Integrated production scheduling and process control: A systematic review

Computers and Chemical Engineering 71 (2014) 377–390

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Computers and Chemical Engineering journal homepage: www.elsevier.com/locate/compchemeng

Review

Integrated production scheduling and process control: A systematic review Michael Baldea a,∗ , Iiro Harjunkoski b a b

McKetta Department of Chemical Engineering, The University of Texas at Austin, 200 East Dean Keeton St., Stop C0400, Austin, TX 78712, USA ABB AG, Corporate Research Germany, Wallstadter Straße 59, 68526 Ladenburg, Germany

a r t i c l e

a b s t r a c t

i n f o

Article history: Received 23 January 2014 Received in revised form 20 August 2014 Accepted 3 September 2014 Available online 16 September 2014 Keywords: Production scheduling Process control Integrated scheduling and control

Production scheduling and process control have the common aim of identifying (economically) optimal operational decisions, and it is reasonable to expect that significant economic benefits can be derived from their integration. Yet, the scheduling and control fields have evolved quite independently of each other, and efforts aimed at integrating these two decision-making activities are quite recent. In this paper, we review progress made thus far in this direction. We identify key elements of control and scheduling, and carry out a systematic investigation of their use as building blocks for the formulation and solution of the integrated scheduling/control problem. On the basis of our review, we define several necessary directions for future development as well as a complement of promising applications. © 2014 Elsevier Ltd. All rights reserved.

Contents 1. 2. 3.

4.

5.

6.

7. 8.

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Motivation and background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Prototype example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1. Problem statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2. Results and discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Key elements of production scheduling and process control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1. Production scheduling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2. Process control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The interaction of scheduling and control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1. The scheduling perspective on control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2. The control perspective on scheduling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3. Time horizons and time representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4. Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The integration of scheduling and control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1. Top-down integration approaches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1.1. Control-defined transition times . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1.2. Dynamic optimization-based scheduling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1.3. Time scale-bridging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2. Bottom-up approaches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3. State-space scheduling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Integrating scheduling and control for batch processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Development needs and avenues for future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

∗ Corresponding author. Tel.: +1 512 471 1281. E-mail address: [email protected] (M. Baldea). http://dx.doi.org/10.1016/j.compchemeng.2014.09.002 0098-1354/© 2014 Elsevier Ltd. All rights reserved.

378 378 379 379 381 381 381 382 383 383 383 383 383 384 384 384 385 385 386 386 386 386

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9.

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8.1. Explicit vs. implicit controllers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2. Mathematical modeling and numerical methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3. Stability and feasibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4. Extension to batch processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.5. Test cases and applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.6. Organizational challenges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1. Introduction Increasingly dynamic market conditions have spurred, over the past decades, significant efforts in operations research, resulting in improvements in tactical and strategic decision-making (i.e., production scheduling and capacity planning) in the chemical industry. Advances in numerical optimization algorithms and the decreasing cost of computer hardware have led to successful implementations of these concepts with clear economic benefits at the enterprise level (Grossmann, 2005). Concurrently, the design of chemical processes has undergone a shift towards process integration. Integrated processes make extensive use of material recycling and energy recovery to reduce raw material and utility demands and to lower operating costs (Westerberg, 2004; El-Halwagi, 2006). While bringing important economic benefits, process integration has a significant impact on process dynamics, giving rise to complex plant-wide interactions (Baldea and Daoutidis, 2012). This has coerced the design of process control systems to adopt a new, multivariable perspective, which accounts for interactions, for state and input constraints, and anticipates the effect of disturbances on the plant operation (Edgar, 2004); the most successful implementation of this new vista is Model Predictive Control (MPC) (Qin and Badgwell, 2003). Scheduling and plant-wide control have the common aim of identifying (economically) optimal operational decisions, and it is reasonable to expect that significant economic benefits can be derived from a tighter integration or a merger of scheduling and supervisory control. Furthermore, one could ideally expect this integration process to be seamless, given that both scheduling and control rely on similar approaches (i.e., the solution of optimization problems using a process model) to accomplish their goals, and that advances in communication technology allow the relevant information to be easily shared within an enterprise. However, due to the various challenges that arose along their development, scheduling and optimal control have, in fact, evolved and continue to evolve largely independently from each other. While an exciting (and apparently easy) undertaking for academic researchers and a potential “quick win” for practitioners, the nexus of scheduling and control faces several modeling, numerical and organizational challenges (Harjunkoski et al., 2009; Shobrys and White, 2002; Engell and Harjunkoski, 2012). The development of a comprehensive theoretical solution and of a transparent framework for the practical implementation of integrated scheduling and control remain important – and still open – problems. In this paper, we review progress made thus far in the integration of scheduling and control. With a main focus on continuous production processes, we determine the key elements of control and scheduling, and investigate their use as building blocks for the formulation and solution of the integrated scheduling/control problem. We introduce a tableau representation of these elements, which we use to create a systematic characterization of the approaches available in the literature. On the basis of our review, we identify several necessary directions

387 387 387 387 388 388 388 388 388

for future development as well as a complement of promising applications. 2. Motivation and background Scheduling and control systems compute and implement operating decisions over different time horizons, ranging, respectively, from several days/weeks to the order of a minute, the latter being the typical sampling and execution frequency for an MPC controller. Traditionally, the decision-making process is hierarchical (Fig. 1), with multivariable supervisory control providing the setpoints of the regulatory control layer, which in turn acts directly on the process. Scheduling calculations provide targets – possibly, but not necessarily, in the form of setpoints – for the supervisory control system. Why is the integration of scheduling and control necessary? In the current economic environment, chemical processes must respond to many more external factors than before. These include rapid changes in the types or products that are manufactured, as well as variations in market demand for these products. Dynamic market conditions are also reflected in fluctuations in raw material and energy prices. Of particular interest in the latter category is the interaction of process systems with the power grid, where they can provide much needed demand response capacity for bridging the gap between electricity consumption and generation rates (Paulus and Borggrefe, 2011; Soroush and Chmielewski, 2013). Under these circumstances, chemical plants must be operated in a dynamic fashion, and the time granularity of production schedules must increase (e.g., synchronizing the production of electricity-intensive processes with the operation of the grid may require that significant production rate changes be made every hour (Ierapetritou et al., 2002; Cao et al., submitted)). As a consequence, production management decisions are made over a time scale in which process dynamics and control become highly relevant. In this context, improving economic performance while meeting safety requirements and environmental mandates, requires exchanging

Fig. 1. The hierarchy of control decisions in the chemical enterprize. Adapted from Seborg et al. (2010).

