# Combined Noncyclic Scheduling and Advanced Control for Continuous Chemical Processes

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## Abstract

**:**

## 1. Introduction

## 2. Literature Review

## 3. Problem Formulation

#### 3.1. Decomposition

#### 3.2. Iterative Method

## 4. Case Study Application

#### 4.1. Process Model

#### 4.2. Scenarios

## 5. Results

#### 5.1. Scenario 1

#### 5.2. Scenario 2

#### 5.3. Return Method: Additional Scenario

#### 5.4. Scenario 3

#### 5.5. Scenario 4

#### 5.6. Scenario 5

## 6. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Abbreviations

NMPC | nonlinear model predictive control |

MILP | mixed-integer linear programming |

MIDO | mixed-integer dynamic optimization |

LP | linear programming |

NLP | nonlinear programming |

NMPC | nonlinear model predictive control |

ISC | integrated scheduling and control |

CSTR | continuous-stirred tank reactor |

MIDO | mixed-integer dynamic optimization |

FBR | fluidized-bed reactor |

SEN | state equipment network |

MLDO | mixed-logic dynamic optimization |

RTN | resource task network |

mp-MPC | multi-parametric model predictive control |

fast MPC | fast model predictive control |

MINLP | mixed-integer nonlinear programming |

MINFP | mixed-integer fractional programming |

MILP | master scheduling problem |

KPI | key performance indicator |

EMPC | economic model predictive control |

HIPS | high impact polystyrene |

SBM | scale-bridging model |

ASU | air separation unit |

PWA | piecewise affine |

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**Figure 1.**Diagram describing the integrated noncyclic scheduling and optimal control problem. For variable descriptions, see Table 4.

Product | ${\mathit{C}}_{\mathit{A}}$ (mol/L) |
---|---|

1 | 0.10 |

2 | 0.30 |

3 | 0.50 |

Start | End Product | ||
---|---|---|---|

Product | 1 | 2 | 3 |

1 | 0.00 | 0.71 | 1.20 |

2 | 0.45 | 0.00 | 0.71 |

3 | 0.94 | 1.57 | 0.00 |

Product | ${\mathit{\tau}}_{\mathit{\theta}}\left(\mathit{h}\right)$ |
---|---|

1 | 0.31 |

2 | 0.43 |

3 | 0.96 |

Variable | Description |
---|---|

$\alpha $ | Current number of slots |

${\tau}_{ss}$ | Matrix of grade transition durations between production steady-states |

${x}_{\theta}$ | Measured process state |

${\tau}_{\theta}$ | Vector of grade transition durations from ${x}_{\theta}$ to each product steady-state |

p | Vector of production steady-states known a priori |

${T}_{M}$ | Prediction horizon duration or makespan |

q | Process flow rate (m${}^{3}$/h) |

${\tau}_{\alpha}$ | Estimation of total grade transition during a prediction horizon for $\alpha $ production slots |

$\delta $ | Vector of maximum demands (${\delta}_{i}$) for products |

${\delta}_{C}$ | Any combination of $\alpha $ product demands |

n | Number of possible products |

$\mathrm{\Pi}$ | Vector of product selling prices |

s | Vector of product storage costs (m${}^{3}$/h) |

${E}_{\alpha}$ | Estimated profit from optimized schedule for $\alpha $ production slots |

${\omega}_{\alpha}$ | Vector of manufactured amount per product (m${}^{3}$) in optimized schedule with $\alpha $ slots |

Parameter | Value |
---|---|

V | 100 m${}^{3}$ |

${E}_{A}/R$ | 8750 K |

$\frac{UA}{V\rho {C}_{p}}$ | 2.09 s${}^{-1}$ |

${k}_{0}$ | 7.2 × 10${}^{10}$ s${}^{-1}$ |

${T}_{f}$ | 350 K |

${C}_{A0}$ | 1 mol/L |

$\frac{\Delta {H}_{r}}{\rho {C}_{p}}$ | −209 K m${}^{3}$/mol |

q | 100 m${}^{3}$/h |

Product | ${\mathit{C}}_{\mathit{A}}$ (mol/L) | Max Demand (m${}^{3}$) | Price ($/m${}^{3}$) |
---|---|---|---|

1 | 0.10 | 2000 | 24 |

2 | 0.15 | 2000 | 29 |

3 | 0.22 | 2000 | 26 |

4 | 0.28 | 2000 | 23 |

5 | 0.34 | 2000 | 21 |

6 | 0.44 | 2000 | 21 |

7 | 0.50 | 2000 | 20 |

Product | ${\mathit{C}}_{\mathit{A}}$ (mol/L) | Max Demand (m${}^{3}$) | Price ($/m${}^{3}$) |
---|---|---|---|

