Process Goose Queue (PGQ) Approaches toward Plantwide Process

Article pubs.acs.org/IECR

Process Goose Queue (PGQ) Approaches toward Plantwide Process Optimization with Applications in Supervision-Driven Real-Time Optimization Hongguang Li and Jingwen Huang* College of Information Science and Technology, Beijing University of Chemical Technology, Beijing100029, China ABSTRACT: Inspired by the biologic nature of flying geese, process goose queue (PGQ) approaches toward plantwide process optimization are explicitly introduced in this paper along with applications in real-time optimization (RTO). Taking advantage of ad-hoc PGQ metrics, process variables associated with a process unit could be accordingly identical with geese positions of a PGQ. Motivated by the self-organization in flight formation of geese, a process unit can achieve such an optimum formation that every goose in the PGQ benefits from the maximum upwash. In this sense, adjustment rules invoked to track the ideal PGQ formulation are accommodated. Followed by this idea, a plantwide process is first decomposed into several hierarchically connected multilayer PGQs. Subsequently, a plantwide PGQ which includes a PGQ-objective and several multilayer PGQs is constructed, which contributes to solving complex plantwide process optimization problems in a novel way. As applications of PGQ approaches, we initially address a supervision-driven RTO issue concerning economic performance deterioration caused by process supervision. A process unit whose variables are shifted by human operators can be regarded as an ill-PGQ which would trigger the autonomous adjustments of the plantwide PGQ. Enabling algorithms concerning adjustment sequence of the plantwide PGQ with an ill-PGQ are constructed, which are generally characterized by ill-PGQ detection, PGQ follow-up, and PGQ-objective achievement. To demonstrate the feasibility and validity of this contribution, the Tennessee Eastman (TE) benchmark process is employed as an extensive case study, showing that the proposed approaches particularly enjoy considerable computational simplicity in contrast with traditional global optimization strategies.

1. INTRODUCTION A plantwide process refers to an industrial plant composed of a series of process units, where downstream units are influenced by upstream units, or the are mutually influential. While plantwide process optimization usually highlights a modelbased process control approach that uses current process information (i.e., plantwide process models and economic data) to predict the optimum operating policies over the next implementing interval.1 Traditionally, plantwide process optimization can be classified into two relatively distinct categories in terms of process model architectures involved: global architecture and decentralized architecture. The global approaches associate a process with one economic objective and optimize it based on rigorous steady-state models. Typical algorithms for global optimization architecture including SQP (sequential quadiatic programming) and IPM (interior point method) have been widely circulated in various research fields.2,3 Additionally, global evolutionary algorithms such as genetic algorithms, particle swarm, and ant colony algorithms could be more attractive to avoiding a possible local optimum.4−6 However, it is acknowledged that global optimization algorithms are easily flung into dilemmas in the presence of model complexity and nonlinearity which are always associated with plantwide processes, eventually resulting in a heavy computational burden. Alternatively, decentralized optimization approaches decompose a plantwide process into several subsystems which are optimized individually before coordinated to achieve an overall optimization performance. The work of Lubomiŕ Bakule7 exemplified that decentralized optimization method© 2012 American Chemical Society

ologies could serve as effective tools to overcome specific difficulties arising in plantwide processes. Relevant to these issues, Dantzig−Wolfe decomposition has been proved quite active with wide applications.8,9 However, limited to linear programming problems, Dantzig−Wolfe decomposition is less adapted for plantwide processes which usually involve severe nonlinearity. Furthermore, concerning physical structures and coupling nature among the subsystems, Sobieski10 presented a generalized multilevel optimization approach termed multidisciplinary design optimization (MDO) which is accommodated to exhibiting challenges of complex systems. Even though MDO has been successfully applied to mechanical design industry,11,12 it gives less straightforward solutions to surviving process model complexity. In a parallel study, hierarchical multilevel optimization algorithms have attracted much attention as well. They first convert an optimization problem into a multisystem and subsequently deal with it using multiobjective programming.13,14 However, demanding for separable or approximately separable objectives, these methods are susceptible to considerable systematic deviations in the presence of severe nonlinear relations between the global objective and subobjectives. Physically, a plantwide process can be identified as a connection of numerous basic process units which are considered as dynamic systems characterized by output and Received: Revised: Accepted: Published: 10848

January 15, 2011 July 4, 2012 July 16, 2012 July 16, 2012 dx.doi.org/10.1021/ie300374g | Ind. Eng. Chem. Res. 2012, 51, 10848−10859