M. Baldea, I. Harjunkoski / Computers and Chemical Engineering 71 (2014) 377–390

379

Table 1 Process parameters for exothermic CSTR (adapted from Davis and Thomson, 2000) value

q V E R UA VCp

1 h−1 104 K 1 h−1 1025 h−1 350 K 1 mol/l -200 Kl/mol

k0 Tf CA0

CA, T

H Cp

0.94

450

P

2

Tc

information and integrating decisions between the production management and control layers of the hierarchy in Fig. 1, in other words, the integration of scheduling and control.1 We believe that understanding the challenges and opportunities of integrating production scheduling and process control should be grounded in a thorough understanding of the individual scope and goals of these activities, which we will describe below. For the moment, we mainly consider continuous, rather than batch processes, and use continuous, rather than discrete-time formulations for both control and scheduling (a more in-depth discussion on the use of continuous and discrete time is provided later in the paper). Batch processes will be discussed further in the manuscript. For both activities, we introduce the key ingredients of the problem formulation, as well as the steps required for identifying a solution.2 We begin with an illustrative example, which we then use as a basis for providing a general formulation. 3. Prototype example 3.1. Problem statement We use a simple CSTR as a prototype system for discussing scheduling, control and approaches for their integration. The reactor (Fig. 2) carries out a reaction converting the feed A into component B. The model (adapted from Davis and Thomson, 2000) consists of a 2-state ordinary differential equation system: dCA q = (CA0 − CA ) − k0 e−EA /RT CA V dt

(1)

dT q 1 UA = (T0 − T ) − k0 e−EA /RT CA H − (T − Tc) V Cp VCp dt

(2)

where CA and T are the concentration and temperature, U is the heat transfer coefficient, A, the heat transfer area,  and Cp are the density and heat capacity of the reaction mixture, V is the reactor volume, q is the feed flow rate, CA0 is the inlet concentration, EA is the activation energy, k0 is the pre-exponential factor in the reaction rate expression, Tc is the coolant temperature and T0 is

1 As a corollary to this observation, integrating scheduling and control may not bring significant benefits for processes whose operation does not require frequent operating point changes and/or whose transition costs (the cost of changing operating point, measured, e.g., in terms of wasted product) do not depend on the production sequence. 2 A by now classic paper by Mayne et al. (2000) pursued a similar path for reviewing the theory of model predictive control, and we acknowledge the inspiration provided by this paper for defining the framework of the systematic analysis in this paper.

Temperature, K

Fig. 2. Continuous tank stirred reactor prototype.

0.92

P3

445

P1

0.9

440

0.88 0.86

435

Conversion

q, CA0

parameter

0.84 430

0.82 0.8

425 335

340

345 350 355 Coolant temperature, K

360

365

Fig. 3. Steady-state conditions in the CSTR prototype as a function of coolant temperature Tc . Vertical lines mark the operating points corresponding to products P1 . . . P3 . Solid lines denote stable steady states while unstable steady states are represented with dashed lines.

the feed temperature. The values of the parameters are provided in Table 1. The reactor can make three products: the low-grade P1 corresponding to 80% conversion (CA = 0.2 mol/l), a medium-grade product P2 (85% conversion, CA = 0.15 mol/l), and a premium product P3 (94% conversion, CA = 0.06 mol/l). The reactor conditions corresponding to each product are shown in Fig. 3. Transitions between product grades can be imposed by changing the coolant temperature Tc , which is the only available manipulated input. The demand for each product Pi , i ∈ {1, 2, 3} is given by ı = [10 30 10] m3 , the prices are  = [100 120 200] $/m3 , the inventory costs cstorage = [20 2 3] $/m3 /h, the feed flow rate (and production rate) is q = 1 m3 /h, the cost of raw material is 20 $/m3 , and the feed concentration CA0 = 1 mol/l. With these data, solving the scheduling problem consists of identifying the optimal sequence in which P1 . . . P3 should be made, that maximizes profit with the lowest makespan Tm . The scheduling problem can be posed mathematically as an optimization problem, typically following a slot-based paradigm: it is assumed that a time interval (“slot”) is assigned to each product (here we assume that there are three time slots, one for each product, but in the general case this number can be larger and a product can be made in two or more non-consecutive slots). Each slot comprises a transition period (where production is changed to the new product) and a production period (Fig. 4). The optimal schedule then aims to maximize a profit objective function of the form:

 1 J= Tm

3  i=1

i ωi −

3  i=1

cstorage,i ωi

3  s=1

 f zi,s (Tm − ts ) − qcrm Tm

(3)

380

M. Baldea, I. Harjunkoski / Computers and Chemical Engineering 71 (2014) 377–390

0.22 PID NLC

0.2

Concentration, mol/l

0.18 0.16 0.14 0.12 0.1 0.08 0.06 0.04 0

10

20

Fig. 4. Slot-based scheduling of continuous processes (Flores-Tlacuahuac and Grossmann, 2006)

ωi =

s=1

f

ts

(4)

zi,s q dt

tss +i i f

tss is the start time of the slot s where product i is made, ts is the end time of the same slot, and i i is the transition time between the product in slot s − 1 and the product i made in slot s. The time points must satisfy the precedence relations f

ts > tss + i i f

tss = ts−1

∀s > 1

∀s = / 1

f

t3 ≤ Tm

(5a)

(5c)

zi,s = 1 ∀i

(5d)





Tsp = 350 + Kc,2

(5e)

which ensure that product i is made once and only once in the three available time slots. The problem formulation must also include constraints related to meeting product demand, i.e.,

∀i

(5f)

Thus, identifying the optimal schedule consists of computing the (continuous) duration of each of the three time slots as well as the (binary) values of zi,k , that maximize J in (3) under constraints (5a)-(5f). However, this problem is not fully defined unless the transition times ii in (5a) and (4) are known. Their values depend on the dynamics of the system, and can, in principle, be computed by carrying out step tests for all possible transitions between products. This approach is, evidently, time consuming. Moreover, for the example under consideration, such tests cannot be carried out for products P1 and P2 because the corresponding steady states are unstable.





t

Tsp − Tds

(6a)

0

1 (CA,sp − CA ) + I,2





t

CA,sp − CA ds

(6b)

0

with Kc,1 = −55 K/mol/l and  I,1 = 0.2 h. For comparison purposes, we also designed a nonlinear inputoutput linearizing controller (Daoutidis and Kravaris, 1992): 2 Tc = (CL Lg Lf CA )

−1

2 2 (CA,sp − CA − 2CL Lf CA − CL Lf CA )

with

q (CA0 − CA ) − k0 e−EA /RT CA V

⎢

f=⎣

and g=



0 UA − VCp

(7)

⎤ ⎥

q 1 UA ⎦ (T0 − T ) − k0 e−EA /RT CA H − T V Cp VCp

i=1

ωi ≥ıi

70

with Kc,1 = 15 K/K and  I,1 = 0.9 h. Then, the outer loop manipulates the temperature setpoint Tsp to track the desired composition setpoint, CA,sp , corresponding to each product.

