a data set is selected, a new local regression model η̂ = Z(y, θ̂) with a set of updated parameters, θ̂, is derived on the basis of a certain predetermined model structure, such as the polynomial, ANN, or support vector machine models. (3) Depending on the updated parameters, θ̂, the estimated value η̂new = Z(ynew, θ̂) is calculated. The new model Z will be used as the CV in the control loop and discarded when a new nonoptimality status is detected, which will be described later. Remark 3: It is worth highlighting that, although the dashed lines of Z(y, θ1) and Z(y, θ2) shown in Figure 3 appear local around s1 and s2, respectively, the JIT approach presented above is actually global within the region specified by the L samples identified with maximum similarities 6. This is because the JIT is the same as other regression-based approaches aiming to minimize the average regression error over the region represented by regression data. In this sense, the JIT is different from other local SOC approaches, where the linearization involved makes an acceptable loss only achievable within a neighborhood area of a nominal operation point irrespective of whether linear or nonlinear CV models are adopted. Therefore, adopting the JIT will ensure the region with acceptable loss to be larger than that obtained by using other local SOC in the CVA. This in turn will reduce the frequency for a nonoptimality to be detected; hence, less CV adaptation will be required. Remark 4: On the basis of the assumption of open-loop stable processes, a single controller can be designed to stabilize the system for all CVs generated by the JIT regression upon the database, provided that the controller gain is sufficiently small according to the small-gain theorem.28 This is because the number of samples in the database is finite; hence, the number of possible CVs to be generated from the database, which equals the number of possible combinations by selecting L samples from the N-sample database, is also finite. For all of these possible CVs, there is a common controller with a gain small enough to stabilize the system if any of these CVs should be adopted. This does not mean that control gains have to be limited to be small because the small-gain theorem is only a sufficient condition. In most practical systems, as shown in the case studies, a relatively large gain, which ensures both stability and reasonable response speed, can be adopted. Therefore, a gain adaptation strategy to accompany the CV adaptation is not necessary. With the proposed adaptation strategy, a satisfactory approximation accuracy for NCO can be ensured within the entire operation range, while the structure of the approximation model Z(y, θ) can remain simple, for example, in linear combinations of measurements y, which can be easily understood by field operators. As described above, a CVA strategy is proposed for RTO of the plant operation. However, frequent adaptation for CVs may not be necessary or desired. This is because of the following: (1) According to Figure 3, if disturbances are within the region where the approximation error between the NCO and the current CVs is small enough, then the corresponding economic performance will still be acceptable without adapting current CVs. (2) Adapting CVs too often may cause too many perturbations to the system, which is not desired from a control performance point of view.

3. A NONOPTIMALITY DETECTION TECHNIQUE In this section, we develop a novel nonoptimality detection technique based on the theory of process monitoring.29 This technique helps to form a prerequisite for adapting CVs. The key idea is to treat optimal statuses as normal operating conditions and nonoptimal statuses as a special kind of process faults. Here, a “status” is defined as the corresponding static system status under specific operation and disturbances (u and d). With a statistical monitoring model built from data collected under various optimal statuses, the nonoptimal statuses can be efficiently detected by certain statistical criteria. Description of Optimal Statuses. When a process is in optimal operations, all of the degrees of freedom of the system have to satisfy the NCO represented in eq 3, which have total nu equations. In other words, once all disturbances are given, the optimal status of the system is determined by satisfying the NCO, which include the process model, active constraints, and the reduced gradient, eq 3. Therefore, for all optimal statuses, the degree of freedom is nd, depending on the number of system disturbances. Once an optimal status is determined, all measurements of the process can be determined correspondingly. In other words, all measurements under an optimal status are solely determined opt by disturbances. Let yopt ̅ and y be the true and measured measurements in an optimal status, and assume the number of measurements is more than the number of disturbances, that is, ny ≥ nd. The relationship between yopt and disturbances can be represented as follows: F(y̅ opt , d) = 0

(9)

where F: ny × nd → ny is the nonlinear mapping function between yopt ̅ and d in the space of optimal statuses. By linearizing 9, the following derivation can be obtained:29 Ay̅ opt + Bd = 0

(10)

y opt = y̅ opt + n

(11)

where A = ∂F /∂y̅ opt(A ∈ ny × ny ), B = ∂F /∂d(B ∈ ny × nd ), and n is the noise as defined in eq 2. Denote B⊥ as the orthogonal complement of B such that BTB⊥ = 0, then left multiplying the term (B⊥)T for eq 10 gives: (B⊥)T Ay̅ opt ≡ Cy̅ opt = 0

(12)

⊥ T

where C = (B ) A. From the above equation, we see that yopt ̅ lies in the subspace spanned by the orthogonal complement of C, that is:

y̅ opt = (CT )⊥ s

(13)

where s denotes the independent components. Inserting the above equation into eq 11 leads to

y opt = (CT )⊥ s + n 14699

(14)



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Equation 14 indicates that with the projection matrix C, the common features of optimal statuses can be equivalently interpreted in a lower-dimensional space. On the basis of eq 14, a nonoptimality detection technique can be developed on the basis of the principal component analysis (PCA), which is the most popular tool for process monitoring. PCA Modeling. Denote the collected samples for PCA modeling in a matrix form Y opt ∈ M × ny storing the measured values of yopt with M samples. Note this matrix is normally scaled to have zero mean values and unit variance for each variable. The PCA form is represented as29 Y

opt

T

T

̃T

̃ = TP + E = TP + TP M×k

T 2 ≤ Tα2 =

SPE = ỹ opt

(15)

SPE ≤

2

(22)

⎛ ⎞1/ h0 2 c h 2 θ h h θ ( − 1) α 2 0 2 0 0 ⎟ = θ1⎜⎜1 + + 2 ⎟ θ θ 1 1 ⎝ ⎠ (23)

ny

θi =

∑

λji , i = 1, 2, 3 (24)

j=k+1

h0 = 1 −

2θ1θ3 3θ22

(25)

and cα is the normal deviate corresponding to the upper 1 − α percentile, where α denotes the confidence level. For those processes where rigorous mathematical models are known, an off-line approach to collect the data for PCA modeling can be adopted; that is, for different random disturbances with expected distributions, the optimization problems are solved using the process model; the obtained data samples in various optimal statuses are then stored for PCA modeling, and the control limits T2α and δα are then calculated correspondingly. For online usage, the process is evaluated for whether it is currently operated in the optimal status by monitoring the T2 and SPE statistics. If either of the control limits is violated, it is considered to be nonoptimal. Hence, we adapt the CVs as developed in the previous section to improve the NCO approximation accuracy. Otherwise, no CV adaptation is required because the current status is believed to be optimal or near optimum. Note, besides the PCA method for nonoptimality detection, various fault detection methods developed in the field of process monitoring are also applicable.30 For example, the kernel PCA is suitable for the fault detection for nonlinear processes, and the independent component analysis is effective for nonGaussian processes. However, in this Article, we only use the simplest linear PCA method as an illustration for the proposed methodology.