Industrial & Engineering Chemistry Research

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input process variables. To imitate nature of a flying goose queue, a certain kind of process units could be regarded as a goose queue. Therefore, achievements of the process units to their optimum set-points are consistent with the mechanism that a flock of geese self-organizes into a V-formation where every goose gets an optimum position with a maximum upwash. In line with this thought, we came up with a novel idea of specifying “process goose queue (PGQ)” approaches to reconfigure plantwide processes, thereby optimizing them in terms of economic perspectives. In this context, plantwide process models are initially decomposed hierarchically into several goose queues, resulting in a multilayer PGQ. Subsequently, by employing PGQ position adjustment algorithms a plantwide process could eventually achieve the optimum economic objectives. In practical process operations, routine process supervision could possibly lead to certain shifted manipulated variables which can be regarded as a kind source of disturbances, resulting in a so-called “supervision-driven real-time optimization (RTO)” problem. According to PGQ metrics, the process unit involving shifted manipulated variables can be considered as an ill-PGQ which triggers position adjustment of the plantwide PGQ. In this sense, PGQ approaches are well adapted to solve the supervision-driven RTO problems. The remainder of this paper is organized as follows. In section 2, we introduce the definitions of a flying goose queue, followed by which theoretical foundations of PGQs as well as position adjustment rules are presented along with an exemplary case. Section 3 specifies implementing procedures of supervision-driven RTO with PGQ methodologies as well as enabling algorithms. In section 4, the Tennessee Eastman (TE) process is employed as a case study which demonstrates the feasibility and validity of the contribution. Section 5 draws conclusion and assesses the future prospective.

Figure 2. Vortices created by pressure difference.

Figure 3. Regions of upwash and downwash created by trailing vortices.

Figure 4. Upwash generated by a flying goose.

power.17 This manifestation could clarify why migrant geese fly from random to V-formations18 as simulated in Figure 5. 2.2. PGQ Fundamentals. 2.2.1. PGQ. From process operation perspectives, plantwide industrial processes can be identified as connections of a variety of basic process units which are considered as dynamic systems characterized by output and input process variables. Generally, the steady-state models of a basic process unit can be described by

2. PGQ APPROACHES 2.1. Nature of Flying Geese. From an aerodynamics perspective, a flying goose can be deemed as a stationary object with airflow moving through its wings above and below. Practically, the airflow over the upper surface moves faster than that below the lower part of the wings, resulting in a lower air pressure on the upper part and making it possible to lift itself by the wings, as shown in Figure 1. Meantime, vortices around the tips are created, as shown in Figure 2. Moreover, the vortices could produce large regions of upwash outboard of the wings and a region of downwash more centrally as shown in Figure 3. In response, Figure 4 shows a corresponding model explored by some researchers.15,16 It is conceivable that the upwash may contribute to the lift for a following bird, thus reducing its requirement for induced

Y = g (S , X )

(1)

where, Y, S, X, and g indicate sets of the output state variables, input state variables, manipulated variables, and steady-state relationship functions, respectively. Accordingly, we introduce PGQ approaches by a series of descriptions as follows. Definition1 (PGQ). A process goose queue (PGQ) is a 3tuple, PGQ = (L, FS, FM), where • L is the leading goose position (LGP), such that {L|L ⊂ . S} ≠ ϕ, represented as • FS is the state following goose position (SFGP), such that . FS ⊂ S represented as • FM is the manipulated following goose position (MFGP), such that {FM|FM ⊂ X} ≠ ϕ, represented as . • We have FS ∩ FM = ϕ, L ∩ FM = ϕ, {L|L ⊂ FS} ≠ ϕ. The graphical description of a PGQ is illustrated in Figure 6, where, L, FS, and FM represent the process variables, Y, the process state variables, S, and the process manipulated variables, X, associated with a process unit, respectively. The dash line circle is used for grouping a PGQ. As an example, Figure 7 shows a Williams−Otto reactor19 which is operating at temperature Tr, reactant flows Fa and Fb,

Figure 1. Lifting force produced by a flying goose. 10849

dx.doi.org/10.1021/ie300374g | Ind. Eng. Chem. Res. 2012, 51, 10848−10859


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Figure 5. (a) Goose position at random. (b) Goose queue in V-formation.

Figure 6. PGQ (process goose queue). Figure 9. Deviation from the optimum formation driven by LGP.

Figure 7. Williams−Otto reactor.

producing a six-component product Z. Figure 8 gives the corresponding PGQ, where, the output state variables Z, input state variables Fa, manipulated variables Fb and Tr are equivalent to L, FS, and FM, respectively.

Figure 10. Deviation from the optimum formation driven by SFGP.

In order to achieve the optimum formation in which every goose of the PGQ can benefit from the maximum upwash, the following optimization tasks should be implemented. ⎞ ⎛ 1 max U = max⎜ = min(L* − L)2 2⎟ * ⎝ (L − L ) ⎠

Figure 8. PGQ for Williams−Otto reactor.