3

zi,s = 1 ∀s

1 (Tsp − T ) + I,1

Tc = 350 + Kc,1

⎡

s=1



60

This suggests that the control problem for this reactor must be addressed prior to solving the scheduling problem. The control objectives are to achieve stable operation over the entire operating envelope, as well as fast response to changes in the composition setpoint. We first address these objectives by designing a linear control structure using two cascaded proportional-integral (PI) controllers. The inner loop of the control structure aims to stabilize the reactor temperature by manipulating the coolant temperature:

(5b)

which require that a time slot be longer than the corresponding transition time, impose the coincidence of the end time of one time slot with the start time of the subsequent time slot and define the relationship between the end time of the last time slot and the total cycle time. Products are assigned to each slot using a set of binary variables, zi,s ∈ {0, 1}, along with constraints of the form 3 

50

Fig. 5. Composition setpoint tracking performance for linear and nonlinear control structures

where ωi stands for the quantity of product i manufactured, 3  

30 40 time, h

(8)

 (9)

where Lf h(x) represents the Lie (or directional) derivative of function h along function f, i.e., Lf h(x) = ∂h f(x). The controller imposes ∂x a critically-damped second-order closed-loop behavior of the form: 2 CL

d2 CA dt

2

+ 2CL

dC A + 1 = CA,sp dt

(10)

with  CL = 0.3 h. We simulated both control structures in the presence of composition setpoint changes that occur every 10 h, such that all possible transitions between the products are considered. The simulation results are shown in Fig. 5 and the corresponding manipulated input profiles are shown in Fig. 6. Both controllers yield excellent

440

Product

M. Baldea, I. Harjunkoski / Computers and Chemical Engineering 71 (2014) 377–390

PID NLC

420 Coolant temperature, K

400

3 2 1 0

10

20

381

30 40 time, h

50

60

70

380 Fig. 7. Optimal schedule for the nonlinear control structure.

360 340 320 300 280 260 0

10

20

30 40 time, h

50

60

70

Fig. 6. Evolution of manipulated input Tc for linear and nonlinear control structures

performance, reflected in the low values of the integral square error (ISE), computed as:



t

(CA,sp − CA )2 ds

ISE =

(11)

0

The linear control scheme has ISEPI = 0.0199 mol2 /l2 h, while the nonlinear controller yields ISENL = 0.0226 mol2 /l2 h. The numerical experiment represented in Fig. 5 then allows us to determine the transition (changeover) times, i i , for the closed-loop process, which we define as the time required for the composition CA to reach a value within 0.1% of the new target value after a setpoint change. These data can be captured in the form of two transition tables, for the linear

while the profit for the nonlinear controller was $3,337, with a makespan Tm = 58.9 h. The Gantt chart for the nonlinear case is shown in Fig. 7 (the production sequence is identical for the two control structures). The results show that, while the performance of the linear control structure is marginally better (in terms of ISE) than that of the nonlinear controller, this performance advantage is not reflected in the process economics. Using the nonlinear controller to carry out the desired product transitions results in a slightly higher profit and lower makespan. This can be attributed to the fact that the controller accounts for the nonlinearity of the process and leads to overall shorter and more consistent changeover times between products (as opposed to the high transition time variability that comes with the linear control scheme). Moreover, Fig. 6 shows that the performance of the linear control scheme is achieved using abrupt changes in the manipulated variable Tc . While not directly accounted for in this example, such aggressive control moves typically carry a cost penalty, which would likely further erode profit. In summary, the findings from this simple example suggest that controller design should be – at the very least – accounted for in decisions regarding production scheduling, and provide an incentive for integrating the two activities.

4. Key elements of production scheduling and process control

(12) and, respectively, nonlinear controllers:

The previous example provided some insight concerning the formulation and solution of the scheduling and control problems for a process system. In this section, we systematize this information and delineate key elements for each problem, which we will then use to define and classify available approaches for integrating scheduling and control.

4.1. Production scheduling

(13) Owing to the behavior of the controllers and to the nonlinearity of / i i . the system, ii = 3.2. Results and discussion With these data, the scheduling problem consisting of objective (3) under constraints (5a)-(5f) can now be solved. We implemented the corresponding equations in GAMS (Rosenthal, 2014) and solved the problem using DICOPT (DIC, 2014) as the MINLP solver (with CPLEX (CPL, 2014) as the MIP solver and CONOPT (CON, 2014) as the nonlinear solver). A setup time of 4 h was allowed for the first slot in each case. Using the linear controller yielded a total profit of $3,292 for the production campaign, with a makespan Tm = 59.8 h,

For a production facility that is capable of manufacturing multiple products (which, in the chemical industry, could contain different chemical compounds or different grades/purities of the same component), production scheduling aims to identify the order (sequence) in which the products should be manufactured, the assignment of tasks to equipment and exact timing of the operations, which maximize profit while meeting the demand (quantity) of each product in a given time frame. Here, we consider an illustrative generalization of the example in the previous section, focusing on a single facility manufacturing NP products; we assume that Ns ≥ NP production slots are available. They key elements of scheduling (ES) using a (cyclic) slot-based approach required to formulate and solve this problem as a MINLP are:

ES1 Objective function. The objective function to be maximized (e.g., in the form of Eq. (3)) captures the cost of raw materials, profit from each product, inventory costs as well as makespan:

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 1 J= Tm

NP 

NP 

i ωi −

Ns 

cstorage,i ωi

i=1

i=1

zi,s (Tm −

f ts )

−

s=1

Ns 

 f qs crm (ts

−

tss )

s=1

(14)

where the binary variables zi,s define the assignment of product i ∈ {1, . . ., NP } to slot s. Product i has inventory cost cstorage,i , price i and production quantity ωi . The production rate in f slot s is qs , the cost of raw materials is crm , and tss and ts are, respectively, the start and end times of slot s. The makespan is Tm . ES2 Production constraints. The production constraints are of the form of Eq. (4), and ensure that product demand ıi is met and production capacity is not exceeded:

∀i

ωi ≥ıi ωi =

Ns 

(15) p

∀i

zi,s ts qs

(16)

s=1 p

ES4 Transition times. The transition time in (4.1) reflects the time required for the process to switch between the products i and i meant to be made in successive time slots. Conventional formulations capture these data in the form of static transition tables (such as those in (12)-(13)), which reflect all possible state transitions. Further information on specific formulation for each of ES1-ES4 can be found in several excellent overview papers on scheduling (e.g., Floudas and Lin, 2004; Méndez et al., 2006a; Maravelias and Sung, 2009; Maravelias, 2012; Harjunkoski et al., 2014). Identifying the optimal schedule consists of computing the (continuous) duration of each of the time slots, the production rate and makespan, as well as the (binary) values of zi,s , that maximize Jsched . As we have shown in the example in Section 3, the scheduling problem can be cast as a mixed integer (non)linear program (MI(N)LP), which can subsequently implemented in specialized modeling environments (e.g., GAMS, AIMMS, AMPL) and solved using existing solvers.

where ts is the production time in slot s. Moreover, the production of product i is completed by due date Td,i :

4.2. Process control

Ns 

Consider a process system such as the one in Section 3, whose states x ∈ Dx ⊂ Rnx can be measured or reconstructed from available measurements and can be influenced by a set of manipulated variables (“control handles”) u ∈ Du ⊂ Rnu . The goal of process control is to compute the time dependent values of u, which ensure that the process outputs y ∈ Dy ⊂ Rny (which are defined as a linear or nonlinear function of the states x) remain at or close to their desired target values (setpoints), ysp ∈ Dy ⊂ Rny , in the presence of (potentially measurable) disturbances d ∈ Dd ⊂ Rnd . Thus, the key elements of process control (EC) are:

f ts zi,s

≤ Td,i

∀i

(17)

s=1

Further constraints are imposed on product assignment, e.g., • exactly one product i is manufactured in time slot s: NP 

zi,s = 1 ∀s

(18a)

i=1

• product i is manufactured at least once within the production cycle

 Ns

zi,s ≥1 ∀i

(18b)

s=1

or, product i is manufactured only once within the production cycle Ns 

zi,s = 1 ∀i

(18c)

s=1

EC1 Control law. The control law, 0 = (y, ˜ ysp , u, d)

(21)

is used to calculate u(t) as a function of the discrepancy between y and ysp and/or the values of the disturbances d; the results of these calculations are termed, respectively, feedback and feedforward control actions. EC2 Process model. Supervisory control systems of the type represented in Fig. 1 are typically model-based, i.e., they rely on a process model of the form

Moreover, the production rate may be modulated such that:

x˙ = f(x, u, d)

(22a)

qmin ≤ qs ≤ qmax

y = h(x)

(22b)

(19)

with qmin and ≤qmax denoting, respectively, the minimum and maximum possible production rates. ES3 Sequencing and timing relations. Timing relations establish the order in which the products are made and the nature of transitions between products. For example, • start time of slot s + 1 is the same as the end time of slot s f

tss = ts−1

∀s = / 1

(20a)

• length of slot s accounts for transition time from slot s − 1 and production time of product i f

ts = tss +

 Np Np   i=1



p

zi ,s−1 zi,s i i + zi,s ts

∀s

(20b)

i =1

where i i is the transition time from product i in slot s − 1 to product i made in slot s. • end time of last slot is less than allowable makespan f

tNs ≤ Tm

(20c)

with f : Dx × Du × Dd → Dx describing the process dynamics and h : Dx → Dy . The model (22a)-(22b) is used together with the control law (21) to obtain u(t). EC3 Operational constraints. Safety or physical limits on the values and/or rates of change of the input variables u, states x and outputs y are captured by constraints of the form: g(x, y, u) ≤ 0

(23)

For the purpose of this paper, it will be useful to further classify EC1 as follows: EC1A Implicit control laws. The control law (21) is formulated in the most general case, that is, as an implicit function of the process outputs, setpoint, disturbances and the value of the manipulated variables u. Optimization-based controllers such as Model Predictive Control (MPC) fall within this category. The formulation of the objective function of an MPC controller, below, ensures that the outputs y track the desired

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setpoints ysp , subject to the system dynamics (22a)-(22b) and additional constraints (23) on the states and inputs.



s.t.

important, and such events should trigger appropriate (scheduling) model update and rescheduling procedures.

t0 +TP

min J = u

383

u R + ysp − y(t) Q dt

(24)

5.2. The control perspective on scheduling

t0

x˙ = f(x, u, d)

(24a)

y = h(x)

(24b)

x, y, u ∈ Dx,y,u

(24c)

where a S = aT Sa, and matrices R and Q are positive semidefinite and, respectively, positive definite, and are both symmetric, and TP is the prediction horizon. EC1B Explicit control laws. Certain control formulations allow for a closed-form expression u = (y, ysp , d)

(24)

to be derived for the manipulated variables u. Conventional linear controllers such as proportional-integral (PI), proportional-integral-derivative (PID) controllers belong to this category. Furthermore, explicit control laws can be obtained for nonlinear inversion-based control systems such as feedback linearizing control (e.g., Daoutidis and Kravaris, 1994) as well as for model predictive control via parametric programming (e.g., Bemporad et al., 2000, 2002). The derivation of PI and feedback linearizing control laws was demonstrated in the example in Section 3. 5. The interaction of scheduling and control Prior to identifying ways to integrate process scheduling and control, it is important to understand how these activities interact. To this end, it is perhaps most useful to consider each activity from the frame of reference of the other. 5.1. The scheduling perspective on control Section 4.1 reveals that at present the scheduling paradigm consists of making integer decisions (i.e., selecting product grades and/or production levels) concerning the operation of the process over a succession of production time slots. This approach makes, in effect, several (implicit) assumptions about and connections with process dynamics and control. Notice that in Equations (4.1), the transition time between different products is defined in terms of the constants i i . As illustrated in the example in Section 3, these terms are closely related to the process dynamics (as described, e.g., by the model (22a)-(22b)) and the control law (21)). A second underlying hypothesis of the static scheduling formulation relates to the process operation within each time slot. The product quantity is defined in terms of the production rate qs and the time that the process spends in the time slot s (see Equations (3) and (4)). In turn, this assumes that the process is actually making the desired product at the desired quality (i.e., it is “on spec”) for the entire production time in time slot s (or that this constraint is satisfied on the average over the time slot). Evidently, this may not always be the case owing to the presence of disturbances d. This emphasizes the importance of the control system (21) in enforcing process stability and rejecting these disturbances during process operations, and, consequently, in implementing the desired production schedule. This is clearly the case in the example presented in Section 3, where disturbances would immediately disrupt the production of products P1 and P2 if the system were operated in open-loop. The communication between the control and scheduling systems at the time when a relevant disturbance does occur is equally