(16)

(17)

T opt Through the above steps, each row vector (yopt can be i ) in Y projected onto the principal and residual spaces:

ŷ iopt = PPTy iopt (18)

opt where ŷopt are the projected vectors onto the principal i and yĩ and residual spaces, respectively, while satisfying:

(ŷ iopt)T ỹ iopt = 0 (19)

Using the PCA, the original data can be described in the reduced k-dimensional uncorrelated principal directions, with most of the variations contained in data retained. T2 and SPE Statistics for Nonoptimality Detection. Through the PCA projection of the original data set, the T2 and squared prediction error (SPE) statistics can be constructed in the principal and residual spaces, respectively. The T2 statistic indicates the variation extent of data in the principal space, which is defined as29 T 2 = (y opt)T PΛ−1PTy opt

= (I − PPT)y opt

where

where Λ is a diagonal matrix consisting of all eigenvalues of Σ, that is:

y iopt = ŷ iopt + ỹ iopt

δα2

ny × k

T̃ = Y optP̃

ỹ iopt = (I − PPT)y iopt

2

Similarly, the corresponding SPE control limit δ2α and the optimality status can also be computed as31

T = Y optP

Λ = diag{λ1, λ 2 , ..., λny}

(21)

Meanwhile, the SPE statistic indicates the distribution of data in the residual space, which is defined as a square of the norm of projected residual vector:29

where T ∈  and P ∈  are the score and loading matrixes of principal components, respectively. E ∈ M × ny is the residual matrix, and T̃ ∈ M × (ny − k) and P̃ ∈ ny × (ny − k) are the score and loading matrixes of residual components, respectively. k is the number of principal components, whose value can be determined from cross validation or cumulative percent variance.30 The various matrixes in eq 15 can be obtained through the symmetric eigenvalue decomposition for the covariance matrix of Yopt, Σ = (Yopt)TYopt/(M − 1): Σ = [ P P̃ ]Λ[ P P̃ ]T

k(M − 1) Fk , M − k ; α M−k

4. FRAMEWORK OF A NEW HIERARCHICAL CONTROL STRUCTURE As has been acknowledged, a traditional HCS suffers from the shortcoming of “slowness” for RTO, which results from the distinct gap between the separated control and RTO layers. With the self-optimizing CVs, it is possible to achieve optimizing control through regulatory control loops. However, most of the current SOC methods were developed locally around the nominal point, which restricted their performances to be only locally valid in a narrow region. The NCO approximation approach proposed in ref 16 is an effort to extend the method to be globally effective. However, the approximation precision may not be satisfactory for a large operation range, or the CV

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If we assume the distributions of process data are Gaussian, then the T2 statistic follows an F-distribution with k and M − k as the degrees of freedom in the optimal operation conditions. Given a significance level α, the control limit of T2 statistic can be calculated, and the optimality status is monitored as29 14700



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Figure 4. Comparison of two different hierarchical control structures: (a) traditional one; (b) CVA-based one.

layer performs an online CV adaptation and passes down new CVs to improve the approximation accuracy for the NCO. A comparison between the proposed HCS and a traditional one is presented in Figure 4, where the major differences are marked in bold. The novelties of the new HCS include: (1) A CVA block. In a traditional HCS, the two steps of disturbance estimation and reoptimization are alternatively and repeatedly performed to achieve new optimum. In the new HCS, they are replaced with a CVA block developed in section 2, the aim of which is to adaptively improve the NCO approximation accuracy, hence improving the economic performance over a wide operation region. (2) A nonoptimality detection block. This is a newly added block for establishing a prerequisite to activate the CVA block, as presented in section 3. It is also worth pointing out that the nonoptimality detection block can also be independently used as a tool for economic performance evaluation; see the reactor case study in the sequel section. (3) A new link between the RTO layer and control layer. Instead of passing down the optimal set-points of CVs to the control layer in a traditional HCS, CV themselves are adapted by updating their model parameters in the new HCS. Their set-points will remain constant, that is, zero during the entire operation period. This is because the “traditional” set-point is included in the new CV structure as a constant term to be updated with other model parameters in this work. Therefore, the new HCS to adapt CV parameters can be seen as a generalization of the traditional set-point updating based HCS. With the generalization, the new HCS provides more degrees of freedom for the upper RTO layer to achieve better process operation. With the above constitutive blocks, some basic characteristics of the two layers in the new HCS can be summarized as follows: (1) The control layer not only rejects the disturbances and stabilizes the system, but also accomplishes near optimal operation through SOC. The convergence to a new optimal status is fast. (2) The RTO layer maintains the economic loss in an acceptable level within the entire operation space via adapting the CVs online, which is conditionally activated by the nonoptimality detection block. Therefore, the action of adapting CVs is nonregular and nonperiodic. Remark 5: Both the data samples contained in the databases for CVA and nonoptimality detection technique are generated

