#
Improving Flexibility and Energy Efficiency of Post-Combustion CO_{2} Capture Plants Using Economic Model Predictive Control

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## Abstract

**:**

## 1. Introduction

## 2. Model Development

#### 2.1. Absorption and Desorption Units

#### 2.1.1. Modeling Assumptions

- Well mixed bulk and liquid phases. Each stage therefore behaves like a continuously stirred tank reactor (CSTR) with no spacial variation in properties.
- Reactions occur only in the liquid film and the influence of the reaction on mass transfer is described using enhancement factor.
- Mass and heat transfer are described by the two film theory [21].
- Pressure drop in the two columns is linear.
- No heat losses to the surrounding area.

#### 2.1.2. Balance Equations

#### 2.1.3. Heat and Mass Transfer Rates

#### 2.2. Heat Exchanger Model

#### 2.2.1. Energy Balance

#### 2.3. Reboiler Model

#### 2.3.1. Material Balance

#### 2.3.2. Energy Balance

#### 2.4. Physical and Chemical Properties

#### 2.5. Model Discretization

## 3. Control Problem Formulation and Design

#### 3.1. Conventional Set-Point Tracking MPC

**Optimization problem 1.**Steady-state optimization problem

**Optimization problem 2.**Tracking MPC optimization problem

#### 3.2. Economic Model Predictive Control

**Optimization problem 3.**EMPC optimization problem

#### 3.3. Implementation

- The continuous time differential and algebraic model equations are discretized by approximating the state and control profiles by a family of polynomials on finite elements. This involves dividing the control horizon into a number of finite elements with the size of each element corresponding to one sampling time. Within each element, the state and input profiles are approximated by a family of polynomials. In this work, Radau orthogonal polynomials in Lagrange form is used.
- The dynamic optimization problem is then formulated as a large scale nonlinear programming (NLP) problem.
- The NLP problem is solved using a computationally efficient solver that exploits the sparsity in the resulting matrix.

**Optimization problem 4.**Reformulated optimization problem.

## 4. Simulations, Results and Discussion

#### 4.1. Simulation Setup

#### 4.1.1. Time-Varying Steam Price and Flue Gas Flow Rate

#### 4.1.2. Tracking MPC Parameter Tuning

#### 4.2. MPC Set-Point Update Strategy

#### 4.3. Operation under Different Carbon Tax Policies

#### 4.3.1. Carbon Tax with Tax-Free Emission Limit

#### 4.3.2. Carbon Tax without Tax-Free Emission Limit

#### 4.4. Operation under Uncertainty

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Abbreviations

CCS | Carbon Capture and Storage |

CSTR | Continuously Stirred Tank Reactor |

EMPC | Economic Model Predictive Control |

eNTRL | electrolyte Non-Random Two Liquids |

DAEs | Differential Algebriac Equations |

MEA | Monoethanolamine |

MPC | Model Predictive Control |

NLP | Nonlinear Programming |

ODEs | Ordinary Differential Equations |

PCC | Post-combustion CO${}_{2}$ Capture |

PDEs | Partial Differential Equations |

SSO | Steady-State Optimization |

ZOH | Zero-Order Hold |

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**Figure 3.**Scaled hourly changes in steam price (Nominal value: $0.01$/\mathrm{kg}$) and flue gas flow rate (Nominal value: 0.0832 ${\mathrm{m}}^{3}/\mathrm{s}$). Price (red), Flue gas flow (blue).

