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Computers and Chemical Engineering 104 (2017) 271–282

Contents lists available at ScienceDirect

Computers and Chemical Engineering

j ourna l ho me pa g e: www.elsev ier .com/ locate /compchemeng

ombined model predictive control and scheduling with dominantime constant compensation

ogan D.R. Beala, Junho Parka, Damon Petersena, Sean Warnickb, John D. Hedengrena,∗

Department of Chemical Engineering, Brigham Young University, Provo, UT, USADepartment of Computer Science, Brigham Young University, Provo, UT, USA

r t i c l e i n f o

rticle history:eceived 10 March 2017eceived in revised form 21 April 2017ccepted 30 April 2017vailable online 5 May 2017

eywords:

a b s t r a c t

Linear model predictive control is extended to both control and optimize a product grade schedule. Theproposed methods are time-scaling of the linear dynamics based on throughput rates and grade-basedobjectives for product scheduling based on a mathematical program with complementarity constraints.The linear model is adjusted with a residence time approximation to time-scale the dynamics basedon throughput. Although nonlinear models directly account for changing dynamics, the model formis restricted to linear differential equations to enable fast online cycle times for large-scale and real-

chedulingodel predictive controlynamic pricingime scalingomplementarity constraints

time systems. This method of extending a linear time-invariant model for scheduling is designed formany advanced control applications that currently use linear models. Simultaneous product switchingand grade target management is demonstrated on a reactor benchmark application. The objective isa continuous form of discrete ranges for product targets and economic terms that maximize overallprofitability.

© 2017 Elsevier Ltd. All rights reserved.

. Introduction

Time-of-day energy pricing for electricity and natural gas pose ahallenge and opportunity for industrial scale manufacturing pro-esses. In many manufacturing processes in which Model Predictiveontrol (MPC) is well-established, such as downstream refiningnd petrochemicals, there is lost opportunity when advanced pro-ess control only operates at certain conditions but must be turnedff when unit production is lowered (Qin and Badgwell, 2003).

challenge with changing production rates is that the dynamicsf a process often change dramatically with throughput. Empiri-al models identified at high throughput rates are often inaccuratend lead to poor control performance at low production rates. Anpportunity with time-of-day pricing is to temporarily reduce theonsumption of energy-intensive processes for periods when thenergy costs are sufficiently high. During off-peak periods, the pro-uction rate is increased or more energy-intensive product gradesre produced to take advantage of low energy costs. With typi-

al daily cycles in energy costs, a cyclical operation forms that theriginal advanced control system may not be designed to follow.hen production targets are not set to maximize to production

∗ Corresponding author.E-mail address: john hedengren@byu.edu (J.D. Hedengren).

ttp://dx.doi.org/10.1016/j.compchemeng.2017.04.024098-1354/© 2017 Elsevier Ltd. All rights reserved.

constraints, MPC may switch to maximize energy efficiency or othersecondary objectives. Set point targets traditionally come from areal-time optimization (RTO) application that optimizes a steadystate operating point for the plant (Soderstrom et al., 2010).

Segregated control and scheduling structure is historically dueto computational factors that limit application complexity (Baldeaand Harjunkoski, 2014). As a result, the control and schedul-ing fields have grown independently and without coordination,leading to a loss of opportunity from combining the applications(Soderstrom et al., 2010; Backx et al., 2000). By combining processscheduling of set points with control, the inefficiencies of applica-tion layering are avoided. One such inefficiency that results fromapplication layering is the infeasibility on the control layer of indi-vidual solutions pass from the supervisory layer (Capón-Garcíaet al., 2013). Scheduling applications frequently do not considergrade transition times because of the large combinatorial look-uptable that would be required to consider all possible transitions.Additionally, objectives of individual solutions can oppose eachother. For example, the controller does not consider the most eco-nomical route to reach a target set point given by the scheduler orsteady-state optimizer (Harjunkoski et al., 2009).

The computational barriers for combined control and schedul-

ing are diminished with improved computer hardware andadaptation of algorithms to the hardware. Algorithmic barriers arebeing overcome with a number of key contributions that are open-

dx.doi.org/10.1016/j.compchemeng.2017.04.024http://www.sciencedirect.com/science/journal/00981354http://www.elsevier.com/locate/compchemenghttp://crossmark.crossref.org/dialog/?doi=10.1016/j.compchemeng.2017.04.024&domain=pdfmailto:john_hedengren@byu.edudx.doi.org/10.1016/j.compchemeng.2017.04.024

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72 L.D.R. Beal et al. / Computers and Ch

ng several fronts of development (Hedengren et al., 2014; Limat al., 2013). Hardware or network resources such as multi-core,loud-based, and graphics processing units (GPUs) provide accesso previously inaccessible computing power. However, advancedrchitectures such as GPUs for optimization impose some limi-ations on the type of problems that can be solved because thelgorithms have not yet been adapted to take full advantage of therchitecture Ma and Agrawal (2010).