The role of the control system in the hierarchy in Fig. 1 is to execute the schedule. In practical terms, this could entail defining the (time varying) setpoint values ysp (t) as a function of the optimum f

values of the decision variables tss , ts , qs , Tc and zi,s , and calculating the value of u(t) to track these setpoints as closely as possible in the presence of the disturbances d. This objective is, evidently, secondary to maintaining process stability and operational safety. Frequently (especially in the case of optimization-based controllers such as MPC), the control system is tasked with satisfying operational constraints, such as maintaining the manipulated inputs u and process state variables x within bounds prescribed by physical considerations. 5.3. Time horizons and time representations One of the salient differences between process scheduling and control lies in the different time horizons that they consider. On the one hand, scheduling calculations account for the entire cycle time (a time horizon that can span hours or days); on the other hand, control actions are executed on a much shorter time scale, spanning seconds or, at most, minutes. As a consequence, the prediction time horizon TP of the MPC controller in (24) is significantly shorter than the cycle time considered in scheduling, Tm . The representation of time constitutes another fundamental (yet often overlooked) difference between scheduling and control models and computations.3 Process control has historically used both continuous-time (e.g., ordinary differential equations) and discrete-time models (e.g., difference equations). Discrete-time models have proven to be advantageous for computational implementations, and, as a consequence, control algorithms such as MPC are largely formulated using sampled-time systems (although we note an increasing level of interest in continuous-time formulations Pannocchia et al. (2014). Continuous-time control models have continuous time, state and input variables. In discrete time systems, the state variables x (and – in the case of a control problem – the system inputs u) are evaluated at discrete time points and are thus discontinuous functions of time. Also, in the continuous time framework, an event can take place at any time instant. Conversely, an event can be construed to occur only at specific values of the discretized time variable (although this could introduce a certain amount of error in modeling and simulation, it is usually not a concern when the discretization of the time variable is sufficiently fine, i.e., when the distance between two consecutive discrete time points is much smaller than the time span considered for simulation or control). The same considerations concerning time representations apply to scheduling, where both continuous-time and discretetime formulations are available (see Floudas and Lin, 2004; Sundaramoorthy and Maravelias, 2011; Mouret et al., 2011 for more details). Here, however, regardless of the representation of time, the state and decision variables are discontinuous functions of time. 5.4. Objectives Another discrepancy between scheduling and control calculations is the nature of their objective functions. Clearly, the objective of scheduling (as shown in Eq. (14)) is to maximize profit (and,

3

We are grateful to the anonymous reviewer for bringing up this subtle point.

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Fig. 8. Integration of scheduling and control.

empirically, the function Jsched has a well-defined “dollar value”); system stability and constraints on manipulated and state variables are not directly accounted for. By contrast, the control objective (and the corresponding objective function, in case we refer to EC2A, model predictive control) is to ensure the stability of the system and track the setpoints ysp (although recent developments in model predictive control extend this functionality to optimizing process economics). In light of the above, from a control perspective (assuming that predictive control is used), accounting for scheduling considerations means, at the very minimum, extending the prediction horizon TP of the controller (24) which, in turn, increases computational complexity. Furthermore, the dynamics of modern integrated chemical plants tend to be highly nonlinear; as such, the models (Eq. (24a)) used to describe the process dynamics (which are linear in most current MPC implementations (Qin and Badgwell, 2003) may not accurately represent the process dynamics at all the operating states that constitute the product portfolio. Consequently, a major challenge arises from the potential need to account for process nonlinearities, and the associated transition from linear to nonlinear MPC. Conversely, from a scheduling perspective, the integration of scheduling and control entails departing from the current paradigm of making integer decisions (i.e., selecting product grades and/or production levels for a succession of production time slots) in a static manner, and accounting for process dynamics. In turn, this requires a transition from the typical mixed-integer programming (MIP) formulation of scheduling problems to a mixed integer dynamic optimization (MIDO) formulation, which significantly increases the complexity (and duration) of the scheduling calculations. Furthermore, the frequency of the scheduling calculations would need to be increased to the execution frequency of the control system, in order to provide closed-loop stability and disturbance rejection, performance characteristics that the control system is, in effect, relied upon.

Fig. 9. Integration tableau for scheduling and control.

the performance of the process and the original schedule cannot be met. Several early papers (see, e.g., Engell et al., 2000; Shobrys and White, 2002; Engell and Harjunkoski, 2012 for a review), have reported ad-hoc solutions and implementations for these concepts. These early results have provided the motivation for establishing a rigorous theoretical basis for this problem. We will review existing developments in this direction in the light of the “key ingredients” enumerated in the previous sections, emphasizing the benefits and challenges of each available approach. In each case, we introduce a summary representation in the form of an “integration tableau” which reflects the combination of the elements ES and EC described in Sections 4.1 and 4.2. We begin by illustrating this concept with the (admittedly uniformative) integration tableau for conventional, “static” scheduling (presented in Row 1 of Fig. 9). Next, the integration tableaus for different process control approaches (respectively, model predictive control, explicit model-based controllers and simple multi-loop control) are presented in the subsequent rows of the figure. Here, the class of explicit model-based controllers encompasses advanced control algorithms, e.g., input-output linearizing multivariable controllers (Daoutidis and Kravaris, 1994), that allow for the derivation of the (nonlinear, multivariable) control law in closed form. 6.1. Top-down integration approaches

6. The integration of scheduling and control The discussion presented above suggests that uniting scheduling with control entails performing calculations and making decisions simultaneously over multiple time scales. Furthermore, a complete integration should contain provisions for accounting and correcting for disturbances, in both scheduling and control. The elements and corresponding information flows for a potential integrated scheduling/control mechanism are shown in Fig. 8. Here, a (supervisory) control system ensures the stable operation of the process, while tracking the time-varying setpoints ysp (t) that are generated by the scheduling mechanism. Complete integration between scheduling and control is accomplished by i) providing the entire schedule to the control system, which relies on this information to improve transition performance between different operating states, and ii) relaying state information back to the scheduling mechanism, which uses these data to perform rescheduling calculations as necessary when disturbances d affect

Referring to the hierarchy in Fig. 1, we use the term “top-down approach” to characterize integration formulations that incorporate dynamics and control elements in a scheduling skeleton. 6.1.1. Control-defined transition times Initial efforts in top-down integration focused on using control concepts to provide a more rigorous characterization of the transition time required to switch product grade. Thus, scheduling ingredient ES4 is defined based on a typically explicit control law (EC1B), with the tableau representation shown in Fig. 10. Here, we note the work by Mahadevan et al. (2002), who used linear control concepts to investigate transition costs based on several metrics, and identifying “difficult” transitions that should be avoided when establishing the final production schedule. More recently, this idea has been extended by Chu and You (2012), who proposed using an off-line calculation to select the optimal tuning parameters for a set of PI controllers, which give the best transition

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Fig. 10. Integration tableau for control-defined transition times.