1 | 0.10 | 1000 | 23 |

2 | 0.15 | 900 | 22 |

3 | 0.22 | 1200 | 29 |

4 | 0.28 | 860 | 26 |

5 | 0.34 | 800 | 25 |

6 | 0.44 | 1100 | 23 |

7 | 0.50 | 1400 | 21 |

Product | ${\mathit{C}}_{\mathit{A}}$ (mol/L) | Updated Price ($/m${}^{3}$) |
---|---|---|

1 | 0.10 | 22 |

2 | 0.15 | 25 |

3 | 0.22 | 29 |

4 | 0.28 | 28 |

5 | 0.34 | 23 |

6 | 0.44 | 21 |

7 | 0.50 | 21 |

Formulation | Selected Production Sequence and Slot Start Times (h) | ||||||
---|---|---|---|---|---|---|---|

Slot 1 | Slot 2 | Slot 3 | Slot 4 | Slot 5 | Slot 6 | Slot 7 | |

Cyclic, Traditional | P1 (0) | P2 (2.88) | P3 (23.6) | P4 (44.4) | P5 (45.1) | P7 (45.9) | P6 (47.2) |

Noncyclic, Iterative | P1 (0) | P2 (6.52) | P3 (27.2) | - | - | - | - |

Formulation | Profit ($) | Off-Spec (m${}^{3}$) | Manufactured Amount per Product (m${}^{3}$) | ||||||
---|---|---|---|---|---|---|---|---|---|

1 | 2 | 3 | 4 | 5 | 6 | 7 | |||

Cyclic, Traditional | 9,984 | 512 | 288 | 2000 | 2000 | 0 | 0 | 0 | 0 |

Noncyclic, Iterative | 18,588 | 148 | 652 | 2000 | 2000 | 0 | 0 | 0 | 0 |

**Table 11.**Alpha iterations: Scenario 1 (MILP:mixed-integer linear programming; NLP: nonlinear programming).

$\mathit{\alpha}$ | MILP | NLP |
---|---|---|

pre-iteration | - | ✓(7 Problems) |

1 | DF | - |

2 | DF | - |

3 | ✓ | - |

4 | ✓ | - |

5 | ✓ | - |

6 | ✓ | - |

7 | ✓ | - |

Formulation | Total Time (s) | NLP | MILP | ||||
---|---|---|---|---|---|---|---|

# | Average (s) | Total (s) | # | Average (s) | Total (s) | ||

Cyclic, Traditional | 18.07 | 7 | 0.735 | 5.15 | 1 | - | 12.92 |

Noncyclic, Iterative | 159.4 | 7 | 0.743 | 5.20 | 5 | 30.58 | 154.2 |

Formulation | Selected Production Sequence and Slot Start Times (h) | ||||||
---|---|---|---|---|---|---|---|

Slot 1 | Slot 2 | Slot 3 | Slot 4 | Slot 5 | Slot 6 | Slot 7 | |

Cyclic, Traditional | P1 (0) | P2 (3.28) | P3 (3.96) | P4 (16.8) | P5 (26.1) | P7 (34.9) | P6 (36.2) |

Noncyclic, Iterative | P1 (0) | P2 (10.0) | P3 (17.1) | P4 (29.9) | P5 (39.2) | - | - |

Formulation | Profit ($) | Off-Spec (m${}^{3}$) | Manufactured Amount per Product (m${}^{3}$) | ||||||
---|---|---|---|---|---|---|---|---|---|

1 | 2 | 3 | 4 | 5 | 6 | 7 | |||

Cyclic, Traditional | 3,824 | 512 | 328 | 0 | 1200 | 860 | 800 | 1100 | 0 |

Noncyclic, Iterative | 7,420 | 302 | 1000 | 638 | 1200 | 860 | 800 | 0 | 0 |

Formulation | Total Time (s) | NLP | MILP | ||||
---|---|---|---|---|---|---|---|