Economic MPC (EMPC) (Ellis et al., 2014; Angeli et al., 2012)ses an objective function that maximizes a profit function ratherhan targeting a set point as in standard MPC. Including the profitunction directly in the MPC application ensures that decisions areirectly driven by economic considerations. The profit function alsorovides guidance on product scheduling, although work on EMPCp to this point has focused on single products. The drawback of thispproach is that EMPC generally requires a short time horizon suchhat the longer horizon required for scheduling constraints andbjectives cannot be met (Ellis et al., 2014). MPC for supply chainanagement (Subramanian et al., 2014) is an alternative strategy

hat extends the control horizon to schedule product movementhrough a distribution network.

Dynamic real-time optimization (DRTO) also has an economicbjective function but augments a steady state real-time opti-ization (RTO) with select differential equations that capture the

alient and dominant dynamics of a process (Pontes et al., 2015;arjunkoski et al., 2013; Biegler et al., 2015). One drawback of RTO

s that the process must be at steady-state (Kelly and Hedengren,013) to perform data reconciliation. RTO has traditionally beenpplied to processes that do not have grade transitions but areominated by changing economics, disturbances, and slow domi-ant dynamics. With dynamics included, DRTO can be solved more

requently than RTO applications and can be solved during periodsf transient disturbances, during startup, or during shutdown peri-ds. RTO calculations are typically performed every hour to everyay while DRTO optimizes the transition between steady-state con-itions. Like EMPC, DRTO does not manage multiple sequentialroduct campaigns as a scheduler.

Complete integration of scheduling and control requires anxtended prediction horizon to plan the production sequence asell as near-term control actions. Two integration approaches are

eferred to as top-down (add control and dynamics to a schedulingpplication) or bottom-up (add scheduling to a control algorithm)Baldea and Harjunkoski, 2014). An early top-down implementa-ion includes differential and algebraic equations in the schedulingpplication (Flores-Tlacuahuac and Grossmann, 2006). Anotherethod is the scale-bridging model (SBM) in which a simpli-

ed model of process dynamics is embedded in the schedulingpplication (Baldea et al., 2015, 2016; Du et al., 2015). A benefitf this method is disturbance rejection (Zhuge and Ierapetritou,012). Algorithms include Benders’ decomposition (Chu and You,013) for problems that have a large-scale structured form andinkelbach’s algorithm (Chu and You, 2012) for non-convex prob-

ems that require global optimization methods. Applications ofombined scheduling and control include batch processes (Capón-arcía et al., 2013; Nie et al., 2012), polymer reactors (Prata et al.,007), parallel Continuously Stirred Tank Reactors (CSTRs) (Flores-lacuahuac and Grossmann, 2010), and an electrical grid thatesponds to current and future price signals (Farhangi, 2010).

Variable electrical pricing incentivizes reduced consumptionuring peak hours (US Department of Energy, 2006). It is desir-ble to match generation to consumption, but the adoption of moreenewable energy requires producers and consumers to respond

o price signals (Safdarnejad et al., 2016, 2015a,b). Energy produc-rs may expose consumers to time-of-day pricing to discourageonsumption during peak hours (Deng et al., 2015). Schedulingperation of chemical processing (Feng et al., 2015; Beal et al.,

l Engineering 104 (2017) 271–282

2017), oil refining (Mendoza-Serrano and Chmielewski, 2013), andair separation (Huang et al., 2011) are some examples of industrialunits that can shed electrical load during peak hours, typically inthe middle of the day. Many cooling-limited processes also operatemore efficiently at night (Beal et al., 2017). Periodic constraints canbe used to optimize a typical daily cycle.

Prior work in scheduling and control integration has been cen-tered around slot-based, continuous-time scheduling formulations.The benefit of discrete-time formulation has been shown (Bealet al., 2017). However, the non-linear discrete-time formulationproved computationally difficult. The purpose of this work is torestrict the dynamic model to linear form while capturing bene-fits of the integration of scheduling and control. There is a largeinstalled base of advanced controls that utilize linear models Qinand Badgwell (2003). A unique aspect of this work is a time-scalingalgorithm that adjusts the linear dynamic model based on residencetime calculations with a theoretical foundation for first-order sys-tems. The time-scaling approximation is applied to higher order,finite impulse response models that are common in industrial prac-tice. These linear models are used in the combined scheduling andcontrol application.

