# Adaptive Nonlinear Model Predictive Control of the Combustion Efficiency under the NOx Emissions and Load Constraints

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Background

## 3. Nonlinear Dynamic Prediction Model

#### 3.1. Data Selection

_{x}). Since the mentioned parameters have a different range, the data should be standardized. This improves the prediction accuracy. All the data are scaled according to the following equation:

_{1}, A

_{2}, A

_{3}, A

_{4}, P

_{a}, S are the most representative control variables in Table 1.

#### 3.2. Classification of Operating Conditions

#### 3.3. Construction of the Predictive Model

^{2}as the regularization and kernel parameters (Np = 500, D = 2, G = 1000, F = 0.7, CR = 0.9 are considered in this paper).

_{h}and b represent model parameters, x

_{k}is the hth input parameter of the training sample, K is the number of training sets, $K({x}_{k},x)$ are the LSSVM kernel parameters, ${\sigma}^{2}$ is the width of the RBF function, and the DE algorithm is used to obtain ${\sigma}^{2}$.

_{k}is the kth true value and f (x

_{k}) is the kth prediction value, and N is the number of data sets.

_{j}

_{,g+1}is the jth individual in the (g + 1)th iteration; x

_{best,g}is the best individual in iteration g; j, r1, r2 are different random integer numbers in the interval [1, Np]; F∈ [0, 2] is the alternating probability;U

^{i}

_{j,g+}

_{1}is the jth individual in the (g + 1)th iteration and the dimension parameter of the ith individual; rand () means a random number witha uniform distribution in the interval [0, 1]; CR∈ [0, 1] is the crossover probability and f(*) is the individual fitness function value.

_{B,h}, Y

_{Nx,h}and Y

_{L,h}are the predicted values for the boiler efficiency, the NOx emissions and the boiler load under the hth operating condition, respectively; Fc, M, A

_{1}, A

_{2}, A

_{3},A

_{4}, P

_{a}, S, O

_{1}, O

_{2},T

_{1}, T

_{2}, T

_{3}, T

_{4}, P are the corresponding symbols for operating parameters given in Table 1; ${\sigma}_{B}{}^{2}$, ${\sigma}_{{N}_{x}}^{2}$ and ${\sigma}_{L}^{2}$ are the kernel parameters of boiler efficiency, NOx emissions and boiler load, respectively;

**w**

_{B},

**w**

_{Nx}and

**w**

_{L}are the weight vectors of boiler efficiency, NOx emissions and boiler load, respectively; e

_{i,b}, e

_{i,Nx}and e

_{i,L}are the loss functions of the boiler efficiency, NOx emissions and boiler load, respectively; a

_{i,B}, a

_{i,Nx}and a

_{i,L}are the lagrange multipliers of boiler efficiency, NOx emissions and boiler load, respectively; b

_{B}, b

_{Nx}and b

_{L}are the constants of boiler efficiency, NOx emissions and boiler load, respectively; N

_{x}*, B* and L* are the previous values for the boiler combustion efficiency, the NOx emissions and the boiler load (at previous sample time), respectively.

#### 3.4. Model Validation and Analysis

#### 3.4.1. Experiment Setup

#### 3.4.2. The Model Performance Evaluation Indicators

#### 3.4.3. Experimental Result Analysis

## 4. Adaptive Nonlinear Model Predictive Control of the Boiler Combustion Efficiency

#### 4.1. The Structure of the Proposed Model Predictive Controller

#### 4.2. Rolling Optimization

_{B}, track the desired value of the boiler combustion efficiency ω

_{B}. The boiler combustion efficiency is considered as the performance index of the MPC controller for achieving the rolling optimization. Thus, the following rolling optimization problem is defined:

_{1}, A

_{2}, A

_{3}, A

_{4}, P

_{a}, S.

_{x}and the L for any particle are calculated according to Equation (5).

_{x}and the L requirements are satisfied, go to Step 5. Otherwise, set the relative particle fitness to 1000 and go to Step 6.

_{B}denotes the predicted boiler combustion efficiency of the kth predictive mode; ${\omega}_{B}^{k}$ is the desired boiler combustion efficiency at time k.