**Figure 4.**Closed-loop trajectories of EMPC and different MPC tuning parameters starting from a higher economic cost to a lower one. EMPC (blue), MPC I (red), MPC II (green), MPC III (black). The average economic performances are 4.700, 4.768, 4.713, 4.692 for EMPC, MPC I, MPC II, MPC III, respectively; (

**top-left**) mass flow rate of CO${}_{2}$ in the treated flue gas; (

**top-right**) economic performance of controller; (

**bottom-left**) volumetric flow rate of solvent; (

**bottom-right**) heat input to reboiler. EMPC: Economic Model Predictive Control

**Figure 5.**Closed-loop trajectories of EMPC and different MPC tuning parameters starting from a lower economic cost to a higher one. EMPC (blue), MPC I (red), MPC II (green), MPC III (black). The average economic performances are 3.654, 3.668, 3.650, 3.685 for EMPC, MPC I, MPC II, MPC III, respectively; (

**top-left**) mass flow rate of CO${}_{2}$ in the treated flue gas; (

**top-right**) economic performance of controller; (

**bottom-left**) volumetric flow rate of solvent; (

**bottom-right**) heat input to reboiler.

**Figure 6.**CO${}_{2}$ emission, input and economic cost trajectories of the PCC plant with carbon tax set at $50.00 per tonne CO${}_{2}$. The set-point of MPC is not updated. EMPC (blue), MPC (red), Emission limit (green); (

**top-left**) mass flow rate of CO${}_{2}$ in the treated flue gas; (

**top-right**) economic performance of controller; (

**bottom-left**) volumetric flow rate of solvent; (

**bottom-right**) heat input to reboiler. PCC: Post-combustion CO${}_{2}$ Capture.

**Figure 7.**CO${}_{2}$ emission, input and economic cost trajectories of the PCC plant for scenario 1 with carbon tax set at $50.00 per tonne CO${}_{2}$. EMPC (blue), MPC (red), Emission limit (green); (

**top-left**) mass flow rate of CO${}_{2}$ in the treated flue gas; (

**top-right**) economic performance of controller; (

**bottom-left**) volumetric flow rate of solvent; (

**bottom-right**) heat input to reboiler.

**Figure 8.**CO${}_{2}$ emission, input and economic cost trajectories of the PCC plant for scenario 1 with carbon tax set at $200.00 per tonne CO${}_{2}$. EMPC (blue), MPC (red), Emission limit (green); (

**top-left**) mass flow rate of CO${}_{2}$ in the treated flue gas; (

**top-right**) economic performance of controller; (

**bottom-left**) volumetric flow rate of solvent; (

**bottom-right**) heat input to reboiler.

**Figure 9.**Average economic performance of the controllers under scenario 1 at different CO${}_{2}$ prices. EMPC (blue), MPC (red).

**Figure 10.**Average heat duty of the controllers under scenario 1 at different CO${}_{2}$ prices. EMPC (blue), MPC (red).

**Figure 11.**CO${}_{2}$ emission, input and economic cost trajectories of the PCC plant for scenario 2 with carbon tax set at $50.00 per tonne CO${}_{2}$. EMPC (blue), MPC (red), Emission limit (green); (

**top-left**) mass flow rate of CO${}_{2}$ in the treated flue gas; (

**top-right**) economic performance of controller; (

**bottom-left**) volumetric flow rate of solvent; (

**bottom-right**) heat input to reboiler.

**Figure 12.**CO${}_{2}$ emission, input and economic cost trajectories of the PCC plant for scenario 2 with carbon tax set at $200.00 per tonne CO${}_{2}$. EMPC (blue), MPC (red), Emission limit (green); (

**top-left**) mass flow rate of CO${}_{2}$ in the treated flue gas; (

**top-right**) economic performance of controller; (

**bottom-left**) volumetric flow rate of solvent; (

**bottom-right**) heat input to reboiler.

**Figure 13.**Average economic performance of the controllers under scenario 2 at different CO${}_{2}$ prices. EMPC (blue), MPC (red).

**Figure 14.**Average heat duty of the controllers under scenario 2 at different CO${}_{2}$ prices. EMPC (blue), MPC (red).