Definition 2 (Upwash). In a pursuit of the optimum formation, the upwash associated with a PGQ is inversely proportional to the distance between the current and ideal positions of the LGP, such as 1 U= * (L − L)2 (2)

s.t. L = f (FS , FM )

(3)

where, f corresponds to the relationship among LGP, SFGP, and MFGP. Motivated by this idea, two types of alternative position adjustment rules associated with a PGQ can be specified as follows. Rule 1 (LGP Driven Adjustments). Once an PGQ operates away from its normal trajectory as shown in Figure 9, LGP would adjust its position autonomously back to an ideal one. At the same time, SFGP and MFGP would operate consistently with the activities of LGP, formulating an adapted operating state. This kind of position adjustment implies solving the following optimization problems:

which reveals that the closer the LPG is to the ideal position, the more upwash can a PGQ obtain. Definition 3 (Optimum Formation). An optimum formation, PGQ* = (L*, F*S , F*M), refers to an ideal PGQ formulation in which every goose of the PGQ can benefit from the maximum upwash. Practically, the optimum formation could be destroyed by uncertain disturbances which drive LGP away from the ideal formation as (L*, FS*, FM * ) → (L, FS, FM), as shown in Figure 9. Another shift arises in that SFGP leaves from the previous upwash maxima position which results in LGP leaving from the previous position as well, i.e., (L*, F*S , F*M) →(L, FS, F*M), as shown in Figure 10. All those scenarios would lead to inferior operating situations in desperate need of adjustment to recover.

max U = min(L* − L)2 FM , FS

s.t. L = f (FS , FM ) FS L ≤ FS ≤ FS U FM L ≤ FM ≤ FM U 10850

(4)



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Rule 2 (SFGP Driven Adjustments). Once the operating state of a PGQ deviates from the optimum formulation due to SFGP failing to follow it as shown in Figure 10, MFGP would try to adjust its position autonomously to create a new optimum formation. At the same time, LGP would slightly shift its position to survive the adjustment while SFGP is kept unchanged. This kind of position adjustment implies solving the following optimization problems: max U = min(L* − L)2 FM

s.t. L = f (FS , FM ) LL ≤ L ≤ LU FM L ≤ FM ≤ FM U

(5)

The new achieved solution of 4 and 5 will be marked **. In summary, once a PGQ deviates from the optimum operating state, both FS and FM can be driven by L to pursuit a new upwash maxima with rule 1, while L would be driven back to the ideal trajectory by manipulating FM but keeping FS constant with rule 2. Referring back to the above-mentioned example, we assume an initial nonoptimum state with Fb = 4.6, Tr = 88.6, Fa = 1.73, and z5 = 0.295. If a new target of z5 is demanded, rule 1 would be launched to implement the position adjustments. As a result, the process variables [FSFM] = [Fa,Fb,Tr] will be triggered to set up an optimum operating state (F*b * = 4.900, T*r * = 89.400, F*a * = 1.980, z*5 * = 0.323), shown in Figure 11. Similarly, if FS = [Fa] deviates from the

Figure 12. Adjustments with rule 2.

variables to minimize or maximize the economic goals subject to constraints of process models. What’s more, the fact that actual formations of process models involved in process optimization depends on the connections of process units accounts for particularly hierarchical decompositions of process models, which could be described as a multilayer PGQ. Definition 4 (Multilayer PGQ). A multilayer PGQ consists of several PGQs organized in a hierarchical architecture. Therein, the PGQs are characterized by PGQi = (Li, FSi, FMi), where, i = 1, ..., m, indicates the depth index. The SFGP of an upper PGQ may serve as the LGP of the neighbored lower PGQ in terms of depth index. Thus, for each PGQ, we have: Li = fi (FSi , FMi) FSi = Li + 1 FSi L ≤ FSi ≤ FSi U

Figure 11. Adjustments with rule 1.

FMi L ≤ FMi ≤ FMi U

optimum operating state (from Fa =1.83 kg/s to Fa = 1.7 kg/s at t = 300 s), one component of z, z5, will decrease from 0.310 because of the mismatch of FM = [Fb,Tr] = [4.78,89.8] which is only suitable for initial operating states. Rule 2 would be triggered to implement the position adjustments, therefore achieving new optimum set-points, FM ** = [Fb**,Tr**] = [4.78,89] at t = 1500 s. New optimum operating states are set up where this component of z increases from 0.053 to 0.302. The adjustments with rule 2 are shown in Figure 12. 2.2.2. Multilayer PGQ. Practically, process optimization could be identified as a procedure of adjusting manipulated

(6)

where FSi = Li+1 denotes the connections among the lower and upper PGQ, i.e., and LGP of lower PGQ acts as the SFGP of upper PGQ. This manifestation is consistent with the practical structure of a plantwide process. Figure 13 shows the cohesive graph of a multilayer PGQ. Under the multilayer PGQ structure, problems 4 and 5 are converted into 7 and 8, respectively, where, j indicates a fixed breadth index. 10851