Fig. 11. Integration tableau for dynamic optimization-based scheduling.

performance for each possible transition in the process. Subsequently, these parameters are implemented online in conjunction with the optimal schedule by changing controller tuning “on the fly” according to the prescribed transition sequence. 6.1.2. Dynamic optimization-based scheduling Utilizing a full-scale dynamic model of the process is a natural means to integrate dynamic and control considerations in the scheduling calculation. The common denominator of works following this idea is the use of a scheduling-like objective (ES1) and constraints (ES2-ES3) along with a full dynamic model of the process (EC2) and the state and input constraints (EC3), resulting in an integration tableau of the form in Fig. 11. Consequently, the solution of the resulting mixed-integer dynamic optimization (MIDO) problem provides not only the optimal production sequence, but also the optimal control moves u(t) required to implement the schedule. Intuitively, this MIDO is large scale and computationally expensive. Solution approaches proposed in the literature fall into two main categories: Sequential solution methods are centered on reformulating the MIDO scheduling problem (using, e.g., orthogonal collocation on finite elements) as a large-scale mixed-integer nonlinear program (MINLP), whose solution yields both the optimal sequence of operating states y and the timevarying values of the manipulated variables u that ensure the optimal transition between them. This approach was proposed in the pioneering work by Flores-Tlacuahuac and Grossmann (2006), and subsequently extended by Terrazas-Moreno et al. (2007), Biegler and Zavala (2009), Zhuge and Ierapetritou (2012). Sequential solution methods are an alternative to the MINLP reformulation of the MIDO problem, in which the control and scheduling problems are solved in a cooperative fashion (Allgor and Barton, 1999; Chatzidoukas et al., 2003a; Nyström et al., 2005; Prata et al., 2008). From an implementation point of view, this involves alternating between the solution of an MLP master problem (to address scheduling) and a dynamic optimization dual problem (which optimizes the transitions between steady-states), until an optimality criterion is met. Dynamic optimization-based scheduling has the advantage of generating a (possibly globally) optimal scheduling and control solution that incorporates detailed economic and dynamic information about the process. However, it relies on calculating the entire schedule and associated control moves off-line, and thus is at a disadvantage the process operation is subject to disturbances, both from a scheduling point of view (e.g., unit breakdowns)

385

Fig. 12. Time scale-bridging model captures the closed-loop input-output dynamics of the process and its control system (Park et al., 2014).

and from a control point of view (e.g., fluctuations in feed quality). Broadly speaking, top-down approaches are equivalent to relying on open-loop control, and possess no explicit feedback structure for ensuring the stability and performance of the process outside of the nominal operating conditions (Mitra et al., 2010; Zhuge and Ierapetritou, 2012). Extending the work of FloresTlacuahuac and Grossmann (2006), Zhuge and Ierapetritou (2012) have recently proposed a rescheduling mechanism, based recomputing the schedule and corresponding control moves over the remaining cycle time, once a disturbance (defined based on the discrepancy between the target and actual process states exceeding a predefined threshold) has occurred. Decomposition-based approaches aimed at reducing the computational burden are discussed in (Terrazas-Moreno et al., 2008; Chu and You, 2012). Another approach for easing the computational difficulties related to the solution of the large-scale MIDO representation of the integrated problem was proposed in (Chu and You, 2013), who used a Benders decomposition based on the observation that, for a singlemachine, multi-product system, the dynamic optimization can be solved separately in each time slot (Fig. 12). Further discussion in this area pertains to the definition of the transition times (ES4). Flores-Tlacuahuac and Grossmann (2006) determined ES4 using an iterative heuristic approach based on obtaining sufficiently smooth transition profiles. A more natural definition, based on the time (measured from the start of the transition) required for the process output parameters to be within certain desired quality bounds, was developed in (Terrazas-Moreno et al., 2007). 6.1.3. Time scale-bridging A recent development introduced by Park et al. (2014) (also see (Du et al., 2014)), time scale-bridging draws upon the ideas used in multi-scale simulation. There, the computational difficulties associated with simulating models that span multiple length scales and time horizons are often dealt with by building a low-order representation of important features and behaviors over a smaller length scale/faster time scale (known as a “bridging model”), which is then incorporated in a higher-level simulation model. Similarly, time scale-bridging for the integration of scheduling and control relies on using a representation of the dynamics of the process and its control system (i.e., a model of the closedloop behavior describing the dependence of the process outputs y to the setpoints ysp ) in the control calculation. For example, Eq. (10), describing the closed-loop behavior of the example reactor in Section 3 could be used as time scale-bridging model. More generally, theoretical developments (e.g., Kumar and Daoutidis, 2002; Baldea and Touretzky, 2013; Baldea and Daoutidis, 2013) and numerous case studies (e.g., Baldea and Daoutidis, 2012) have shown that the input-output dynamic behavior of process systems is quite slow, and evolves over a time scale that is relevant to scheduling calculations and, moreover, that it can be described using a low dimensional model. In light of the above, scheduling using time scale-bridging models proceeds in a similar manner with dynamic-optimization-based scheduling described in the previous subsection, with the exception that the dynamic model of the process is replaced by the low-order

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Fig. 13. Integration tableau for scheduling using a time scale-bridging model.

scale-bridging model, and the scheduling calculation will produce a time-varying setpoint signal ysp (t) for the controller, rather than the equivalent trajectories of the system inputs u. The corresponding integration tableau is shown in Fig. 13. The time scale-bridging approach described here presents two significant advantages: first, (off-line) MIDO scheduling calculations are significantly facilitated by replacing the (likely large-scale) dynamic model of the process with a lower dimensional system of equations, and, second, the system remains under feedback control during normal operations. We note that extensive research efforts have been dedicated on time scale-bridging in the context of integrating production planning and scheduling (e.g., the use of “big-bucket,” discrete time grids for planning activities, linked via appropriate constraints to “small-bucket” continuous time representations for scheduling (Maravelias and Sung, 2009)). These ideas can potentially be translated into time scale-bridging approaches for integrating scheduling and control, provided that the feedback control and real-time execution requirements outlined above can be met. 6.2. Bottom-up approaches The process control community appears to favor the converse “bottom-up,” perspective to the integration of scheduling and process control. Specifically, process control researchers propose incorporating economic considerations in the design of plant-wide control systems, either at the level of the distributed control system (Skogestad, 2004) or as part of the supervisory controller (Kadam and Marquardt, 2007; Engell, 2007, 2009). A notable advance in this direction is economic model predictive control (EMPC) (Engell, 2009; Amrit et al., 2011; Heidarinejad et al., 2012), where the tracking objective function of conventional MPC represented in is replaced with an economics-based (nonlinear) objective Eq. (24)  t +T Je = t 0 P (x, u)dt. We direct the readers to the recent review 0

paper by Ellis et al. (2014) for an in-depth overview of EMPC. While not explicitly accounting for scheduling (most published developments concern linear systems with continuous variables – which are potentially ill-suited for accounting for multiple operating steady states), the EMPC framework has implicitly introduced the necessary feedback mechanisms for ensuring closed-loop stability and economic optimality in the presence of disturbances. Noting that hybrid systems formulations that include discrete decision variables – required to deal with scheduling problems – have thus far received very little consideration in the EMPC context (Ellis et al., 2014), we can still regard EMPC as holding potential for integrating scheduling and control, with the resulting structure having the tableau presented in Fig. 14.