expressions may have to be too complex to be adopted. To address this limitation, this Article presents a strategy of adapting CVs online to increase the optimizing precision over the entire operation range. On the other hand, although the operation region with an acceptable loss level corresponding to current CVs in the control layer is limited, the actual loss of current operation may still be acceptable. Therefore, it is not necessary to adapt the CVs continuously but depending on current economic performance. Motivated by this requirement, a sophisticated strategy is proposed to conditionally adapt the CVs only when the economic performance is not satisfactory by applying the nonoptimality detection technique proposed in the previous section. To fully utilize the advantages of new techniques proposed in this work, a new type of HCS structure is further constructed on the basis of the aforementioned developments. The new structure is termed the CVA-based HCS in this Article. The proposed new HCS forms a general framework for control synthesis and systematic RTO solutions as illustrated in Figure 4b. In the CVA-based HCS, the control layer is similar to a traditional one except that the CVs in regulatory loops are selected for self-optimizing purpose, while the RTO layer is composed of several major blocks: reconciliation, stationary conformation, nonoptimality detection, and CVA. The blocks of reconciliation and steady-state conformation are similar to those in a traditional HCS, which serve as pretreatment steps for RTO. Note we still include the stationary conformation step in the optimization layer to ensure correctness of nonoptimality detection, which is developed in the steady state. However, the new HCS does not suffer from the “slowness” shortcoming of RTO performance because the constructed CVs in the control layer have the potential converging to a near optimal status from a nonoptimal deviation caused by disturbances quickly. The distinct features of the new HCS are the inclusion of the nonoptimality detection block and CVA block, which differentiate it from a traditional HCS, where the procedures of estimating disturbances, updating models, and reoptimizing are carried out in turn as shown in Figure 4a. In the new HCS, after a steady state has been confirmed, the optimality of current system status is first judged by the nonoptimality detection block. In practice, one can also choose to carry out the stationary conformation and nonoptimality detection blocks in parallel. If both of the two conditions are satisfied, the RTO 14701



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generated in the following way: Dividing the two independent variables, CAi and Ti, into 10 equal parts in their own regions, we obtain a total number of 121 sampling points, and their corresponding measurements and gradients are collected to form a database. The resulting CV is given as

using a process model. Therefore, an accurate process model is responsible for their performances. The advantage of using a process model is that corresponding information can be archived off-line prior to plant operation. However, in the case of a model−plant mismatch, performance degradation is inevitable, as shown in the CSTR case study in the next section.

Z0 = −1020.8 − 81.6035C B + 2.4846Ti

5. CASE STUDIES For illustrations of proposed methodologies, two case studies are provided in this section. The first case study with an exothermic reactor investigates the individual aspects of the proposed techniques, the online CVA strategy and nonoptimality detection, and then examines the new HCS as a whole via dynamic simulations. The advantages of proposed methodologies are verified through comparisons with other RTO methods. The second evaporator example is adopted to demonstrate a successful application of the new HCS to a more realistic plant. Exothermic Reactor. Process Description. This is a continuous stirred-tank reactor (CSTR)12,32 where a reversible exothermic reaction A ⇌ B occurs, where A is the raw material and B is the desired product. Table 1 gives physical descriptions

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whose RMSE index is calculated to be 17.5318 with 200 randomly generated testing samples. For CV adaptation, the JIT regression approach is applied using the same database adopted in the off-line global regression approach, and the balancing weight ρ in eq 6 is set to be 0.5 in all cases; that is, the distance and angle information are equally considered for evaluating the similarity between samples. The RMSE indices for the same 200 testing samples with different number of relevant samples L, ranging from 5 to 15, are plotted in Figure 5. The results show that the

Table 1. Process Variables and Nominal Values for Exothermic Reactor variable

physical description

nominal value

unit

CA CB T Ti CAi CBi J

outlet A concentration outlet B concentration outlet stream temperature inlet stream temperature inlet A concentration inlet B concentration economic objective

0.498 0.502 426.803 424.292 1.0 0 −5149.3

mol L−1 mol L−1 K K mol L−1 mol L−1 10−4 $

of various process variables in this CSTR; a detailed process model can be found elsewhere.12,16 For the reactor, the inlet stream temperature Ti is the sole manipulated variable, u = [Ti]. A choice of y = [CBTi]T is considered as the measurements with Gaussian distributed noises, whose means are both 0, and the standard deviation is considered as 0.01 and 0.5 for CB and Ti, respectively. A single disturbance of d = [CAi] falls into an expected range of 0.5 ≤ CAi ≤ 1.5. The operational objective is to minimize a cost function:16 J = −20090C B + (0.1657Ti )2

Figure 5. RMSE values against L ranging from 5 to 15.

approximation accuracy can be significantly enhanced by using JIT local regression technique, and the RMSE index is reduced to less than 3 for all tested values of L. The smallest RMSE (2.46) occurs when L = 7, indicating only a few samples are needed for extracting the local feature of the gradient. Therefore, L = 7 will be used in later developments. Nonoptimality Detection. To extract the common feature of various optimal statuses, we solve the optimization problem minimizing J using the process model, with 500 randomly generated disturbances in the defined range. The data samples containing measured CB and Ti are stored and scaled to be 0 means and unit variances for PCA modeling, and the principal number is chosen as 1, which is determined from the fact that the first principal component contributes 96.88% of the total variance. With a significance level of α = 0.99, the control limits of T2 and SPE statistics are calculated to be 6.6992 and 0.4111, respectively. The distribution of CB and Ti is plotted in a 2-D space, marked as “○”, in Figure 6. It can be observed that these optimal data samples approximately fall along a one-dimensional principal component direction captured by the PCA method, as illustrated by the dashed line in the figure. Those samples falling outside either the T2 region or the SPE region will be considered as the nonoptimal statuses. For example, samples A

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This optimizing control problem has only one degree of freedom. Thus, the NCO is to keep the gradient dJ/du at zero. Online CVA with JIT Regression. Because the self-optimizing performance depends on how well the CV approximates the NCO, the root-mean-square error (RMSE) is used to evaluate the performance, which is defined as n

RMSE =

∑i =te1 [Z(y) − dJ /du]i2 nte

(27)

where nte is the number of testing samples, and subscript i denotes corresponding values of the ith sample. In this example, we consider the linear function of measurements for illustrative purpose. For comparison, an off-line global regression approach16 is first carried out. The regression is performed over a database with 121 samples, which are 14702



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in monitoring the optimal data set than the nonoptimal data set; hence it is more likely to assign a sample to be optimal to avoid a false alarm. This may be caused by the nonlinearity of the CSTR process. Moreover, it also shows that the SPE statistic is more efficient in detecting the nonoptimal statuses for this CSTR case. A similar trend can also be observed in the dynamic simulation results to be presented below. Online Implementation. In the online implementation, we compare the real-time performances of different strategies, the traditional two-step RTO approach (denoted as E1), the offline global approach16 (E2), and the CVA approach proposed in this Article. In the CVA approach, we further differ two cases depending on whether the nonoptimality block is incorporated as a condition for adapting CV. In the first case, E3I, the CV is adapted periodically without taking whether current operation is optimal into consideration, while in the second case, E3II, the new HCS proposed is examined entirely. Descriptions of different experiment cases are given in Table 2. Figure 6. Distribution of optimal samples: (A) falls outside T2 region, (B) falls outside SPE region, and (C) falls outside both of the two regions.