#### 4.3. Feedback Correction

#### 4.4. Validation of the Model Predictive Control Results

#### 4.4.1. Experiment Setup

#### 4.4.2. Experimental Result Analysis

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Acronyms

NOx | Nitrogen Oxides | CFD | Computational fluid dynamics |

ARIMA | Autoregressive integrated moving average | PLS | Partial least squares |

ANN | Artificial neural networking | BP | Back-propagation |

MLP | Multilayer perceptron | LSSVM | Least square support vector machine; |

DE | Differential evolution | SIS | Supervisor information system |

DCS | Distribution control system | PCA | Principal component analysis |

KNN | Kth Nearest Neighbor | PSO | Particle swarm optimization |

GA | Genetic algorithm | ADRC | Active disturbance rejection controller |

MFC-VRFT | Model-Free control and virtual reference feedback tuning | MPC | Model predictive control |

B | Boiler efficiency | L | Boiler load |

N_{x} | NOx emissions | DELSSVM | DE-based LSSVM |

MAE | Mean absolute error | MAE | Mean absolute error |

RMSE | Root mean square error | MAPE | Mean absolute percentage error |

DDMMF | A data-driven modeling method with feature selection capability | BMPC | Model predictive controller |

PID | Proportion-integral-derivative | RBF | Radial based function |

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**Figure 2.**Flowchart of the differential evolution-based least square support vector machine (DELSSVM) algorithm.

**Figure 4.**The prediction errors obtained from different prediction models for operating condition 1.

Name | Nomenclature | Symbol | Unit | Classical | |
---|---|---|---|---|---|

Input variable | Controllable variable | Coal feed | Fc | t/h | B/L/N_{x} |

Main feed-water flow | M | t/h | B/L/N_{x} | ||

Coal mill inlet airflow A | A_{1} | t/h | B/L/N_{x} | ||

Coal mill inlet airflow B | A_{2} | t/h | B/L/N_{x} | ||

Coal mill inlet airflow C | A_{3} | t/h | B/L/N_{x} | ||

Coal mill inlet airflow D | A_{4} | t/h | B/L/N_{x} | ||

Primary air flow | P_{a} | t/h | B/L/N_{x} | ||

Secondary air flow | S | t/h | B/L/N_{x} | ||

State variable | The flue gas oxygen content A | O_{1} | % | B/N_{x} | |

The flue gas oxygen content B | O_{2} | % | B/N_{x} | ||

Coal mill inlet temperature A | T_{1} | °C | B | ||

Coal mill inlet temperature B | T_{2} | °C | B/N_{x} | ||

Coal mill inlet temperature C | T_{3} | °C | B | ||

Coal mill inlet temperature D | T_{4} | °C | B/N_{x} | ||

Furnace pressure | P | Pa | B | ||

Output variable | NOx emissions in flue gas | N_{x} | mg/m^{3} | ||

Boiler combustion efficiency | B | % | |||

Boiler load | L | MW |

Parameter | Number (Condition 1) | Number (Condition 2) | ||||
---|---|---|---|---|---|---|

N_{x} | B | L | N_{x} | B | L | |

Train data (samples) | 800 | 800 | 800 | 600 | 600 | 600 |

Test data (samples) | 300 | 300 | 300 | 270 | 270 | 270 |

Sample Interval (min) | 1 | 1 | 1 | 1 | 1 | 1 |

Input variable | 13 | 16 | 9 | 13 | 16 | 9 |

Output variable | 1 | 1 | 1 | 1 | 1 | 1 |

Metric | Definition | Equation |
---|---|---|

MAE | Mean absolute error | $MAE=({\displaystyle {\sum}_{k=1}^{K}\left|{Y}_{k}-{\widehat{Y}}_{k}\right|})/K$ |

MAPE | Mean absolute percentage error | $MAPE=({\displaystyle {\sum}_{k=1}^{K}\left|{Y}_{k}-{\widehat{Y}}_{k}\right|}/{Y}_{k})/K$ |

RMSE | Root mean square error | $RMSE=\sqrt{({{\displaystyle {\sum}_{k=1}^{K}\left|{Y}_{k}-{\widehat{Y}}_{k}\right|}}^{2}/(K-1))}$ |