**Figure 15.**CO${}_{2}$ emission, inputs and economic cost trajectories of the PCC plant under uncertainty in flue gas flow rate. Nominal value =1.0. Uncertainty generated by adding Gaussian white noise with zero mean and standard deviation $\sigma $ = 0.1, to the flue gas flow rate. EMPC (blue), MPC (red); (

**top-left**) mass flow rate of CO${}_{2}$ in the treated flue gas; (

**top-right**) economic performance of controller; (

**bottom-left**) volumetric flow rate of solvent; (

**bottom-right**) heat input to reboiler.

**Figure 16.**CO${}_{2}$ emission, inputs and economic cost trajectories of the PCC plant under uncertainty in flue gas CO${}_{2}$ concentration. Uncertainty generated by adding Gaussian white noise with zero mean and standard deviation $\sigma $ = 0.06, to the flue gas CO${}_{2}$ concentration. EMPC (blue), MPC (red); (

**top-left**) mass flow rate of CO${}_{2}$ in the treated flue gas; (

**top-right**) economic performance of controller; (

**bottom-left**) volumetric flow rate of solvent; (

**bottom-right**) heat input to reboiler.

**Table 1.**PCC plant configuration and flue gas condition. Adapted from [20]. PCC: Post-combustion CO${}_{2}$ Capture.

Property | Value |
---|---|

Packing properties (Absorber and desorber) | |

Column internal diameter $\mathrm{m}$), ${D}_{c}$ | 0.43 |

Packing height ($\mathrm{m}$) | 6.1 |

Packing type | IMTP #40 |

Nominal packing size ($\mathrm{m}$) | 0.038 |

Specific packing area (${\mathrm{m}}^{2}/{\mathrm{m}}^{3}$) | 143.9 |

Flue gas condition | |

Temperature ($\mathrm{K}$) | 319.7 |

Volumetric flow rate (${\mathrm{m}}^{3}/\mathrm{s}$) | 0.0832 |

CO${}_{2}$ mole fraction | 0.15 |

N${}_{2}$ mole fraction | 0.80 |

MEA mole fraction | 0.00 |

H${}_{2}$O mole fraction | 0.05 |

Lean-rich heat exchanger | |

Volume of tube side (${\mathrm{m}}^{3}$), ${V}_{tube}$ | 0.016 |

Volume of shell side (${\mathrm{m}}^{3}$), ${V}_{shell}$ | 0.205 |

Overall heat transfer coefficient ($\mathrm{J}/\mathrm{Ks}$), $UA$ | 1899.949 |

MPC | a | b | c | d |
---|---|---|---|---|

I | 0.0001 | 0.01 | 0 | 0 |

II | 0.0001 | 0.01 | 0.05 | 0.05 |

III | 1.0 | 0.01 | 0.05 | 0.05 |

**Table 3.**Different MPC set-point update strategies and percent decrease in economics and heat duty. Heat duty is defined as the ratio of the amount of reboiler heat input to the mass of CO${}_{2}$ absorbed.

Strategy | Description | Avg. Cost (%) | Avg. Heat Duty (%) |
---|---|---|---|

1 | No update/Fixed operating point | 5.83 | 6.52 |

2 | Update every hour | 2.02 | 4.27 |

3 | Update every 30 min | 0.79 | 2.99 |

4 | Update every 10 min (sampling time) | 0.03 | 2.32 |

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**MDPI and ACS Style**

Decardi-Nelson, B.; Liu, S.; Liu, J.
Improving Flexibility and Energy Efficiency of Post-Combustion CO_{2} Capture Plants Using Economic Model Predictive Control. *Processes* **2018**, *6*, 135.
https://doi.org/10.3390/pr6090135

**AMA Style**

Decardi-Nelson B, Liu S, Liu J.
Improving Flexibility and Energy Efficiency of Post-Combustion CO_{2} Capture Plants Using Economic Model Predictive Control. *Processes*. 2018; 6(9):135.
https://doi.org/10.3390/pr6090135

**Chicago/Turabian Style**

Decardi-Nelson, Benjamin, Su Liu, and Jinfeng Liu.
2018. "Improving Flexibility and Energy Efficiency of Post-Combustion CO_{2} Capture Plants Using Economic Model Predictive Control" *Processes* 6, no. 9: 135.
https://doi.org/10.3390/pr6090135