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process reaches the steady states and no additional disturbance occurs, optimization algorithms will be carried out over the time interval [t1, t2]. On the contrary, if the process is unsteady at time t1 or the continuous disturbance occurs, RTO can not be implemented because of the failure to reach steady state. Alternatively, PGQ approaches partition the waiting time into N equal time intervals (Δt), thereby characterizing the dynamic process into several pseudosteady states. Therein, measurements will be collected between two time intervals and then adopted by PGQ to follow-up the previous optimum formation and achieve the optimal PGQ-objective in sequence. After N optimization calculations, continuous improvements of manipulated variables [FMij,1 FMij,2, ..., FMij,k, ..., FMij,N] are achieved. In regard to problem 7, assuming that an LGPi of PGQi changes from an initial point Lij , k = f ij,k( FSij , k , FMij , k) to Lij,k = f ij,k(FSij,k,FMij,k) at the kth Δt, we can employ a first-order Taylor expansion to express an approximation:

Figure 13. Multilayer PGQ. m

m

min ∑ σij = min i=1

FSij , FMij

∑ (Lij* − Lij)2 i=1

s.t. Lij = fij (FSij , FMij)

f (FSij , k , FMij , k) = f ( FSij , k , FMij , k) +

FSij = L(i + 1)j FSij L ≤ FSij ≤ FSijU FMij L ≤ FMij ≤ FMijU m

[FSij , k − FSij , k] +

(7)

m

i=1

FMij , k = FMij , k

(9)

where υk is the Taylor remainder. Additionally, we specify

i=1

s.t. Lij = fij (FSij , FMij)

f (FSij , k , FMij , k) − f ( FSij , k , FMij , k)

FSij = L(i + 1)j FMij L ≤ FMij ≤ FMijU

∂f ∂FMij , k

FSij , k = FSij , k

[FMij , k − FMij , k] + υij , k

min ∑ σij = min ∑ (Lij* − Lij)2 FMij

∂f ∂FSij , k

≜

(8)

2.2.3. Algorithms. To solve problems 7 and 8, conventional nonlinear programming (NLP) algorithms usually wait for steady state and pay little attention to the objective’s variation with respect to time horizon during optimization. In order to overcome the deficiencies of long-waiting for steady states associated with conventional plantwide RTO, PGQ algorithms divide the waiting process for steady state into small pseudosteady states. Figure 14 shows the behaviors of conventional RTO and PGQ algorithms for solving problems 7 and 8 with solid and dashed lines, respectively. Consider that an uncertain disturbance occurs at time t0 . For conventional RTO strategies, there is no adjustment until the process reaching the steady states at time t1. At time t1, if the

∂f ∂FSij , k

[FSij , k − FSij , k] + FSij , k = FSij , k

∂f ∂FMij , k

FMij , k = FMij , k

[FMij , k − FMij , k] + υij , k

(10)

Accordingly, we definite dij(k) = f (FSij , k , FMij , k) − f ( FSij , k , FMij , k) +

∂f ∂FSij , k

FSij , k + FSij , k = FSij , k

∂f ∂FMij , k

FMij , k + υij , k

(11)

ωij , k = [FSij , kFMij , k]T

uij , k

⎡ ⎢ ∂f =⎢ ⎢ ∂FSij , k ⎢⎣

FSij , k = FSij , k

FMij , k = FMij , k

(12)

∂f ∂FMij , k FMij , k =

⎤ ⎥ ⎥ ⎥ ⎥ FMij , k ⎦

(13)

The initial value of SFGP and MFGP at time k can be obtained from the previous estimate ωk−1, i.e., FSij , k = e 2 Tωij , k − 1

(14)

FMij , k = e1Tωij , k − 1

(15)

where em is a vector with 1 at position m and 0 elsewhere. Therefore, the optimization task of SFGP and MFGP turns out to be the following:

Figure 14. Strategies of RTO and PGQ approaches. 10852


Industrial & Engineering Chemistry Research min σij = min |dij(k) − uij , kωij , k|2 ωij

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(16)

uij , k

where σ is an additive source of disturbances, which includes the Taylor remainder υ as well. In eq 11, f(FSij,k,FMij,k) is the optimum PGQ formation observed so far, marked as Lij,k * , while f( FSij , k , FMij , k ) corresponds to the output of current PGQ formation, marked as Lij,k. It is easy to verify that with this specification, 16 presents the same implication as 7. To solve 16, steepest descent methods20 could be adopted so that ωk = ωk − 1 − α∇ω [d(k) − ukωk − 1]×

FMij , k = ωij , k − 1

(17)

FSij , k = L(** i + 1)j , k

(26)

(27)

3. PROCESS OPTIMIZATION 3.1. Plantwide PGQ. Usually, the economic objective function of a plantwide optimization problem could be built with respect to local process state variables and manipulated variables. In this sense, an additional description about the objective of a multilayer PGQ is introduced as follows. Definition 5 (PGQ-Objective). A PGQ-objective is equivalent to the economic objective function of a plantwide process, characterized by

(18)

(19)