2. Time-scaling with first-order systems

It is well known that linear MPC performance degrades withchanges in the actual process time constant or gain (Hedengrenand Eaton, 2017). This effect has been quantified for MPC wherethere is model mismatch in the time constant or gain of a first ordersystem. A simple example is where the actual system is describedby a single differential equation as �p(dy/dt) =− y + Kpu with a pro-cess time constant of �p = 1 and a process gain of Kp = 1. An MPC

controller with objective∑20

i=1|yi − 5| drives the response from aset point of 0 to 5. The controller model is similar to the processbut with variable model time constant of �m and gain Km in theequation �m(dy/dt) =− y + Kmu. Common industrial practice is thatacceptable MPC performance can be achieved with gain mismatchless than 30% (0.7 ≤ Km ≤ 1.3) and time constant mismatch less than50% (0.5 ≤ �m ≤ 1.5) (Hedengren and Eaton, 2017). The dominanttime constant for many industrial processes is characterized by thevolume (V) divided by the volumetric flowrate (q) as �p = (V/q). Theexplicit solution to the first order equation is given by Eq. (1).

y[k + 1] = exp(

−�t�p

)y[k] +

(1 − exp

(−�t

�p

))Kpu[k] (1)

where y is the output, u is the input, k is the discrete time step, andt is the time. The integrated sum of absolute errors is computedfor combinations of Km and �m, each between 0.1 and 5.0. The 3Dcontour plot shows the control performance over the range of timeconstant and gain mismatch (see Fig. 1a). A mismatch in the gain(x-axis) and time constant (y-axis) are plotted versus the error. Thevertical axis (z-axis) is the integrated absolute value of the objectivefunction for a set point change from 0 to 5. A lower sum of absoluteerrors equates to better control performance with a minimum atKm = 1 and �m = 1 (no model mismatch) as shown in Fig. 1b.

Especially poor performance occurs when the model has ahigher time-constant (slower) than the actual process. The model inMPC predicts that changes happen slower than are actually the case,leading to a controller that is more aggressive. This aggressivenesstranslates into overshoot of the set point or even instability. Like-wise, a model with a lower gain than the actual process also exhibitspoor control performance. The model predicts that larger changes

in the Manipulated Variable (MV) are required to drive the processto the new set point. In reality, a smaller adjustment is required andthe over-reaction of the controller leads to overshoot and possiblyinstability. A contour plot of the performance profile gives insight

L.D.R. Beal et al. / Computers and Chemical Engineering 104 (2017) 271–282 273

Fig. 1. Performance degradation of MPC with model mismatch.

Fig. 2. Contour plot of performance degradation of MPC. An objective below 7 is acceptable, between 7 and 30 is marginal, and above 30 (red line) is poor performance. (Fori to the

omt�afnap

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nterpretation of the references to color in this figure legend, the reader is referred

n the performance as shown in Fig. 2a. Fig. 2b shows the perfor-ance with the time-scaled model. The parallel contour lines show

hat there is no performance degradation of the controller whenm changes such as a production rate decrease or increase. Thebility of the MPC to function over all production rates is requiredor processes that respond to utility price or product demand sig-als. This simple example shows the increased effectiveness over

wider range of operating conditions with time-scaling while stillreserving the linear model.

This result nearly agrees with industry observations, shownith dashed box in Fig. 2. Another feature of this result is that

he combination of high mismatch in time-constant (�m) and lowismatch in gain (Km) combine to form a region of poor control

erformance. Acceptable or marginal performance is also possiblef both �m and Km are both too high or too low. If there is low mis-

atch in time-constant (�m) and high mismatch in gain (Km), theontroller degradation is manifest as sluggishness and only incre-ental moves in the MV but not instability.The time scaling approach adjusts either the controller cycle

ime or the discrete model time step based on the change in unithroughput q relative to the nominal throughput q̄. �̄p is the nom-nal time constant associated with q̄. The modified process timeonstant is �p = q × (�̄p/q̄) which now has a linear relationship to q.

web version of this article.)

If the process model is not easily adjusted, the cycle time �t of thecontroller is adjusted to �t × (q/q̄) to compensate for the chang-ing process dynamics. For first-order systems, this gives an exactrepresentation of the nonlinear dynamics without modifying theoriginal linear model.

3. Selective time-scaling

Multi-variate and higher-order systems may have certain MVto CV relationships that are known to scale with changing unitthroughput while others are invariant to throughput changes. Priorwork has focused on decomposition of fast and slow dynamicsZhang et al. (2014) or variable time-delay of measurements (Jiand Rawlings, 2015; Srinivasagupta et al., 2004). For systems withmultiple MVs and CVs, only the relationships that are sensitiveto throughput are scaled. These can be identified with a dynamicprocess simulator or else by repeating plant identification testsat low and high production rates. A method to scale higher order

systems is to transform the linear time-invariant (LTI) model intodiscrete form. In discrete form, the sampling time is scaled by (q̄/q)and resampled to preserve the overall model sampling time. Asan example of this time scaling approach, consider the 7th order

274 L.D.R. Beal et al. / Computers and Chemical Engineering 104 (2017) 271–282

F

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G

cactmMc(b

4

tsooiaCiva

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I(af

ig. 3. Time-scaling of a 7th order system when the feed rate is reduced to half.

ystem given by Eq. (2) as a transfer function in terms of Laplaceariable s.

(s) = CV(s)MV(s)

= 1.5(s2 + 0.6s + 1)(0.5s + 1)5

(2)

Suppose that the dynamics of this system depend on the produc-ion rate and that the feed rate to the unit is reduced to half of theate where the model is originally identified. When a time-scalingransformation of q/q̄ = 2 is applied, the new transfer function islso a 7th order system but with shifted dynamics. The steadytate gain of the transfer function is preserved with this methodf dynamic transformation. The resulting transfer function is Eq.3).