# | Average (s) | Total (s) | # | average (s) | Total (s) | ||

Cyclic, Traditional | 22.43 | 7 | 0.776 | 5.43 | 1 | - | 17.00 |

Noncyclic, Iterative | 115.2 | 7 | 0.791 | 5.54 | 3 | 36.57 | 109.7 |

$\mathit{\alpha}$ | MILP | B |
---|---|---|

pre-iteration | - | ✓(7 Problems) |

1 | DF | - |

2 | DF | - |

3 | DF | - |

4 | DF | - |

5 | ✓ | - |

6 | ✓ | - |

7 | ✓ | - |

Product | ${\mathit{C}}_{\mathit{A}}$ (mol/L) | Max Demand (m${}^{3}$) | Price ($/m${}^{3}$) |
---|---|---|---|

1 | 0.10 | 1000 | 23 |

2 | 0.15 | 900 | 24 |

3 | 0.22 | 1200 | 29 |

4 | 0.28 | 1200 | 26 |

5 | 0.34 | 800 | 25 |

6 | 0.44 | 4000 | 21 |

7 | 0.50 | 4000 | 21 |

$\mathit{\alpha}$ | MILP | Predicted Profit ($) |
---|---|---|

pre-iteration | - | - |

1 | DF | - |

2 | ✓ | 4,957 |

3 | ✓ | 7,077 |

4 | ✓ | 8,113 |

5 | ✓ | 12,273 |

6 | ✓ | 9,973 |

7 | ✓ | 7,601 |

Formulation | Selected Production Sequence and Slot Start Times (h) | ||||||
---|---|---|---|---|---|---|---|

Slot 1 | Slot 2 | Slot 3 | Slot 4 | Slot 5 | Slot 6 | Slot 7 | |

Noncyclic, Iterative | P1 (0) | P2 (3.98) | P3 (13.7) | P4 (26.5) | P5 (39.2) | - | - |

Formulation | Predicted Profit ($) | Off-Spec (m${}^{3}$) | Manufactured Amount per Product (m${}^{3}$) | ||||||
---|---|---|---|---|---|---|---|---|---|

1 | 2 | 3 | 4 | 5 | 6 | 7 | |||

Noncyclic, Iterative | 12,273 | 302 | 398 | 900 | 1200 | 1200 | 800 | 0 | 0 |

Formulation | Selected Production Sequence and Slot Start Times (h) | ||||||
---|---|---|---|---|---|---|---|

Slot 1 | Slot 2 | Slot 3 | Slot 4 | Slot 5 | Slot 6 | Slot 7 | |

Cyclic, Traditional | P3 (0) | P2 (12.0) | P1 (12.7) | P7 (16.3) | P6 (18.1) | P5 (29.9) | P4 (38.7) |

Noncyclic, Iterative (Initial) | P3 (0) | P4 (12.0) | P5 (21.4) | P2 (30.1) | P1 (37.4) | - | - |

Noncyclic, Iterative (Re-calc) | - | P5 (3.00) | P4 (11.6) | P3 (20.9) | P2 (31.6) | P1 (37.4) | - |

Noncyclic, Iterative (Actual) | P3 (0) | Disturbance | P5 (3.00) | P4 (11.6) | P3 (20.9) | P2 (31.6) | P1 (37.4) |

Formulation | Profit ($) | Off-Spec (m${}^{3}$) | Manufactured Amount per Product (m${}^{3}$) | ||||||
---|---|---|---|---|---|---|---|---|---|

1 | 2 | 3 | 4 | 5 | 6 | 7 | |||

Cyclic, Traditional (Actual) | −2096 | 762 | 291 | 0 | 988 | 860 | 800 | 1100 | 0 |

Noncyclic, Iterative (Actual) | 4993 | 440 | 1000 | 500 | 1200 | 860 | 800 | 0 | 0 |

Formulation | Total Time (s) | NLP | MILP | ||||
---|---|---|---|---|---|---|---|

# | Average (s) | Total (s) | # | Average (s) | Total (s) | ||

Cyclic, Traditional | 24.17 | 7 | 0.859 | 6.01 | 1 | - | 18.16 |

Noncyclic, Iterative (Initial) | 112.0 | 7 | 0.836 | 5.05 | 3 | 35.63 | 106.9 |

Noncyclic, Iterative (Re-calc) | 147.8 | 7 | 5.77 | 40.42 | 4 | 26.84 | 107.4 |

Formulation | Selected Production Sequence and Slot Start Times (h) | ||||||
---|---|---|---|---|---|---|---|

Slot 1 | Slot 2 | Slot 3 | Slot 4 | Slot 5 | Slot 6 | Slot 7 | |

Cyclic, Traditional | P5 (0) | P4 (8.00) | P3 (17.3) | P2 (30.1) | P1 (30.8) | P7 (34.4) | P6 (36.2) |