The eventual eq 19 implies that geese positions in PGQs can be updated through iteration so that they follow up the individual objectives. SFGP and MFGP involved in the multilayer PGQ will be affected by their neighbors during follow-up, resulting in coupling issues in gradient calculations. We adopt the following equations to exchange information from each other. ⎧ψ b ω = ⎪ ij , k − 1 ∑ ij , l l , k − 1 l ∈ N i ⎨ ⎪ × ⎩ ωij , k = ψij , k − 1 + αuij , k [dij(k) − uij , kψij , k − 1]

(25)

The optimization formulation along with its achievement can be of the same form as 16 and 21, respectively. The achieved LGP in lower PGQ, L*(i+1)j,k * , will serve as the SFGP of in upper layer PGQ, characterized by

Therefore, 17 turns out to be the following: ωij , k = ωij , k − 1 + αuij , k ×[dij(k) = uij , kωij , k − 1]

FMij , k =

⎤ ⎥ ⎥ FMij , k ⎥ ⎦

The initial value of a PGQ, ( FSij , k , FMij , k ) at time k can be obtained from the previous estimate ωij,k−1, i.e.,

where, α is the step length, and ∇ and × denote gradient and conjugate transposition, respectively. Consider that ∇ω [d(k) − ukωk − 1]× ≈ uk ×ukωk − 1 − d(k)uk ×

⎡ ⎢ ∂f =⎢ ⎢⎣ ∂FMij , k

P = min φ(PS1 , PS2 , ..., PSn , PM )

(28)

where, PSj (j = 1,2,...,n) and PM are local process state variables and manipulated variables, respectively. Referring back to definition 1, PSj and PM could be similarly considered as SFGP and MFGP of a PGQ, respectively. Therefore, a graphical description of the PGQ-objective could be presented in Figure 15.

(20)

Consider an N × N matrix B with individual non-negative real entries bij,l such that bij , l = 0 if l ∉ Mi × Nj

ITB = IT

(21)

where I denotes the N × 1 vector with unit entries. The first and second equations of 20 are known as the PGQ adaptation and combination updates, respectively. In this regard, every PGQi adapts its current estimate, ωij = [FSijFMij], using its individual measurements {Lij*(k),uij,k} available at time k to obtain ψij,k. All PGQs exchange their pre-estimates ψij,k−1 with neighbors. Every node combines the pre-estimates to achieve new estimate ωij,k. The resultant SFGP, FS(ij,k) ** , in the upper layer PGQ will sever as the objective of the LGP in lower PGQ, which is characterized by * FS** (ij , k) = L(i + 1)j , k

Figure 15. PGQ-objective.

Generally, the procedures toward establishing a multilayer PGQ for process optimization are summarized as follows: (1) The objective function consistent with economic demands is converted to a PGQ-objective involving several process state variables, PSj, which equivalently imply the LGP (L1j) as well as a couple of manipulated variables, PM. (2) Starting off with each PSj (L1j) along the reverse traveling paths of mass and energy flows, a plantwide process is decomposed into several PGQs in terms of key process units. Thereafter, LGP, SFGP, and MFGP associated with the PGQs in separate depth and indexes are accordingly formulated. (3) PGQs with same breadth index are hierarchically connected according to the increasing depth indexes, where the SFGP of an upper PGQ serves as the LGP of the neighbored lower PGQ, i.e., FSij = L(i+1)j. Consequently, several PGQ groups with different breadth indexes are similarly established.

(22)

Meanwhile, to solve 8, we adopt algorithms similar to the formulations by rule 1. Specially, we specify dij(k) = f (FSij , k , FMij , k) − f ( FSij , k , FMij , k) +

∂f ∂FMij , k

ωij , k = [FMij , k]T

FMij , k + υij , k FMij , k = FMij , k

(23) (24) 10853



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(4) According to the metrics of PGQ-objective, the PGQ groups with different breadth indexes can be linked with the established PGQ-objective. An exemplary graphical description of a plantwide PGQ is shown in Figure 16.

Figure 17. Hierarchical architecture of process performances.

human operators and RTO implementation aiming at optimizing economy performance. Supported by measurement and control systems, human operators identify the smoothness and safety performances of the processes operating in normal situations, detect and analyze the disturbances before employing the supervisory manipulated variables (SMV), which pertain to the set of MFGP, to keep processes as safe and smooth as possible. The nominal execution of process supervision may cause shifted SMV, possibly leading to changes of economic situations further. In other words, process supervision improves the safety and smoothness performances on the one hand, but degrades the economy performance on the other hand. Accordingly, the process unit containing SMV could be regarded as an ill-PGQ of the plantwide PGQ, usually represented with red dashed-line circle as shown in Figure 18. The impact of an ill-PGQ will

Figure 16. Plantwide PGQ.