Table 2. Experiment Descriptions experiment E1

and B fall outside of the T2 and SPE regions, respectively, while sample C falls out of both regions. A data set with 1000 test samples is generated to verify the effectiveness of the proposed nonoptimality technique. The first 500 samples are generated in optimal statuses, and the remaining 500 samples are generated in nonoptimal statuses, whose economic cost is larger than the minimal cost by more than 1%. The detection results are shown in Figure 7, which verifies the efficiency of the proposed detection scheme. The false alarm rate for optimal statuses is 0% for T2 statistic and 0.6% for SPE statistic, while the missing rate for nonoptimal statuses is 8% for T2 statistic and 2.2% for SPE statistic. The overall detection results are also calculated, as 0.6% false rate and 2.2% missing rate for optimal and nonoptimal statuses, respectively. It shows that the algorithm exhibits a better ability

E2 E3I E3II

description traditional two-step RTO approach; disturbance estimation and reoptimization are alternatively performed for RTO off-line global approach for self-optimizing control; no additional RTO actions online CVA; CVs are adapted periodically at each RTO instant, without taking whether current operation is optimal into consideration CVA-based HCS; CVs are conditionally adapted only when the system is judged to be nonoptimal by the nonoptimality detection block

To facilitate later comparisons, an average economic cost J ̅ in a time range [t−,t+] is defined as J̅ =

1 + t − t−

∫t

t+ −

J (t ) d t

(29)

Figure 7. Detection performance: (a) the first 500 testing samples; (b) the total 1000 testing samples. 14703



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Furthermore, a comparing index is also defined to quantify the improved economic performance of Ei over Ej (in percentage) as φEi | Ej = 1 −

JE̅ i − Jopt JE̅ j − Jopt

=

JE̅ j − JEi JE̅ j − Jopt

× 100% (30)

where Jopt denotes the optimal cost function one can ideally achieve in an operational phase. Therefore, a positive φEi|Ej implies the economic behavior of Ei is better than Ej, the bigger the better, whereas a negative φEi|Ej implies the economic behavior of Ei is worse than Ej, the smaller the worse. The following arrangements are considered: (1) The same disturbance scenarios will be tested, where CAi experiences large magnitude step changes to simulate abrupt operating condition changes. The total simulation time is 7500 s, and CAi is set to be 1.0, 1.5, 0.75, 1.25, and 0.5 every 1500 s, respectively, as shown in Figure 8. (2) For the stationary conformation, we simply

Figure 9. Online implementation results of E1: (a) trajectories of Ti and its optimal value, (b) online monitoring result of T2 statistic, and (c) SPE statistic.

The dynamic response of the manipulated variable Ti and the optimality monitoring results are shown in Figure 9 and summarized in Table 3. As shown in Figure 9, E1 starts with the

Figure 8. Disturbance trajectory.

wait for the system to attain a steady state. An RTO interval of 1000 s after the disturbance change is adopted. At each RTO instant, E1 performs disturbance estimation and reoptimization for RTO, while E3I (periodically) and E3II (conditionally) adapt the CV. E2 imposes no particular RTO actions, except for enforcing self-optimizing Z0 to 0 with a feedback controller. (3) The nonoptimality detection technique will be used for all four experiments, with the aim of demonstrating its own usage. Therefore, E1, E2, and E3I simply utilize it as an independent monitoring tool for evaluating the economic performance during operation, whereas E3II integrates it into the HCS and allows it to interact with the CVA block. Traditional Two-Step RTO Approach (E1). Because E1 places no particular emphasis on CV selection, we first need to design the regulatory control loop. Specifically, we choose to select Ti as the CV, which corresponds to an open-loop control policy for this case. Note that Ti was actually confirmed to be the best CV with a single measurement.12,33 Furthermore, we make another assumption that the disturbance will be estimated accurately and the reoptimization will be solved precisely, although these may not be realistic in a real situation. The open-loop control policy is also in favor of improving the RTO performance of E1, because the manipulated variable is immediately adjusted to its optimal value once reoptimization is accomplished (corrupted by some noise in this example), rather than gradually adjusted to its desired position through a feedback control loop (usually ramped changes are adopted in real applications, however, not adopted here for the ease of comparisons). Because of the above reasons, E1 in this particular experiment provides the best scenario of RTO performance that one can actually achieve in a real case, if a traditional two-step RTO approach is considered.

Table 3. Summaries of Online Implementation Results of E1 control limit variation

phase

deviation extent from optimality (observing Ti)

T2

SPE

J̅

I (0−1500) II (1500−3000) III (3000−4500) IV (4500−6000) V (6000−7500)

minor moderate→minor big→minor big→minor big→minor

no no no no no

no yes→no yes→no yes→no yes→no

−5149.23 −9986.67 −2983.69 −7419.85 −472.96

nominal case, which is the true optimum, and hence a good RTO performance is attained in phase I (0−1500 s). However, in the coming phases II−V (1500−7500 s), the system in each phase first deviates from the optimality (observing Ti) in a certain extent, then it gets back to the optimum once RTO is performed, where perfect disturbance estimation and precise reoptimization are assumed. The results also justify the usage of the proposed nonoptimality detection technique as an effective tool for performance evaluation; one can find that when the system deviates from the optimality, the SPE control limit is exceeded, and once RTO is performed and implemented (the “minor” deviations are due to noises of Ti), the SPE statistic goes back to normal again. However, the T2 statistic shows inferior performance for detecting nonoptimal operations; this is because the control system is operated not far away from the optimum, where the T2 statistic is not effective. It can be verified that for an arbitrary Ti that far away from the optimum, the T2 statistic will trigger an alarm. 14704



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Off-Line Global Regression Approach (E2). E2 incorporates no RTO layer, and an invariant CV of Z0 in eq 28 is configured. A single PI control loop for enforcing Z0 to be 0 is used for the entire simulation time. The dynamic responses are shown in Figure 10 and summarized in Table 4. It can be found that

approaches will either increase the capital cost or result in a CV too complex to be understood by the field operators. Online CVA Approach (E3I and E3II). The overall simulation results of E3I and E3II are shown in Figure 11 in connection

Figure 11. Online implementation results of E3I: (a) trajectories of Ti and its optimal value, (b) online monitoring result of T2 statistic, and (c) SPE statistic.