**Table 4.**The prediction evaluation indicators of different objects for different models in operating condition 1.

NOx Emissions | Boiler Load | Boiler Efficiency | |||||||
---|---|---|---|---|---|---|---|---|---|

Model | MAE (mg/m^{3}) | RMSE (mg/m^{3}) | MARE (%) | MAE (MW) | RMSE (MW) | MARE (%) | MAE (%) | RMSE (%) | MARE (%) |

DDMMF | 1.697 | 13.466 | 4.9 × 10^{−4} | 0.001 | 4.244 | 0.001 | 0.010 | 0.162 | 7.3 × 10^{−}^{5} |

MLP | 1.766 | 14.138 | 9.4 × 10^{−4} | 0.029 | 11.567 | 0.011 | 0.058 | 0.184 | 2.0 × 10^{−4} |

ARIMA | 2.559 | 28.249 | 1.9 × 10^{−}^{3} | 3.520 | 21.563 | 0.789 | 0.040 | 0.712 | 5.1 × 10^{−4} |

PLS | 2.321 | 15.844 | 3.0 × 10^{−}^{3} | 0.963 | 35.940 | 0.631 | 0.118 | 1.628 | 2.8 × 10^{−4} |

**Table 5.**The prediction evaluation indicators of different objects for different models in operating condition 2.

NOx Emissions | Boiler Load | Boiler Efficiency | |||||||
---|---|---|---|---|---|---|---|---|---|

Model | MAE (mg/m^{3}) | RMSE (mg/m^{3}) | MARE (%) | MAE (MW) | RMSE (MW) | MARE (%) | MAE (%) | RMSE (%) | MARE (%) |

DDMMF | 1.518 | 1.674 | 0.329 | 0.519 | 1.789 | 0.003 | 0.021 | 0.474 | 1.5 × 10^{−4} |

MLP | 1.883 | 7.832 | 1.899 | 4.683 | 6.649 | 0.012 | 0.149 | 2.975 | 8.4 × 10^{−4} |

ARIMA | 5.405 | 15.062 | 1.054 | 0.001 | 6.634 | 0.005 | 0.128 | 0.981 | 3.7 × 10^{−4} |

PLS | 2.321 | 16.946 | 2.443 | 0.115 | 4.630 | 0.005 | 0.065 | 0.684 | 5.4 × 10^{−4} |

Parameter | Number (Condition 1) | Number (Condition 2) | ||||
---|---|---|---|---|---|---|

N_{x} | B | L | N_{x} | B | L | |

Test data (samples) | 300 | 300 | 300 | 270 | 270 | 270 |

Sample Interval (min) | 1 | 1 | 1 | 1 | 1 | 1 |

Input variable | 13 | 16 | 9 | 13 | 16 | 9 |

Output variable | 1 | 1 | 1 | 1 | 1 | 1 |

Description | The Predictive Control Error of NOx Emissions / Boiler Load (%) | The Percentage of NOx Emissions Decline by BMPC | The Percentage of Boiler Efficiency Improvement by BMPC | Number of Samples |
---|---|---|---|---|

the operation condition 1 | −4.911~4.969 | 1.887 | 1.613 | 300 |

the operation condition 2 | −4.989~4.995 | 2.166 | 1.136 | 270 |

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## Share and Cite

**MDPI and ACS Style**

Tang, Z.; Wu, X.; Cao, S.
Adaptive Nonlinear Model Predictive Control of the Combustion Efficiency under the NOx Emissions and Load Constraints. *Energies* **2019**, *12*, 1738.
https://doi.org/10.3390/en12091738

**AMA Style**

Tang Z, Wu X, Cao S.
Adaptive Nonlinear Model Predictive Control of the Combustion Efficiency under the NOx Emissions and Load Constraints. *Energies*. 2019; 12(9):1738.
https://doi.org/10.3390/en12091738

**Chicago/Turabian Style**

Tang, Zhenhao, Xiaoyan Wu, and Shengxian Cao.
2019. "Adaptive Nonlinear Model Predictive Control of the Combustion Efficiency under the NOx Emissions and Load Constraints" *Energies* 12, no. 9: 1738.
https://doi.org/10.3390/en12091738