Along the depth direction, the more PGQs are obtained, the more decentralization of the process models as well as optimization algorithms are achieved. However, a bullwhip effect which introduces distortion could degrade optimization performance with more depth PGQs. Empirically, the depth number of a plantwide PGQ could be the same as that of the key process units involved. In addition, another measure to alleviate the bullwhip effect is exchanging information among PGQs with eq 20 in section 2. Accordingly, the plantwide process optimization is equivalent to implementing the adjustment rules in corresponding plantwide PGQ. As mentioned in section 2.2.1, an alternative adjustment rule is suitable for different case in optimization; therefore resulting in 7 and 8 in multilayer PGQ. In eq 7, Lij* = f(FSij,k,FMij,k) is the objective of LGP which comes from the resultant SFGP of upper layer PGQ. In eq 8, L*ij is the optimal position of LGP in previous optimum formation which severs as the objective in the PGQ follow-up periods. In the plantwide PGQ, 7 and 8 are used for process steady state optimization and RTO, respectively. The process of implementing steady state optimization with rule 1 has been introduced in a previous work.21 In regard to RTO, rule 2 is implemented to the plantwide PGQ in sequence starting from the so-called ill-PGQ which is trigged by uncertain disturbances consequently leading to the economic performance deterioration. 3.2. Supervision-Driven RTO. According to Abnormal Situation Management (ASM),22 an industrial process can possibly operate in normal, abnormal, or emergency situations. A process operating in the normal situation should demand three aspects of process performance which are hierarchically placed as shown in Figure 17. Therein, the safety performance which aims to guarantee safe productions is of the paramount priority. This is followed by the smoothness performance which is responsible for the smooth-running control and operations against disturbances. The top layer in the hierarchy refers to the economy performance that targets maximum production benefits. Practically, two types of operations could contribute to the process performance improvement: process supervision concerning process safety and smoothness performed by

Figure 18. Plantwide PGQ with an ill-PGQ.

propagate along the traveling stream in a plantwide process. It is seen that if PGQ g1 is detected as the ill-PGQ, the impact will propagate along the traveling stream: FSg1 → FS(g−1)1/Lg1 → FS21/L31 → FS11/L21 → PS1/L11 → PGQ-objective, eventually leading to deterioration of the economic performance, where g is the depth index of the ill-PGQ. A new superior situation needs to be recovered from process adjustments, which becomes a major activity of a supervisiondriven RTO strategy, specified as follows. Definition 6 (Supervision-Driven RTO). Supervision-driven RTO refers to an RTO scheme whose manipulated process variables besides SMV are invoked to keep the process going on ideally in terms of economy performance. Consider that a supervision-driven RTO event trigged by an ill-PGQ happens. According to PGQ metrics and the SFGP driven adjustment rule (rule 2), a position adjustment sequence of plantwide PGQ is suggested to implement the plantwide process RTO starting with ill-PGQ detection. Figure 19 presents the schematic of the PGQ-RTO implementing 10854



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PGQij is followed and tracked by adjusting FMij. Starting off with PGQ(g−1)j, Rule 2 (SFGP driven adjustment) is stepwise applied at a decreased index i.

procedures in one iteration interval, including ill-PGQ detection, PGQ follow-up as well as PGQ-objective achievement.

1

min

∑

1

∑

σij = min FMij

i=g−1

(Lij* − Lij)2

i=g−1

s.t. Lij = fij (FSij , FMij) FSij = L(i + 1)j Lij L ≤ Lij ≤ LijU FMij L ≤ FMij ≤ FMijU

(29)

(3). PGQ-Objective Achievement. On the basis of the structures of the PGQ-objective and plantwide PGQ, the process state variables involved in the PGQ-objective can be substituted by all LGPs with depth index 1. Thereafter, the optimum value of PGQ-objective, P*, can be achieved by applying rule 2 (SFGP-driven adjustment), i.e., solving the following optimization problem: min σ0 = min|P* − P| PMx

Figure 19. Schematic of PGQ-RTO algorithms.

s.t. P = ϕ(PSj , PM ) L1j = PSj

Initially, the detection of an ill-PGQ is implemented along with a judgment of whether its upper PGQ serving as the PGQobjective. In the presence of a PGQ-objective, the PGQobjective achievement algorithms would be launched; otherwise, the PGQ follow-up activities would be performed. If a stopping criterion is met, the algorithms terminate; otherwise, they go back to perform PGQ follow-up. The detailed procedures are presented as follows. (1). Ill-PGQ Detection. Once the PGQ-objective is changed, ill-PGQ detection would start along the ascending depth indexes of PGQs. If Lgi shifts but FSgi remains constant, PGQgj would be identified as the ill-PGQ. (2). PGQ Follow-up. On the basis of the relations among LGP, SFGP, and MFGP described in Figure 18, PGQ follow-up approaches start off with the upper layer of PGQgj. The optimization problem behind PGQij follow-up approaches is formulated by eq 29, in which, the previous optimum value of

PM L ≤ PM ≤ PM U

(30)