Figure 10. Online implementation results of E2: (a) trajectories of Ti and its optimal value, (b) online monitoring result of T2 statistic, and (c) SPE statistic.

with Table 5, and in Figure 12 in connection with Table 6, respectively. Finally, their economic improvements over E1 and E2 are given in Table 7. For E3I, the following results can be observed: (1) In each phase, the system will eventually converge to near optimum with CV adaptation, no matter whether it has been already operated near optimum. Hence, the CVA strategy is effective. (2) Comparing E3I with E1 (Table 7), the φ indices suggest there are economic improvements in phases II, IV, and V (φE3I|E1 = 88.3%, 77.7%, 68.4%). In phase I, because E1 is in the nominal point, which leads to 0 loss, the φ index calculated from eq 30 goes to −Inf and hence is not comparable. It can be easily checked that the absolute value of loss actually does not suffer much. In phase III, E3I performs worse than E1 (φE3I|E1 = −117.9%), because in the first 1000 s of phase III, Ti deviates far from the optimal value, which leads to a big loss. The corresponding CV was adapted in former phase II, implying that the CV obtained in phase II is no longer good enough for phase III. It is very interesting to note that in phase II, the system’s behavior has not been improved much with adapted CV, because it has already been very good in the first 1000 s of phase II. Although the two CVs around time instant t = 2500 s behave similar in phase II, the latter is proved to perform poor in the coming phase III. In this situation, we actually do not need to adapt the CV (see later E3II case). (3) Comparing E3I with E2, the φ indices are significantly improved in all phases except phase III, with the same reasons as presented before. For E3II, in the whole 7500 s of simulation time, the CV is only adapted twice as listed in Table 6. The first time occurs at t = 1000 s, because the exceeded SPE monitoring statistic indicates a suboptimal operation. The second time of adapting CV

Table 4. Summaries of Online Implementation Results of E2 control limit variation phase


T2

SPE

J̅

I (0−1500) II (1500−3000) III (3000−4500) IV (4500−6000) V (6000−7500)

moderate big small big small

no yes no no no

yes yes no yes no

−5134.81 −9873.07 −2984.47 −7393.03 −497.77

using Z0 as the CV does not generate good self-optimizing performance all of the time. In phase I (0−1500 s), the converged Ti deviates from its desired value moderately. Nevertheless, the economic loss is not negligible. In phases II (1500−3000 s) and IV (4500−6000 s), the deviations are relatively big, indicating poor self-optimizing performances. In phases III (3000−4500 s) and V (6000−7500 s), the selfoptimizing performance is good because the converged Ti is close to its desired positions. Again, all of these performances can be evaluated by the proposed monitoring statistics, as shown in Figure 10b and c. Phases I, II, and IV can be identified by either T2 or SPE statistics, through observing whether their control limits are exceeded. Phases III and V are reported to be optimal because the T2 or SPE statistics are both within the limits, which are in accordance with the true situation. From the obtained results, it can be concluded that the economic performance of E2 is not satisfactory. Although increasing the number of measurements or adopting a nonlinear combination CV may improve the self-optimizing performance, these 14705



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Table 5. Summaries of Online Implementation Results of E3I control limit variation phase

adapted CV (time instant)


T2

SPE

J̅

I (0−1500) II (500−3000) III (3000−4500) IV (4500−6000) V (6000−7500)

−995.0 − 13.47CB + 2.24Ti (t = 1000) −1231.2 + 26.1CB + 2.8Ti (t = 2500) −634.9 − 23.3CB + 1.53Ti (t = 4000) −1126.6 − 41.8CB + 2.7Ti (t = 5500) −552.6 − 79.5CB + 1.37Ti (t = 7000)

moderate→minor minor big→minor minor moderate→minor

no no no no no

yes→no no yes→NO no no

−5141.68 −9992.71 −2962.62 −7433.80 −503.64

Table 7. Economic Performance Improvements of E3I and E3II As Compared To E1 and E2 E3I (%)

E3II (%)

phase

φE3I|E1

φE3I|E2

φE3II|E1

φE3II|E2

I (0−1500) II (1500−3000) III (3000−4500) IV (4500−6000) V (6000−7500)

−Inf 88.3 −117.9 77.7 68.4

47.6 99.3 −127.9 91.0 29.2

−Inf 89.2 66.1 69.2 66.8

47.6 99.4 64.5 87.7 25.7

In addition to step change disturbances shown above, the response of the new HCS to slowly drafting disturbances and its performance under some model−plant mismatches are further studied below. Slow Varying Disturbance. Besides the disturbance scenario with abrupt changes, as investigated above, the real plant may also suffer from disturbances with different dynamics, such as the slow varying disturbances. As an illustration, a slow varying disturbance scenario is considered. In the first 1500 s, the disturbance CAi keeps at its nominal value 1, while after 1500 s to the end 7500 s, CAi slowly ramps from 1 to 0.75 in the remaining simulation time; note that CAi actually does not settle to stationary since 1500 s. The response of proposed HCS is plotted in Figure 13. In the first 1500 s, the system behavior is identical to the previous case above when CAi runs at its nominal value; the CV is adapted to achieve satisfactory optimizing performance at t = 1000 s. With this action, the detection indicators fall below the alarm levels. After that, it is interesting to note that with the occurrence of slow varying disturbance, the SPE index begins to increase as CAi slowly drifts. As the SPE accumulates and finally exceeds its limit at about 6500 s, the detection scheme triggers a CVA action, which then eliminates the alarm. After that (6500−7500 s), the detection indicator continues to gradually increase, but this time, the T2 shows a more obvious effect on the trend. For the manipulated variable, one can observe that Ti is not able to track its optimal value precisely until an occurrence of CVA to correct its trajectory. It should be noted that under slow varying disturbance, the system is not able to achieve stationary after 1500 s; however, the dynamic of CAi is much slower than the CSTR process, and the system status can