(s) = CV(s)MV(s)

= 1.5(4s7 + 21.2s6 + 47s5 + 57s4 + 42s3 + 20s2 + 6.2s + 1) (3)

The continuous transfer function is first converted to dis-rete form with a sufficiently small sampling time. For this case,

sampling time of 0.5 s is chosen for the continuous to dis-rete transformation. The discrete model sampling time is seto 2 s × 0.5 s = 1.0 s based on the time-scaling factor and then the

odel is resampled to 0.5 s to be consistent with other unscaledV/CV models. Continuous to discrete transformations and dis-

rete model resampling for LTI models are standard methodsSeborg et al., 2010) and are not repeated here for the sake ofrevity. A graphical demonstration of this method is shown in Fig. 3.

. Scheduling and control formulation

An innovation of this work is to combine scheduling and con-rol with linear models and quadratic objective functions for fastolution with Quadratic Programming (QP) Wang and Boyd (2010)r Nonlinear Programming (NLP) solvers. There are multiple meth-ds for formulating tiered price structures. The focus of this works on creating mathematical expressions that have continuous firstnd second derivatives and fit within the QP or a QP with Quadraticonstraint (QPQC) framework. In contrast, most modern schedul-

ng applications utilize integer variables, often as binary decisionariables. Including integers requires MILP or MINLP solvers, whichre significantly slower and more complex.

.1. MPCC steps for product pricing

The method uses a continuous formulation to logical decisions.

t uses a mathematical programs with complementarity constraintsMPCC) formulation to avoid binary variables that would otherwiseccommodate tiered pricing structures Powell et al. (2016). Thisormulation uses a step function MPCC, using constraints v0y = 0

Fig. 4. Individual MPCC step functions are combined to create a continuous differ-entiable expression of switching conditions.

and v1(1 − y) = 0. v0 and v1 are positive slack variables. The com-plementarity constraints force variables to their bounds – resultingin y being a binary variable at its bounds [0, 1].

This condition can be difficult for solvers to find a solution sothe condition is typically either solved as an equivalent inequality(v0y ≤ 0), a relaxed inequality (v0y ≤ �), or included in the objec-tive function (min v0y) Baumrucker et al. (2008). The step functionMPCC that turns y from 0 to 1 at switching point xp is shown in Eq.(4a)–(4d).

minv0,v1,yv0y + v1(1 − y) (4a)x − xp = v1 − v0 (4b)v1, v0 ≥ 0 (4c)0 ≤ y ≤ 1 (4d)

To preserve the QP structure, the complementarity constraintsare included in the objective function. If the complementarity con-straints are included as inequalities a QPQC or NLP solver must beused. With sufficient weighting, the complementarity constraintsin the objective function are zero at a final optimal solution butnot necessarily along the search path to the solution. Being zeroat the solution, the complementarity constraints do not influenceother objective terms such as minimizing energy consumption ormaximizing profit. The MPCC switching conditions are combinedto create a tiered product pricing structure as shown in Fig. 4.

Each product has a different value and potentially a differ-ent width of specification limits. The positive and negative stepfunctions at different switching points are combined with pricinginformation to create an objective function with multiple products.The individual steps are shown in the upper subplot of Fig. 4 whilethe summation of all steps is shown in the lower subplot. ProductA ranges from 0 to 1 and has a price of 1 per unit production. Prod-uct B ranges from 2 to 3 and has a price of 2 per unit production.Product C ranges from 2 to 3 and has the highest price of 3 per unitproduction. The facility has a capacity of two units of productionper hour and the minimum amount of production for each prod-uct over a 7 h time window is 2 units of A, 5 units of B, and 3 unitsof C. Because C is the most valuable product, any spare capacityshould favor the production of C. In switching between products,a schedule should account for transition material between gradesthat does not have value. The speed of a transition is limited by the

maximum move rate of the Manipulated Variable (MV) of 1.6 perhour and the dynamics of the process. The dynamics of the processare simply a linear first-order system as �(d(CV)/dt) =− CV + MV witha time constant � of 1.0.

L.D.R. Beal et al. / Computers and Chemical Engineering 104 (2017) 271–282 275

Fig. 5. The control and scheduling optimization are combined to determine optimal MV movement along with the optimal order and quantity of production for each grade.

F , and fia

lpAAc6tfs

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ig. 6. Optimized schedule with periodic condition, intermediate production targetsnd maximize profit (product C).

The grade-specific objective function with a quadratic objective,inear equation, and simple inequality constraints is added to therocess model to optimize the timing of grade transition switches.

complete statement of the QP optimization problem is shown inppendix A. The series of three products demonstrates a simpleombined control and scheduling problem as shown in Figs. 5 and

. The cases have different initial conditions and constraints buthe same underlying model. The first case has an initial conditionor CV of 0.0 at the lower specification limit for product A while theecond case starts at CV of 4.5.