The actual solution end condition is given by the following: m

n

∑ ∑ σij 2 ≤ δ i=1 j=1

(31)

In summary, in order to implement the supervision-driven PGQ-RTO algorithms, a plantwide process should first be formulated as a plantwide PGQ, where corresponding process variables are grouped as ω0 = [PM] and ωij = [FMij]. For each PGQij (i = 1, 2, ..., (g − 1)), Lij*(k) and Lij(k) are created to formulate eq 22. After ill-PGQ detection, the detailed algorithm steps are launched as follows. Step 1. From the upper PGQ of the ill-PGQ, PGQ(g−1)j, to PGQ1j, update every PGQij with its current estimates

Figure 20. TE process. 10855



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Figure 21. Plantwide PGQ of the TE process.

ωij = [FMij] and use measurements {dij(k),uij,k} to obtain ψij,k. All PGQs exchange their pre-estimates ψij,k with neighbors. An estimate ωij,k can result in a new LGP, **, which will serve as the SFGP of the upper PGQ(i−1)j Lij,k according to eq 29. Step 2. From the PGQ with the largest depth index, g − 1, to the PGQ-objective, update ω with respect to the structure of the multilayer PGQs. Step 3. Obtain PS and optimize PGQ-objective based on the relationship of eq 30. Step 4. If a stopping criterion eq 31 is met, then terminate; otherwise, we go back to step 1.

of the purge and product streams, steam cost, and compressor power cost and are measured by eq 33. Therein, the objective function corresponds to the hourly operating cost (Ctot) in $/h which is aimed to be minimized. H

C tot = F 9

F

∑ Ci ,cstXi ,9 + F11 ∑ Ci ,cstXi ,11 + 0.0536Wcmp i=A i≠B

i=D

+ 0.0318Fsteam

(33)

Setting the last two items in (33) to the values around those reported by Ricker,24 the TE process could be reconfigured into a plantwide PGQ which includes two multilayer PGQs, shown in Figure 21. The optimization steps of the plantwide PGQ start with ill-PGQ detection, followed by tracking and updating the objective value of every individual PGQ. The detailed implementing procedures are listed as follows. (1) Ill-PGQ Detection. Assume that TE process operates under a certain optimum condition initially. At time t = 4 h, through process supervision operations, human operators decrease the reactor temperature from 122.9 to 120.4 °C by manipulating TCR. The trajectory of the reactor temperature is shown in Figure 22. As the LGP of PGQ21, FS11/L21 = [Ps] is affected by this step change. Therefore PGQ21 changes its optimum formation, eventually leading to the shift from t = 4 to

4. CASE STUDIES The TE (Tennessee Eastman) process23 shown in Figure 20 was proposed by Downs and Vogel (1993),24 which has been widely circulated in the literature as a case study due to its attractive challenging features. The TE process involves five major process units, including a two-phase reactor, a partial condenser, a separator, a stripper, and a compressor. Two products are created from four reactants, an inert component B and a byproduct F, denoted by a total of eight components, A, B, C, D, E, F, G, and H, respectively. Relevant reactions are listed as follows: A(g) + C(g) + D(g) → G(liq) A(g) + C(g) + E(g) → H(liq) A(g) + E(g) → F(liq) 3D(g) → F(liq)

(32)

There are 12 manipulated variables and 41 state variables involved in the process. Specifically, the manipulated variable vector FM contains 10 variables, FM = [F1, F2, F3, F4, F8, F9, F10, F11, TCR, TCS], where Fi is the molar flow rate of stream i [kmol/h] (i = 1, 2, ..., 11) and TCR and TCS are temperatures of the reactor and separator. The measurement noise is included in the simulation. The reactant and product losses are in terms

Figure 22. Reactor temperature caused by process supervision. 10856



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promptly collected between two time intervals and the new estimates of ω21,k = ω32,k = [TCS] are achieved by the PGQ enabling algorithm. The obtained L21(k) = L32(k) = Ps will serve as SFGPs of PGQ11 and PGQ22 as shown in Figure 26.

30 h, as shown in Figure 23. Successively, as the LGP of PGQ11 PS1/L11 = [Xi,9] is affected by the propagated information.

Figure 23. Separator pressure.

Figure 26. Trends of PGQ21 follow-up.

Instead of using all components, material balance of component “C” around PGQ11 has been used as a typical case. The trajectory of XC,9 from t = 4 to 30 h is shown in Figure 24. In

(3) PGQ11 Follow-up. PGQ11 receives FS(11, k) = Ps from PGQ21. The new estimates of ω11,k = [F10 F8] are obtained by adapting and exchanging their information using eq 20 by communication with PGQ22. The selected follow-up history of the LGP of PGQ11 (i.e., Xi,9, i = C) are shown in Figure 27. The resultant L11(k) = Xi,9 will serve as SFGP of PGQ0.