Figure 12. Online implementation results of E3II: (a) trajectories of Ti and its optimal value, (b) online monitoring result of T2 statistic, and (c) SPE statistic.

occurs at t = 4000 s, with the SPE monitoring statistic exceeded again. In the above, every time the CV is adapted, the system gets back to near optimality. As shown in Table 7, the economic improvement of E3II is similar to E3I in phases I, II, IV, and V. In phases I and II, the φ indices (−Inf and 47.6% in phase I, 89.2% and 99.4% in phase II) are almost exactly the same as those in E3I (−Inf and 47.6% in phase I, 88.3% and 99.3% in phase II). In phases IV and V, where the CV is constant without adaptation, the φ indices (69.2% and 87.7% in phase IV, 66.8% and 25.7% in phase V) are only slightly smaller than those in E3I (77.7% and 91.0% in phase IV, 68.4% and 29.2% in phase V). However, in phase III, E3II achieves a great economic improvement, where the φ indices increase to 66.1% and 64.5%, as compared to the fact that the economic performances are negative in E3I. Table 6. Summaries of Online Implementation Results of E3II

control limit variation phase

adapted CV (time instant)


T2

SPE

J̅

I (0−1500) II (1500−3000) III (3000−4500) IV (4500−6000) V (6000−7500)

−945.0 − 13.47CB + 2.24Ti (t = 1000)

moderate→minor minor moderate→minor small small

no no no no no

yes→no no yes→no no no

−5141.68 −9992.78 −2995.50 −7432.28 −502.92

−662.3 − 31.37CB + 1.60Ti (t = 4000)

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Figure 13. Online implementation results of proposed HCS under slow varying disturbances: (a) trajectories of Ti and its optimal value, (b) online monitoring result of T2 statistic, and (c) SPE statistic.

Figure 14. Online implementation results of E3I under a model−plant mismatch.

accuracy of process model. In the case of a model−plant mismatch, the ultimate solution will stay somewhere away from the true optimum. The deviated distance relates to the extent of the particular model−plant mismatch. (2) Every time the CV is adapted, the detection alarms disappear. This is also reasonable because the CVA and detection algorithm consistently inherit from the same process model, although the model is not exactly accurate. Therefore, the proposed HCS is robust (at least to some reasonable extent) to model−plant mismatches. Evaporator. Process Description. In the evaporator, the concentration of dilute liquor is increased by evaporating solvent from the feed stream through a vertical heat exchanger with circulated liquor. The process contains 3 state variables and involves 20 process variables, as tabulated in Table 8 with their nominal values. For the process diagram and rigorous process model, we refer to ref 16. The manipulated variable and disturbances are

be considered as a pseudo steady state. The CVA is performed in such a pseudo steady state and yet obtains satisfactory results. The simulation results can be expected because the JIT regression is based on a local region, which is determined from the selected similar samples. The slow varying disturbance results in the gradual drift of operating condition, whose impact on the economic performance is partially addressed by the selfoptimizing CV. For the remaining effect that cannot cope by the locally constructed CV, it accumulates in the system and is simultaneously reflected in the detection scheme. As the disturbance drifts to such an extent that the accumulated effect in the detection scheme exceeds the control limit, a CVA is triggered. The experiment again demonstrates the virtue of the nonoptimality detection scheme; that is, one does not need prior knowledge of disturbance dynamics but instead just relies on detection results, which “feedback” the current economic performance to the control system. If the economic loss is acceptable, then no action needs to be performed, as indicated in Figure 13 (1000−6500 s); otherwise, the CVA is triggered. Model−Plant Mismatch. In real applications, there may exist a model−plant mismatch in the problem formulations, which is not considered in the framework of the present work. To investigate this effect, we introduce significant +20% biases for the two kinetic coefficients of positive and reverse reactions, to mimic a not well-modeled plant. The disturbance scenario with step changes shown in Figure 8 is applied to this plant. The simulation results show that the performance of proposed HCS degrades (Figure 14). In the whole simulation time, the CV is adapted three times at 1000, 2500, and 4000 s, respectively. Regarding the simulation results, two phenomenons can be observed: (1) When the detection block reports no alarm, which indicates the system is optimally operated, the manipulated variable actually deviates from its true optimal value to a certain extent. This is expected because the performance of the model-based RTO scheme relies heavily on the

u = [ F2 P100 F3 F200 ]T d = [ F1 X1 T1 T200 ]T

where the variation ranges for disturbances are defined as ±20% of their nominal values. The following measurements are considered: y = [ F2 F3 F100 F200 ]T

all of which are with Gaussian noises of 0 mean and 0.05 standard deviation. The operation objective is formulated as minimizing the following cost function [$ h−1], which relates to steam, cooling water, and pump work: J = 600F100 + 0.6F200 + 1.009(F2 + F3)

(31)

The constraints related to product specification, safety, and design limits are as follows:

X 2 ≥ 35 + 0.5% 14707

(32)



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0 kg min−1 ≤F3 ≤ 100 kg min−1

Table 8. Process Variables and Nominal Values for Evaporator Process variable

physical description

nominal value

unit

F1 F2 F3 F4 F5 X1 X2 T1 T2 T3 L2 P2 F100 T100 P100 Q100 F200 T200 T201 Q200 J

feed flow rate product flow rate circulating flow rate vapor flow rate condensate flow rate feed composition product composition feed temperature product temperature vapor temperature separator level operating pressure steam flow rate steam temperature steam pressure heat duty cooling water flow rate inlet cooling water temperature outlet cooling water temperature condenser duty operational cost

10 1.4085 28.0 8.5915 8.5915 5 35.5 40 91.216 83.608 1 56.426 10.017 151.52 400 366.626 230.525 25 45.498 330.775 6178.2

kg min−1 kg min−1 kg min−1 kg min−1 kg min−1 % % °C °C °C m kPa kg min−1 °C kPa kW kg min−1 °C °C kW $ h−1

40 kPa ≤ P2 ≤ 80 kPa

(33)

P100 ≤ 400 kPa

(34)

F200 ≤ 400 kg min−1

(35)

(36)