The optimizer minimizes the amount of products A and B whileeeting the minimum requirements before transitioning to the

ext grade. This allows the schedule to favor the production of C,he most profitable product. By minimizing the amount of products

and B, the scheduler creates a plan that produces an extra unitf product C. The combined controller and scheduler anticipatesrade transitions by shifting the production specification to the

nal production targets. The schedule and control actions adjust to meet constraints

upper limit to minimize the transition time to the next grade. This isapparent at time 1.0 h and 4.2 h where the product is already tran-sitioning to the new grade just as the minimum required amountfor products A and B are produced. The combined scheduler andcontroller adjusts both the control actions as well as the order inwhich the products are produced. In the next example, the initialcondition is shifted to a different starting location to demonstratethis feature of the approach.

The computational time is an important factor for control prob-lems where the algorithm must return a solution within the cycletime necessary to rejected disturbances and maintain processstability. The fully discretized combined scheduling and controlproblem with 3 products and discrete time points has 2730 vari-ables, 1820 equations, and 910 degrees of freedom. The problem is

solved on a Dell R815 server with an AMD Opteron 6276 Processorand 64 GB of RAM. The problem requires 8.6 s to converge with theAPOPT solver (version 1.0) or 4.0 s with the IPOPT solver (version

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76 L.D.R. Beal et al. / Computers and Ch

.10). MPC commonly uses a warm-start from a prior solution tomprove efficiency. With a warm-start from the prior time-step, theptimizer requires 0.5 s with APOPT and 3.0 s with IPOPT. Active-setolvers (such as APOPT) and interior point solvers (such as IPOPT)re both effective for large-scale MPC applications although inte-ior point solvers have better performance as degrees of freedomncrease Hedengren et al. (2014).

.2. Dynamic cyclic schedule

A common method to optimize a schedule in current practices to place products on a fixed grade wheel where each products visited sequentially in a rotation. The cyclic schedule visits allrades in a forward and then a reverse order to end up at the orig-nal grade and begin the cycle again. If there is a sudden demandor a particular grade out of sequence, the product wheel is stillypically traversed in the grade wheel order but there may be a

inimal amount of the less desirable grades in favor of the desiredrade production. In some processes, such as polyolefin produc-ion, there are multiple versions of the grade wheel. One version ofhe grade wheel may include only commodity products while otherheels or sub-wheels may be a more complex cycles that include

ess frequently produced products.The contribution of this work is the discrete time, extended

ontroller and scheduler that is also able to produce a cyclic sched-le. However, this cyclic schedule is not fixed but adjusts theequence of products automatically when updated economic oronstraint information is available. The schedule and control actionay update every controller cycle (e.g. every minute). Some con-

traints for scheduling are periodic boundary constraints where thenal condition must be equal to the initial condition, contracteduantities that must be produced by a certain date or time, partic-lar equipment limitation, or time-of-day transition constraints. Anxample of a time-of-day constraint is that certain grade transitionseed start-up or shut-down of auxiliary equipment. A constraintay be that the grade transition should happen only during a week-

ay day-time shift where there is adequate operator support.The prior example problem is augmented with intermediate

roduction targets and a periodic constraint. The scheduler andontroller, shown in Fig. 6 produces an optimized schedule and con-rol actions. It is a dynamic rather than a static grade wheel becausehe order of product production and quantity is re-optimized everyycle of the controller. The product order or quantity may changeased on changing customer demand, price signals for electricityr feed costs, or disturbances that drive the system to a differenttate.

The optimized solution meets constraints as well as maximizeshe production of product C, the highest value product. The inter-

ediate constraints originate from contracted delivery times andtorage capacity of a particular product. In this case, the storageapacity of all products is less than 5 units of production. Thisequires two deliveries of product B that are scheduled at twoqually spaced intervals of 5 h. The production of product C exceedshe required delivery amount because it is the highest value prod-ct. Although the initial condition is at product C, the controller

mmediately targets product B to meet the required delivery of 5nits of production at 5 h. Without the periodic constraint, produc-ion of product C would be maximized before transitioning becausehe scheduler evaluates the alternative that a transition back toroduct C results in lost profit potential. However, with the requiredransition back to product C, the scheduler puts excess productionf product C at the end.

While this method is capable of producing cyclic schedules, theptimizer should begin from current conditions rather than steady-tate product conditions to fully integrate control and scheduling.yclic schedules combined with online control may lead back to off-

l Engineering 104 (2017) 271–282

spec conditions because of disturbances or because the controlleris in a transition. Instead, this method is better suited to a differ-ent set of constraints – production amounts and due dates. Theseconstraints give more freedom to the optimizer so the economicobjective will improve or be equal to the solution with periodicconstraints.

As with adding any constraint, there is potential to make theproblem infeasible. Multi-objective optimization statements, suchas those used to refine a dynamic grade wheel sequence, can alsobe posed with a rank-ordered set of constraints with an explicitprioritization of objectives. The �1-norm dead-band formulationis discussed in more detail in Hedengren et al. (2014) and relatedto combined scheduling and control production targets in Section4.3. Posing the constraints as multi-objective penalties versus hardconstraints allows the problem to remain feasible yet still meet themost important objectives in order of priority.