Fig. 15. Tableau representation of state-space scheduling.

6.3. State-space scheduling In a different vein, we note the development of state-space (i.e., control-oriented) formulations of the scheduling problem (Subramanian et al., 2012). These formulations account for the presence of disturbances and the need to re-compute the control moves and the schedule upon their occurrence. The use of hybrid MPC for scheduling problems has been reported, e.g., by Gallestey et al. (2003), and multiparametric solutions to scheduling have also been discussed by several authors (e.g., Poncet and Stothert, 2006; Kopanos and Pistikopoulos, 2014). While state-space scheduling does not, strictly speaking, constitute an integration of scheduling and control in the sense of this paper, it does lend itself to a tableau representation as presented in Fig. 15. The objective function in the optimal control formulation (EC1) may in effect be replaced by the scheduling objective (ES1). 7. Integrating scheduling and control for batch processes Extensive efforts have been expended on developing models and algorithms for batch process scheduling, as well as on the dynamic optimization of batch operations. Comparatively, the body of theory and applications for integration of scheduling and control for batch systems appears to be very limited. An early contribution by Bhatia and Biegler (1996) demonstrated the economic benefits that stem from accounting for dynamics at the batch scheduling stage. Computational difficulties appear to have slowed subsequent progress until recently. Nie et al. (2012) present a modeling framework and solution algorithm for integrating scheduling with dynamic optimization of batch processes, reformulating the associated mixed-logical dynamic optimization as a mixed-integer nonlinear program using a Big M approach and orthogonal collocation on finite elements. The reference trajectories for the unit operations in the process are obtained together with the optimal schedule. We note that this approach parallels to some extent the developments described in Section 6.1.2 in the context of continuous processes. Similar developments are reported in Capón-García et al. (2013). Working along the same lines but within the limited scope of sequential batch processes, Chu and You (2014) also introduce a rescheduling framework based on re-solving the scheduling problem, when needed, over a time horizon that may be shorter than the remaining makespan. A different vista on the use of moving horizon techniques was recently proposed by Zhuge and Ierapetritou (2014), who discuss the use of multiparametric MPC controllers to facilitate the integration of scheduling and control for batch processes. The proposed approach relies on using the constraints arising from the parameterized MPC controller in the formulation and solution of the integrated scheduling and control problem. The authors also deal with the computational challenges of solving the integrated problem by proposing linear approximations of the process dynamics and the nonlinear objective function. 8. Development needs and avenues for future work

Fig. 14. Integration tableau of economic model predictive control (EMPC).

Realizing the promise of integrating scheduling and control requires reconciling a tight coupling (potentially, solving as a single problem) of the control layer with the scheduling layer (and the

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ensuing time scale multiplicity issues), with ensuring closed-loop stability, robust performance and real-time solvability. In this section, we highlight a set of development directions that should be pursued to this end. The choice of a top-down or a bottom-up approach (in the sense defined above) for integrating scheduling and control, remains a central research question. We believe that this question is, in effect, open ended. Given the plethora of scheduling formulations and control algorithms available, it is unlikely that a single solution will be found for integrating scheduling and control. Rather, the discussion refers to the choice of a top-down or bottom-up starting point in the quest for a solution, and the discussion below is intended to be, except as specified, agnostic to this starting point.

8.1. Explicit vs. implicit controllers As we have highlighted above, top-down integration approaches have considered largely explicit control laws. However, the vast majority of industrial controllers are now of the model predictive (i.e., implicit) type (Qin and Badgwell, 2003). Thus, it is imperative that future work address the use of MPC in an integrated scheduling/control framework. Here we note two recent development directions in MPC. First, the work in decentralized, cooperative MPC (Rawlings and Stewart, 2008; Liu et al., 2009), which relies on several, interacting MPC controllers for the supervisory control of large plants. Creating a tighter integration between such control systems and plant-level scheduling is of elevated interest, especially in view of the additional flexibility provided by the partial dynamic decoupling of the plant sections via controller cooperation. The integration of distributed MPC and production scheduling could have significant benefits in terms of lowering computational loads and/or allowing the associated optimization calculations to be carried out in parallel. Second, economic MPC (EMPC) is, as mentioned above, in itself an attempt at accounting for economic considerations in the process of making control decisions. Thus, EMPC appears to be an excellent starting point for future bottom-up approaches to integrating scheduling and control. Here, key aims for future research include a comprehensive framework for dealing with nonlinear systems, extending the prediction horizon to timescales relevant for scheduling and incorporating discrete or integer decisions in the formulation of the control problem.

8.2. Mathematical modeling and numerical methods Complementary to these topics are issues related to modeling and numerical solution methods. Intuitively, the integrated control/scheduling formulation will require the use of nonlinear (rather than linear) process models, that are capable of accurately describing the process behavior at all the requisite operating points, as well as of accurately characterizing the dynamic transitions between these points. Performing online, real-time optimization calculations based on such models is typically difficult (a difficulty compounded by the presence of integer decisions). Further progress towards accurate model reduction methods and fast numerical solution algorithms is thus required. Additionally, the availability of an integrated development environment that facilitates model formulation and implementation, and the solution of the corresponding optimization problems, would greatly contribute to extending the practice of integrating modeling and control to non-expert users.