Figure 24. Profiles of XC,9.

this sense, PGQ31/PGQ42 is detected to be the ill-PGQ which successively caused the variations of PGQs and PGQ-objective traveling along the following paths: PGQ31 → PGQ21 → PGQ11 → P and PGQ42 → PGQ32 → PGQ22 → PGQ12 → P. As a result, the shifted economic objective values are shown in Figure 25. (2) PGQ21 Follow-up. In this phase, PGQ21 and PGQ32 represent the same process unit which tries to bring L21/L32 = [Ps] back to its previous operating points. The optimization process from t = 4 to 30 h is divided into small time intervals so as to the real-time measurements of PGQ21 and PGQ32 are

Figure 27. Trends of PGQ11 follow-up.

(4) PGQ-Objective Achievement. PGQ-Objective updates its value from Xi,9 and Xi,11 which is achieved in terms of PGQ11 and PGQ12, respectively. If a stopping criterion is met, then terminate; otherwise, we go back to the PGQ21 follow-up. The other width depth multilayer PGQ follow the same steps as stated above. The trends of PGQ-objective achievement are shown in Figure 28. The eventual optimum solutions are listed in Table 1, showing an achieved objective value of $116/h.

5. CONCLUSIONS Inspired by biologic nature of a flying goose queue, this paper has explicitly introduced the novel PGQ approaches together with the strategies able to deal with supervision-driven RTO for plantwide processes, showing the capabilities of overcoming the algorithmic deficiencies associated with conventional optimization approaches such as huge process models, enormous manipulated variables, and long waiting time for detecting steady states. Specifically, we particularly stressed on the PGQ foundations by providing basic relevant definitions as well as enabling algorithms. Extensive case studies on supervisiondriven RTO have been performed with PGQ approaches. It was

Figure 25. Profiles of the economic objective. 10857



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It should be pointed out that, the same with gradient methods, PGQ approaches are inevitably trapped into a local optimum when more than one extremum exists. In response, in depth theoretical investigations to avoid this problem are underway

■

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Tel.: 86-13436380711. Fax: 86-10-64442932. Notes

The authors declare no competing financial interest.

Figure 28. PGQ-Objective achievement.

■

Table 1. Final Optimum Solutions name of PGQ PGQ -objective

PGQ11 PGQ21 PGQ31 PGQ41 PGQ12 PGQ22 PGQ32 PGQ42 PGQ52

SFGP/LGP PS1 **/L11 ** = [Xi,9(i=A,C,D,E,F,G,H) = 31.07, 12.24, 1.30, 20.00, 3.94, 5.61, 2.70] PS2 **/L12 ** = [Xi,11(i=D,E,F) = 0.02, 0.92, 0.19] FS11 **/L21 ** = [Ps = 2702.33, Ts = 86.74] F*S21*/L*31* = [TR = 120.40] FS31 **/L41 ** = [PR =2797.96, LR = 61.25] FS41 ** = [0] F*S12*/L*22* = 61.67] F*S22*/L*32* = 86.74 ] FS32 **/L42 ** = FS42 **/L52 ** = 61.25] F*S52* = [0]

[Pstr = 3329.99, Tstr = [PS = 2702.33, TS = [TR = 120.40] [PR =2797.96, LR =

MFGP PM3 ** = [F9 = 26.75, F11 = 45.99]

FM11 ** F8 F*M21 * FM31 **

= = = =

[F10 = 36.73, 0] [TCS = 14.46] [TCR = 35.54]

FM41 ** = [F1 = 26.44, F2 = 62.42, F3 = 52.33] F*M12 * = [F4 = 59.93] F*M22 * F8 FM32 ** FM42 **

= = = =

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[F10 = 36.73, 0] [TCS = 14.46] [TCR = 35.54]

F*M52 * = [F1 = 26.44, F2 = 62.42, F3 = 52.33]

reported that an optimization problem associated with the TE process had been successively solved with an NLP solver (OptControlCentre)25 which uses IPOPT. The calculation time for the set point change studies is 20 CPU s. Duvall and Riggs26 have solved an RTO problem of the TE process via SQP, where the real-time optimizer takes about 5 min to converge. Instead, it only takes 11.8 CPU s for PGQ to complete a supervision-driven RTO routine. These manifestations allegedly demonstrate the following benefits of the contribution: (1) In contrast to conventional process optimization methods which usually optimize a complex objective function involving enormous manipulated variables, PGQ approaches decompose an optimization problem into several PGQ follow-up implementations, which contribute to considerable computational simplicity. (2) It is found that the follow-up of PGQs as well as PGQobjective achievement could launch independently, which makes the options for appropriate optimization algorithms more flexible. (3) The PGQ-based RTO approach is able to utilize more real-time measurements which are beneficial for process model updates. Additionally, with characteristics of evolutionary algorithms, it can achieve more satisfied economy performances in terms of average objective profits. 10858



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Process Goose Queue (PGQ) Approaches toward Plantwide Process

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