Noncyclic, Iterative (Initial) | P5 (0) | P4 (8.00) | P3 (17.3) | P2 (30.1) | P1 (37.4) | - | - |

Noncyclic, Iterative (Re-calc) | - | P5 (4.00) | P4 (8.00) | P3 (23.3) | P2 (44.1) | - | - |

Noncyclic, Iterative (Actual) | P5 (0) | P4 (8.00) | P3 (23.3) | P2 (44.1) | - | - | - |

Formulation | Profit ($) | Off-Spec (m${}^{3}$) | Manufactured Amount per Product (m${}^{3}$) | ||||||
---|---|---|---|---|---|---|---|---|---|

1 | 2 | 3 | 4 | 5 | 6 | 7 | |||

Cyclic, Traditional (Actual) | 2,816 | 546 | 294 | 0 | 1200 | 860 | 800 | 1100 | 0 |

Noncyclic, Iterative (Actual) | 16,024 | 220 | 0 | 320 | 2000 | 1460 | 800 | 0 | 0 |

Formulation | Total Time (s) | NLP | MILP | ||||
---|---|---|---|---|---|---|---|

# | Average (s) | Total (s) | # | Average (s) | Total (s) | ||

Cyclic, Traditional | 23.15 | 7 | 0.950 | 6.65 | 1 | - | 16.50 |

Noncyclic, Iterative (Initial) | 122.9 | 7 | 0.964 | 6.75 | 3 | 38.71 | 116.1 |

Noncyclic, Iterative (Re-calc) | 132.0 | 7 | 0.994 | 6.96 | 5 | 25.00 | 125.0 |

Formulation | Selected Production Sequence and Slot Start Times (h) | ||||||
---|---|---|---|---|---|---|---|

Slot 1 | Slot 2 | Slot 3 | Slot 4 | Slot 5 | Slot 6 | Slot 7 | |

Cyclic, Traditional | P1 (0) | P2 (2.88) | P3 (23.6) | P4 (44.4) | P5 (45.1) | P7 (45.9) | P6 (47.2) |

Noncyclic, Iterative (Initial) | P1 (0) | P2 (6.52) | P3 (27.2) | - | - | - | - |

Noncyclic, Iterative (Re-calc) | - | - | P3 (8.00) | P4 (28.8) | - | - | - |

Noncyclic, Iterative (Actual) | P1 (0) | P2 (6.52) | P3 (8.00) | P4 (28.8) | - | - | - |

Formulation | Profit ($) | Off-Spec (m${}^{3}$) | Manufactured Amount per Product (m${}^{3}$) | ||||||
---|---|---|---|---|---|---|---|---|---|

1 | 2 | 3 | 4 | 5 | 6 | 7 | |||

Cyclic, Traditional (Actual) | 9,760 | 512 | 288 | 2000 | 2000 | 0 | 0 | 0 | 0 |

Noncyclic, Iterative (Actual) | 20,820 | 224 | 652 | 80 | 2000 | 1844 | 0 | 0 | 0 |

Formulation | Total Time (s) | NLP | MILP | ||||
---|---|---|---|---|---|---|---|

# | Average (s) | Total (s) | # | Average (s) | Total (s) | ||

Cyclic, Traditional | 19.64 | 7 | 0.827 | 5.79 | 1 | - | 13.85 |

Noncyclic, Iterative (Initial) | 181.2 | 7 | 0.892 | 6.24 | 5 | 35.01 | 175.0 |

Noncyclic, Iterative (Re-calc) | 145.3 | 7 | 0.642 | 4.50 | 6 | 23.47 | 140.8 |

© 2017 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Petersen, D.; Beal, L.D.R.; Prestwich, D.; Warnick, S.; Hedengren, J.D.
Combined Noncyclic Scheduling and Advanced Control for Continuous Chemical Processes. *Processes* **2017**, *5*, 83.
https://doi.org/10.3390/pr5040083

**AMA Style**

Petersen D, Beal LDR, Prestwich D, Warnick S, Hedengren JD.
Combined Noncyclic Scheduling and Advanced Control for Continuous Chemical Processes. *Processes*. 2017; 5(4):83.
https://doi.org/10.3390/pr5040083

**Chicago/Turabian Style**

Petersen, Damon, Logan D. R. Beal, Derek Prestwich, Sean Warnick, and John D. Hedengren.
2017. "Combined Noncyclic Scheduling and Advanced Control for Continuous Chemical Processes" *Processes* 5, no. 4: 83.
https://doi.org/10.3390/pr5040083