Note a 0.5% back-off has been imposed on X2 to ensure the variable remaining feasible in all circumstances. Application of Proposed HCS. At the nominal point, two process constraints, X2 − 35.5 = 0 and P100 − 400 = 0, are active. It can be seen that these two constraints will keep active within the whole disturbance region.22 Because P100 is a manipulated variable, P100 −400 = 0 can be implemented by keeping P100 at its maximum in a feed-forward manner. Besides, the separator level L2, which has no steady-state effect, is openloop unstable, and hence must be stabilized and consume one degree of freedom. Therefore, the dimension of reduced gradient is one, which can be analytically derived as22 ∇r J = 0.6 − 0.5538

T201 − T200 F200

⎡ 0.16(F1 + F2) + 0.07F1 42.F1 ⎤ × ⎢6.306 + ⎥ T100 − T2 36.6 ⎦ ⎣

(37)

Decentralized control loops are designed to configure the regulatory control layer. In particular, X2 − 35.5 = 0 and L2 are paired with F2 and F3, respectively. The remaining reduced gradient is approximated with y as the CV, which will be adapted during the operation. This CV is controlled by manipulating F200 to track 0 set-point. Both the database for CVA and the nonoptimality detection are constructed through Monte Carlo experiments. For the CVA, both the manipulated variable F200 and the disturbances are allowed to vary in reasonable ranges. 2000 samples are

Figure 15. Disturbance trajectories. 14708



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Figure 16. Dynamic simulation results.

the CV is adapted two times. At 1000 min, the CV is adapted to be Z1 = −0.0207 − 0.0442F2 + 0.0045F3 − 0.1527F100 + 0.0064F200 as the SPE statistic exceeds. At 10 000 min, the CV is adapted to be Z2 = 0.0922 − 0.2466F2 + 0.0030F3 − 0.0998F100 + 0.0052F200 as the SPE statistic exceeds again. Afterward, the CV is no longer adapted as indicated by both T2 and SPE statistics. As expected, we note that every time the CV is adapted, the manipulated variable F200 will be near its desired optimal value and the detection alarms disappear, which demonstrate the efficiency of the proposed HCS. During the simulated time, the total cost of applying proposed HCS is calculated to be $1,534,658. We also conducted dynamic simulations for the traditional two-step RTO (with F200 in open loop) and the global regression approach (details not shown here), and the total costs are calculated to be $1,536,059 and $1,534,682, respectively. Consider an ideal case for directly controlling the reduced gradient eq 37, where the total cost is $1,534,613. Using the definition of performance index φ, it can be easily evaluated that the proposed HCS gives 96.7% and 34.8% improvements over the traditional two-step RTO and the global regression approach, respectively.

collected containing the information on measurements and the reduced gradient. The balancing weight ρ in eq 6 for calculating similarity is set to be 0.5. The number of relevant samples L is chosen as 10, which is determined by the same routine as the previous reactor case. The off-line performance verification for JIT regression gives an RMSE value of 0.0093 for 500 testing samples, as compared to the fact that the RMSE value is 0.0425 when a global regression method is used. For the nonoptimality detection, only the disturbances are allowed to vary to construct the database; other related process variables are determined by solving the optimization problem under random disturbance scenarios. Finally, 2000 samples are collected. Applying the PCA modeling, the number of principal components is set to be 3, which explains more than 99% of the information. The T2 and SPE control limits for nonoptimality detection are calculated as 11.39 and 0.021, respectively. The overall detection results for 500 random samples are 0.6% false rate for optimal ones and 0.2% missing rate for nonoptimal ones, which are quite satisfactory. Because various RTO methods are compared and discussed in detail in the previous reactor example, in the evaporator case, we will only focus on the dynamic behavior of the proposed HCS, due to the length of this study. The tested disturbances are arranged to be step changing scenarios within an overall simulation time of 15 000 min. In the first 3000 min, all disturbances stay at their nominal values. In the next 3000 min, all four disturbances simultaneously change to their minimum values, nominal values, or maximum values randomly, as shown in Figure 15. The CVA interval is set to be 1000 min, which is sufficient for the process to settle down. The adapted CV starts with Z0 = −0.2845−0.0741F2 − 0.0069F3 − 0.0546F100 + 0.0047F200, which results from the global regression approach. Figure 16 plotted the response of F200 and detection results of T2 and SPE. The dynamic simulation shows that, in the entire 15 000 min,

6. CONCLUSIONS In this Article, a new type of CVA-based HCS was proposed for the control synthesis and RTO of process operation, which is different from a traditional HCS with the set-points of CVs updated. In the proposed HCS, neither disturbance estimation nor reoptimization are explicitly required. The optimizing task is mainly handled by the constructed CVs in regulatory control loops. As a big extension of methodologies, we proposed to online adapt the CVs by the RTO layer to improve the economic performance. Furthermore, a nonoptimality detection technique was also developed to provide a prerequisite to activate the CVA. 14709