4.3. Acceptable range of production quantity

One drawback to the prior examples is that all spare produc-tion capacity is typically placed on the highest value product.Over-production of any product can have the effect of loweringthe selling price because of supply and demand market forces. Inscheduling, there is often a range of production quantity that isacceptable instead of just a single hard limit. To accommodate this,the scheduling and control algorithm can use an �1-norm objectivefunction to give a target region for the production quantity, ratherthan one specific hard limit. Eq. (5) shows a generalized �1-normcontrol formulation used in this work.

minx,CV,MV

� = wThi

ehi + wTloelo + xT cx + MVT cMV + �MVT c�MV

s.t. 0 = f(

dx

dt, x,

dCV

dt, CV, MV

)

ehi ≥ x − dhielo ≥ dlo − x

(5)

In this formulation, � is the objective function, x is the pro-duction quantity per grade. Parameters wlo and whi are penaltymatrices for solutions outside of the production target region. Theslack variable elo and ehi define the error of the dead-band low andhigh limits. Parameters cx, cMV, and c�MV are cost vectors for theproduction quantity (positive values minimize production withinregion, negative values maximize production within region), theMVs (positive values minimize MV quantities such as energy use,negative values maximize MV quantity such as pump speed cor-related to higher efficiency), and change of MVs, respectively. Thefunction f is an open-equation set of equations as functions of x, CV,MV, and time derivatives of x and CV. The demand targets dlo anddhi define lower and upper target limits for production.

This range formulation is not used in this work but is presentedto demonstrate one of many ways that this problem structure canbe expanded to meet various scheduling needs.

5. Application: continuously stirred tank reactor

A continuously stirred tank reactor (CSTR) is a common bench-mark used in nonlinear model predictive control and schedulingapplications Beal et al. (2017), Baldea and Harjunkoski (2014),Kelly and Hedengren (2013). This nonlinear application is used to

demonstrate the strength of the approach to time-scale based onthroughput and combined scheduling and control.

The linear time-scaled model changes dynamics based on reac-tor flow rate, allowing linear MPC to be used instead of nonlinear

L.D.R. Beal et al. / Computers and Chemica

Fe

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oatreTtli

and quantities are shown in Table 2.

TR

ig. 7. Diagram of the well-mixed and liquid-full CSTR. The A → B reaction isxothermic and controlled by a cooling jacket fluid.

PC. The multi-product scheduling objective is used instead of aimple target tracking to combine the scheduling and control intone application. Rapid convergence is ensured with a linear modelnd quadratic objective because the problem is convex and becauseP solvers are efficient for large-scale systems.

The CSTR application is highly nonlinear because of an exother-ic reaction that has the potential to cause rapid reaction of stored

eactant and thereby cause a temperature run-away. The CSTRhares characteristics of many industrial processes such as poly-er reactors or many refining processes but with much simplerathematics that are amenable to demonstrating a new approach

or control and scheduling. The liquid full reactor is used to con-ert compounds A → B with constant liquid density (�) and heatapacity (Cp) as shown in Fig. 7.

The reaction kinetics are first order and irreversible. Reactionf A to B is exothermic with the potential for temperature run-way because of the exponential dependence of reaction rate onemperature, typical of an Arrhenius form for reaction rates. Theeactor is well-mixed with reactor concentration and temperaturequally distributed and also equal to the outlet measured values.he reactor temperature is regulated with a cooling jacket liquid

emperature, Tc. The cooling jacket temperature is normally regu-ated by adjusting the rate of cooling or the coolant flow rate butn this model the jacket temperature is assumed to be controlled

able 1eactor initial conditions and parameter values.

States Description

CA Concentration of reactant A CB Concentration of product B T Reactor temperature

Manipulated variables Description

Tc Cooling jacket temperature

Parameters Description

q Volumetric flowrate V Tank volume CA0 Feed concentration of reactant A UA Overall heat transfer coefficient Cp Heat capacity of reactor fluid � Density of reactor fluid Tf Feed temperature �Hr Heat of reaction (exothermic) k0 Pre-exponential factor, rate constantEA/R Activation energy divided by R

l Engineering 104 (2017) 271–282 277

directly and the dynamics are approximated by a maximum rate ofchange.

The dynamics of the CSTR are dictated by a species and energybalance as shown in Eqs. (6)–(7) and in Fig. 8.

VdCAdt

= q(CA0 − CA) − k0Ve−EART CA (6)

�CpVdT

dt= q�Cp(Tf − T) − k0Ve

−EART CA�Hr − UA(T − Tc) (7)

where V is the volume of the reactor, CA is the concentration of reac-tant A, q is the volumetric flowrate, CA0 is the inlet concentrationof reactant A. The energy balance includes terms UA as an overallheat transfer coefficient times the tank surface area, Cp as the reac-tor fluid heat capacity, � as the fluid density, Tf as the temperatureof the feed stream, T as the temperature of reactor, and Tc as thetemperature of cooling jacket fluid. Terms related to the reactioninclude �Hr as the heat of reaction, EA as the activation energy, Ras the universal gas constant, and k0 as the pre-exponential factor.Table 1 lists the CSTR parameters and the associated values.