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8.3. Stability and feasibility Preserving the stability of a process under an integrated scheduling/control system is evidently of primary concern, and theoretical developments in this direction are necessary. The concept of stability has been central to systems and control theory research since its inception, and rigorous classifications and definitions of, and criteria for, stability are now available in standard texts (see, e.g., Khalil, 2002). From a scheduling perspective, the concept of stability is new (Subramanian et al., 2012). Using a state-space scheduling formulation, Subramanian et al. (2012) showed that a solution that is acceptable from a scheduling point of view (i.e., delivering the requisite products in the desired time frame) can be in effect dynamically unstable, with a product inventory (i.e., an extensive quantity) showing unbounded growth over the cycle time. This is reflective of a design flaw which makes the system unsuited for the demand structure, and highlights the need to account for the interaction of process design and operational capabilities when closing the loop in scheduling. From a control-theoretical perspective, the integration of scheduling and control can potentially be regarded as a composite, fast-slow control system, and extension of the developments available in this area (see, e.g., Khalil, 2002; Kokotovic et al., 1986) would appear as a fruitful starting point for stability analysis. However, available composite control theory (see, e.g., Khalil, 2002) places more stringent (i.e., asymptotic or exponential) stability requirements on the fast and slow subsystems, and further developments are required to establish these properties for scheduling algorithms. Secondly, the issue of performance under integrated scheduling and control should receive close attention. Intuitively, here we should consider the effect of disturbances that occur over multiple time scales: process disturbances occur over a short time horizon, and are typically dealt with by the control system. However, the bottom-up effect (e.g., a disturbance that leads to the production of off-spec product for an extended period of time) must be accounted for and effective rescheduling algorithms should be developed; the latter should also account for scheduling disturbances (e.g., changes in order quantity, unit breakdown, changes in task yields and task delays). Progress in this area will likely be enabled by state-space formulations to scheduling (e.g., Subramanian et al., 2012) but computational challenges still hinder real time-applications to the most general, monolithic case considered in this paper. Another problem comes from feasibility of integrated models. This is especially a concern in highly non-linear problems such as crude-oil blending, where finding a set of feasible setpoints for the control system is itself a challenge. Ensuring recursive feasibility may be handled by using deviation variables (Subramanian et al., 2012), yield variables (Kelly and Mann, 2003) or iterative corrective loops (Méndez et al., 2006b), where the optimization is performed off-line, only releasing solutions after feasibility has been proven. Providing a fully integrated scheduling and control scheme for this class of problems remains, however, a major challenge. 8.4. Extension to batch processes Most developments in the batch realm parallel the research available form continuous processes, by simultaneously considering the dynamic optimization of all batch units together with the scheduling problem. This approach is computationally costly and thus has limitations in practical applications. Batch processing is a major component of the chemical industry, and significant more effort should be dedicated to developing methods for integrating scheduling and control in this sector. Clearly, difficulties here are related to the specific modeling and control challenges of batch processes, including, e.g., their

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inherently non-stationary nature, the nonlinearity of their models, and the dynamic complexity that arises from the potential need for coordinate multiple units and stages that operate in parallel. As a consequence, it is very likely that most integrated scheduling/control approaches developed for continuous processes will no longer migrate directly to the batch processing realm. Along similar lines, the distribution of the optimization tasks between the scheduler and a local control system should receive further attention. Developments to date (e.g., Bhatia and Biegler, 1996; Chu and You, 2014) appear to favor transmitting state (e.g., temperature) setpoint trajectories from the scheduling mechanism to a local controller, rather than the corresponding time-varying values of the manipulated inputs. 8.5. Test cases and applications Two decades ago, the publication of a realistic large scale process model (the Tennessee Eastman challenge process proposed by Downs and Vogel (1993)) catalyzed significant developments in plant-wide control. We argue that the availability of a similar challenge problem would have an equally beneficial impact on the emerging field of control/scheduling integration. Interestingly, a fairly large quota of the publications reviewed above use a small number of prototype systems to develop case studies. Continuous polymerization processes dominate, with the reactor model of Congalidis et al. (1989) (later simplified under the assumption of isothermal operation by Mahadevan et al., 2002) receiving significant attention. More complex polymerization process models were considered by, e.g., Chatzidoukas et al. (2003b), Terrazas-Moreno et al. (2008), Prata et al. (2008) and Biegler and Zavala (2009). Yet (and in spite of the frequent use of the model of Congalidis et al., 1989), there is still no “yardstick” by which different control/scheduling integration approaches can be uniformly evaluated. The aforementioned control/scheduling integration challenge problem need not be limited to polymerization systems. Applications for integrating scheduling and control abound in the chemical and energy industries, including metal production, wastewater treatment and power generation (Engell and Harjunkoski, 2012). Also of interest is the interaction between the chemical industry and the grid, particularly in energy intensive applications such as air separation, where production and storage of cryogenic products can be scheduled as a means for balancing grid load (see, e.g., Cao et al., 2011; Miller et al., 2008; Pattison and Baldea, 2014). Commercial and residential buildings are another significant energy consumer that can benefit optimally integrating energy storage with temperature control (see, e.g., Oldewurtel et al., 2010; Ma et al., 2012b, 2012a; Mendoza-Serrano and Chmielewski, 2012; Henze, 2013; Touretzky and Baldea, 2013, 2014). Furthermore, integrated control/scheduling algorithms may find a beneficial application in power firming – mitigating the inherent variability of power generated from renewable energy sources (e.g., wind, solar) via energy storage (using, e.g., batteries).

respectively, business performance and stable and safe operation (Shobrys and White, 2002; Engell and Harjunkoski, 2012). According to Shobrys and White (2002), achieving a tighter coordination and integration of scheduling and process control requires breaking “organizational silos” and changes in human behavior. Thus, operators with higher skill levels will be needed to earn acceptance for more sophisticated (and likely more complex) operating “tools” (i.e., scheduling and control software). The situation in academia is very similar to that in industry: scheduling and control have been the domain of separate groups of researchers, and efforts to foster interactions between the two communities are quite recent. Here, we highlight the oneday joint process operations/process control meeting at the 2012 FOCAPO/CPC conference as a first-of-a-kind initiative relevant to this area. Further hope comes from the emergence of smart manufacturing, defined as “a design and operational paradigm involving the integration of measurement and actuation; environment, safety and environmental protection, regulatory control, high fidelity modeling, real-time optimization and modeling, and planning and scheduling.” (Edgar and Davis, 2009) Smart manufacturing initiatives are generally sponsored by industry-government-academia consortia (Davis et al., 2012, also, see, e.g., a recent list of such projects supported by the U.S. Department of Energy (U.S. Department of Energy, 2013)), and constitute a very propitious and fertile ground for research and applications focused on the integration of scheduling and control (Christofides et al., 2007; Edgar and Davis, 2009). 9. Conclusions In this paper we have reviewed literature contributions in the emerging field of integrating production scheduling and process control, with a focus on multi-product continuous processes. We adopted a systematic perspective based on identifying the key elements of both control and scheduling, and categorizing existing approaches from the perspective of the use of a (combination of) these elements. Based on our findings, we identified future development needs and pitfalls – both technical and related to human factors – as well as a complement of promising applications where the integration of scheduling and control can yield substantial economic benefits, turning today’s waste into tomorrow’s profit. It is our sincere hope that this material will serve as the basis for vigorous future developments in this exciting and economically interesting emerging field. Acknowledgements Financial support from ABB Corporate Research through the ABB RFP program is acknowledged with gratitude.

8.6. Organizational challenges

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