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(4) Mercangoz, M.; Doyle, F. J., III. Real-time optimization of the pulp mill benchmark problem. Comput. Chem. Eng. 2008, 32, 789− 804. (5) Chachuat, B.; Srinivasan, B.; Bonvin, D. Adaptation strategies for real-time optimization. Comput. Chem. Eng. 2009, 33, 1557−1567. (6) Skogestad, S. Plantwide control: The search for the selfoptimizing control structure. J. Process Control 2000, 10, 487−507. (7) Sequeira, S. E.; Graells, M.; Puigjaner, L. Real-time evolution for on-line optimization of continuous processes. Ind. Eng. Chem. Res. 2002, 41, 1815−1825. (8) Sequeira, S. E.; Herrera, M.; Graells, M.; Puigjaner, L. On-line process optimization: parameter tuning for the real time evolution (RTE) approach. Comput. Chem. Eng. 2004, 28, 661−672. (9) Ying, C. M.; Joseph, B. Performance and stability analysis of LPMPC and QP-MPC cascade control systems. AIChE J. 1999, 45, 1521−1534. (10) Halvorsen, I. J.; Skogestad, S.; Morud, J. C.; Alstad, V. Optimal selection of controlled variables. Ind. Eng. Chem. Res. 2003, 42, 3273− 3284. (11) Alstad, V.; Skogestad, S. Null space method for selecting optimal measurement combinations as controlled variables. Ind. Eng. Chem. Res. 2007, 46, 846−853. (12) Kariwala, V. Optimal measurement combination for local selfoptimizing control. Ind. Eng. Chem. Res. 2007, 46, 3629−3634. (13) Kariwala, V.; Cao, Y.; Janardhanan, S. Local self-optimizing control with average loss minimization. Ind. Eng. Chem. Res. 2008, 47, 1150−1158. (14) Alstad, V.; Skogestad, S.; Hori, E. S. Optimal measurement combinations as controlled variables. J. Process Control 2009, 19, 138− 148. (15) Jaschke, J.; Skogestad, S. Optimal controlled variables for polynomial systems. J. Process Control 2012, 22, 167−179. (16) Ye, L.; Cao, Y.; Li, Y.; Song, Z. Approximating necessary conditions of optimality as controlled variables. Ind. Eng. Chem. Res. 2013, 52, 798−808. (17) Jaschke, J.; Skogestad, S. NCO tracking and self-optimizing control in the context of real-time optimization. J. Process Control 2011, 21, 1407−1416. (18) Srinivasan, B.; Bonvin, D.; Visser, E.; Palanki, S. Dynamic optimization of batch processes - II. Role of measurements in handling uncertainty. Comput. Chem. Eng. 2003, 27, 27−44. (19) Cheng, C.; Chiu, M.-S. A new data-based methodology for nonlinear process modeling. Chem. Eng. Sci. 2004, 59, 2801−2810. (20) Fujiwara, K.; Kano, M.; Hasebe, S.; Takinami, A. Soft-sensor development using correlation-based just-in-time modeling. AIChE J. 2009, 55, 1754−1765. (21) Ge, Z. Q.; Song, Z. H. A comparative study of just-in-timelearning based methods for online soft sensor modeling. Chemom. Intell. Lab. Syst. 2010, 104, 306−317. (22) Cao, Y. Direct and indirect gradient control for static optimisation. Int. J. Autom. Comput. 2005, 2, 13−19. (23) Francois, G.; Srinivasan, B.; Bonvin, D. Use of measurements for enforcing the necessary conditions of optimality in the presence of constraints and uncertainty. J. Process Control 2005, 15, 701−712. (24) Francois, G.; Srinivasan, B.; Bonvin, D. Comparison of six implicit real-time optimization schemes. J. Eur. Syst. Autom. 2012, 46, 291−305. (25) Gros, S.; Srinivasan, B.; Bonvin, D. Optimizing control based on output feedback. Comput. Chem. Eng. 2009, 33, 191−198. (26) Francois, G.; Srinivasan, B.; Bonvin, D. On the equivalence between neighboring-extremal control and self-optimizing control for the steady-state optimization of dynamical systems. Ind. Eng. Chem. Res. 2014, 53, 7470−7478. (27) Francois, G.; Bonvin, D. Use of transient measurements for the optimization of steady-state performance via modifier adaptation. Ind. Eng. Chem. Res. 2014, 53, 5148−5159. (28) Skogestad, S.; Postlethwaite, I. Multivariable Feedback Control: Analysis and Design, 2nd ed.; John Wiley & Sons: Chichester, UK, 2005.

An exothermic reactor was extensively studied to explore various features of techniques proposed in this Article and verify the contributions. A further evaporator case was also provided to demonstrate the successful application to a more realistic plant. It should be noted that, in the present work, a process model is required to construct the databases for both CVA and nonoptimality detection. In the presence of a model− plant mismatch, the performance of proposed HCS will deteriorate as investigated in the CSTR example. However, it has been shown that the proposed HCS has some robustness to such model errors. Besides model−plant mismatches, another remaining problem is the assumption of an invariant set of active constraints. In the case of the set of active constraints changes, the structure of NCO components varies, and thus the CVs should be approximated differently. In a traditional HCS where the optimization problem is explicitly resolved, the active set change can be naturally solved because the constraints are mathematically considered every time solving the optimization problem. However, in the new HCS, this problem cannot be solved straightforward because one needs to specify the active constraints and reduced gradients in prior. Recently, several solutions have been proposed to deal with active constraint changes.22,34 How to find an effective solution in the new framework is a consideration of future work.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS L.Y., X.M., and Z.S. gratefully acknowledge the financial support from the National Natural Science Foundation of China (NSFC) (61304081), Zhejiang Provincial Natural Science Foundation of China (LQ13F030007), Postdoctoral Science Foundation of China (2013M541778), National Project 973 (2012CB720500), and Ningbo Innovation Team (2012B82002, 2013B82005).

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ABBREVIATIONS ANN = artificial neural network CV = controlled variable CVA = controlled variable adaptation HCS = hierarchical control structure JIT = just-in-time MPC = model predictive controller NCO = necessary conditions of optimality NEC = neighboring-extremal control RMSE = root mean square error RTO = real-time optimization PCA = principal component analysis SOC = self-optimizing control SPE = squared prediction error

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REFERENCES

(1) Marlin, T. E.; Hrymak, A. N. Real-Time Operations Optimization of Continuous Processes; 1997. (2) Skogestad, S. Control structure design for complete chemical plants. Comput. Chem. Eng. 2004, 28, 219−234. (3) Engell, S. Feedback control for optimal process operation. J. Process Control 2007, 17, 203−219. 14710



Article

(29) Qin, S. J. Statistical process monitoring: basics and beyond. J. Chemom. 2003, 17, 480−502. (30) Ge, Z. Q.; Song, Z. H.; Gao, F. R. Review of recent research on data-based process monitoring. Ind. Eng. Chem. Res. 2013, 52, 3543− 3562. (31) Jackson, J. E.; Mudholkar, G. S. Control procedures for residuals associated with principal component analysis. Technometrics 1979, 21, 341−349. (32) Alstad, V. Studies on selection of controlled variables. Ph.D. thesis, Norwegian University of Science and Technology, 2005. (33) Ye, L.; Cao, Y.; Li, Y.; Song, Z. A Data-Driven Approach for Selecting Controlled Variables; Proceedings of 8th International Symposium on ADCHEM, Singapore, 2012; pp 904−909. (34) Manum, H.; Skogestad, S. Self-optimizing control with active set changes. J. Process Control 2012, 22, 873−883.

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A Novel Hierarchical Control Structure with Controlled Variable

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