A regression is shown with varying orders for an Output Error(OE) time-series model in Fig. 8. Second and third order modelshave nearly the same fit to the nonlinear regression while a firstorder model is insufficient in capturing the process dynamics. Asecond simulation is performed with the production rate reducedfrom 12 m3/h to 6 m3/h.

The concentration response of the reactor at half production rateis shown in the bottom subplot of Fig. 8 with the second order time-scaled model that originally fit with simulated data from the fullproduction rate simulation. The time-scaled approach is effectiveat capturing the essential process dynamics without re-fitting aprocess model at low rates.

One reactor makes multiple products by varying the concentra-tions of A and B in the reactor. The cooling jacket temperature Tc isthe manipulated variable in this optimization. The production rateof the reactor changes throughout a typical 24-h period because oftime-of-day pricing that necessitates a cut-back of production dur-ing peak energy prices and when cooling capacity is limited. Theproduction rate is modified by adjusting the total flow through thereactor (q) between half-rates at 6 m3/h and full-rates at 12 m3/h.

There is demand for three products with quantities that must bemet in the schedule over a 48-h horizon. The product descriptions

The most valuable is product P2 while the least valuable prod-uct is P3. Although P3 has the lowest price, it also has the highestrequired quantity. Spare capacity in the production facility favors

Initial condition

0.1 mol/L0.9 mol/L386.82 K

Initial value

300 K

Value

12 m3/h40 m3

1 mol/L5000 W/K0.239 J/kg K1000 kg/m3

350 K−11.95 MJ/mol

1.8e10 1/h8750 K

278 L.D.R. Beal et al. / Computers and Chemical Engineering 104 (2017) 271–282

Fig. 8. Step tests in the jacket cooling and linear model regression of the step response.

Fig. 9. Combined control and sch

Table 2Product summary with demand and price.

Product CA (mol/L) Demand (m3) Price ($/m3)

P1 0.12 ± 0.01 120 9

pettocac

P2 0.25 ± 0.01 130 11P3 0.35 ± 0.01 150 6

roduct P2. A potential drawback to always switching to P2 at thend of a campaign is that there is lost material during the transi-ion back to P2. An improved strategy is to make excess P2 whenhe schedule requires it to meet a minimum target demand instead

f transitioning back to P2 near the end of the time period. Theombined control and scheduling solution is shown in Fig. 9 over

48 h time period with 6 min time intervals. The problem is dis-retized with orthogonal collocation on finite elements. Each 6 min

edule optimization results.

segment is integrated with Radau quadrature. The resulting QPor QPQC problem is solved with a nonlinear programming solverwith a simultaneous solution of the objective and equations. If spe-cific control action is needed at more frequent intervals, the firststeps of the horizon could be adjusted to meet a required con-troller cycle time. This would develop a near-term move plan andsimultaneously solve the scheduling optimization problem withone application. The feed flow rate is decreased to half each daybetween the hours of 08:00 and 18:00 as is done with some energyintensive processes that exploit time-of-day electricity pricing. Theaddition of production rate as a decision variable and the associatedcooling constraints is outside the scope of this work because the

model becomes nonlinear. The simultaneous control and schedul-ing of production rate and product grade sequence is the topic ofa future publication (see Beal et al. (2017) for preliminary results).

L.D.R. Beal et al. / Computers and Chemical Engineering 104 (2017) 271–282 279

idual

TtTtomta

iabTittsdmcimtct

oo

tast

Fig. 10. Profit function and indiv

he top sub-plot is the sequence of control moves to drive the sys-em to produce on-spec products and transition between products.he middle sub-plot shows the grade specifications and the concen-ration in the reactor. The bottom sub-plot is the total productionf each grade with the minimum required as indicated by the circlearkers at hour 48. The production rate is non-zero during transi-

ion periods because the total rate includes production of off-specs well as on-spec grade material.

The control influences the scheduling solution and the schedul-ng solution gives the controller target values. The controllerdjusts the jacket temperature (Tc) to minimize the transition timeetween grade specifications and remain within the grade limits.he controller response improves with knowledge of the schedul-ng targets because pro-active pre-transition movement shows thathe product specification are pushed to an upper limit right beforehe transition begins. One factor that affects the schedule is thepeed of the transition from one product to another. The controlynamics are due to limitations in the manipulated variable move-ent and the process dynamics affecting the speed of attaining

oncentration range. The speed of the control response is factorednto the schedule as lost production time when transition (off-spec)

aterial is produced. Blending of transition material is one strategyo dilute the off-test material in prime products. This strategy is notonsidered in this approach but could be included as a constraint onhe amount of reblend fraction that is allowed in the final product.

The scheduling profit function is an application and adaptationf Eqs. (4a–4d). Fig. 10 displays the slack and step function for eachf the product limits along with the overall profit function.

The slack variables and complementarity conditions combine

o create discrete steps with a function that has continuous firstnd second derivatives. For this problem, the complementarity con-traints were included as constraints and in the objective functiono assist the optimizer and ensure binary decision variables. The

step functions for each product.

transition time to each product is not specified beforehand but is aresult of the optimization solver finding a sequence of grade tran-sitions that maximize overall profit. The individual steps of Fig. 10occur when one of the product grade limits is crossed. One of eachof the slack variables for each step is non-zero before or after thestep. The slack variables guide the solver with a continuous form ofa discrete grade switching function. The overall solution is intuitivebecause the optimizer produces excess product P2 (highest volu-metric price) but meets only the minimum production quantitiesfor the lower value products (P1 and P3).

A weakness of this method is that the CSTR is nonlinear butthe solution is computed with a linearized model. Linearizationneglects features of the process that may be optimized if a suitableprocess model more accurately represented the physical process. Akey contribution of this work is that the solution to the combinedcontrol and scheduling problem is relatively computationally inex-pensive in comparison to a full nonlinear solution. In this case, thelinearized model with time-scaling has a total of 22,048 variablesand 16,640 equations and is solved on a Dell R815 with an AMDOpteron 6276 Processor and 64 GB of RAM. After initialization, theproblem requires 23.7 s to converge with the IPOPT solver. Thisis representative of the cycle time of the application as it repeat-edly solves to reject disturbances and as new objective informationor demand constraint information is available. This speed, cou-pled with the inclusion of a process model sufficient for control,allows this formulation to be utilized for on-line control, on top ofproviding an advanced schedule. This is the epitome of combin-ing scheduling and control – a fully unified optimization that canreplace both layers.

A nonlinear version of this application is future work that willbe reported in a subsequent publication. A key difference with thefull nonlinear solution is that the solution in this work is severalorders of magnitude faster because of the quadratic objective and

2 emica

ltatAlM

6

aotwrettttdefc

80 L.D.R. Beal et al. / Computers and Ch

inear time-scaled model. Although combined scheduling and con-rol with linear models or feedback linearization does not alwaysccurately predict a highly nonlinear system, the linearized solu-ion is a valuable starting point to initialize a nonlinear solution.lso, many processes are not highly nonlinear and this approach is

ikely suitable for systems that are already controlled with linearPC.

. Conclusion

A combined scheduling and control application is enabled byn MPCC objective function, discrete-time, and linear time-scalingf process dynamics based on production rate changes. The objec-ive of this work is to extend traditional linear MPC applicationsith a scheduling objective that allows for rapid convergence for

eal-time applications. One drawback of this work is that nonlin-arities are not included in the application. These nonlinearities arehe subject of future work. The method is tested on a CSTR applica-ion that includes three grades over a 48 h time horizon with 6 minime intervals. The embedded controller simulates realistic transi-ion times between each of the products. The scheduling objective

etermines the order and quantity of production at each gradeven with half-rate reduction during peak electricity demand. Theormulation is sufficiently fast enough, and includes enough pro-ess dynamics, to be utilized in on-line control. This presents a

l Engineering 104 (2017) 271–282

fully unified optimization that fulfills the roles of, and can replace,both control and scheduling for a comparable system. Although themethod is demonstrated on the CSTR application, this formulationcan be applied to other systems by replacing the model, pricingstructure, and constraints of the scheduler.

Acknowledgments

Financial support from the NSF Award 1547110, EAGER: Cyber-Manufacturing with Multi-echelon Control and Scheduling, isgratefully acknowledged.

Appendix A.

A.1 Combined scheduling and control example

The following APMonitor model and Python script detail thevariables, equations, and commands necessary to reproduce thecombined scheduling and control presented in Section 4.1. Theapplication uses two elements including the model file (sched-ule.apm) and a data file (schedule.csv). The data file is a list oftimes between 0 and 7 with time increment 0.1 and another col-umn labeled last that is 0 everywhere except the end point as 1. Theparameter last is to enforce the constraints that a certain amountof each product should be produced. All source files are availablefrom https://github.com/APMonitor.

Listing 1: Constants and parameters

https://github.com/APMonitorhttps://github.com/APMonitorhttps://github.com/APMonitorhttps://github.com/APMonitor

emical Engineering 104 (2017) 271–282 281

R

A

L.D.R. Beal et al. / Computers and Ch

Listing 2: Variables and equations

Python Script for combined control and scheduling

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Combined model predictive control and scheduling with dominant time constant compensation1 Introduction2 Time-scaling with first-order systems3 Selective time-scaling4 Scheduling and control formulation4.1 MPCC steps for product pricing4.2 Dynamic cyclic schedule4.3 Acceptable range of production quantity

5 Application: continuously stirred tank reactor6 ConclusionAcknowledgmentsA.1 Combined scheduling and control exampleReferences

References