# An Efficient Robust Predictive Control of Main Steam Temperature of Coal-Fired Power Plant

^{*}

## Abstract

**:**

## 1. Introduction

- The excessively high MST results in serious damage of the superheater and inlet pipe of turbine;
- The excessively low MST decreases the net efficiency of power plant, and moreover, the steam in the last stage of the low pressure turbine may become wet under low MST condition, which endangers the turbine blades;
- The frequent temperature fluctuation worsens the heat exchanging in superheater and increases thermal stress of the superheater and turbine cylinder, which will bring material damage to the plant.

- Numerical optimization problem must be solved at each sampling interval; the online computational effort of these control algorithms is too heavy;
- MST system is operated under complicated working circumstance, such as aging of equipment, complicated combustion process involved in boiler and unpredictable disturbance, however, the robustness of control strategies is rarely involved in the control design.

- An improved offline RMPC approach is proposed by introducing two extra parameters for a better convergence of the recursive algorithm;
- A manipulated variable target observer is developed based on the center parameter of zonotope-type prediction model, which can help the RMPC achieve an offset-free control of the MST.

## 2. Main Steam Temperature System

#### 2.1. System Description

#### 2.2. Simulation Model

## 3. Problem Formulation

**Theorem**

**1**

## 4. An Improved Offline Robust Model Predictive Control Approach for MST System

#### 4.1. The Offline Design for RMPC Control Law

- Partition ${\Phi}_{0}$ into several simplexes ${S}_{i}$, $i\in \{1,2,\cdots ,l\}$ applying Delaunay triangulation [37], i.e., ${\Phi}_{0}=\{{S}_{1},\cdots ,{S}_{l}\}$, calculate and store respectively $V{V}_{i}\stackrel{\Delta}{=}{\left[\begin{array}{ccc}V{}^{\ast}\left({x}_{p1}\right)& \cdots & {V}^{\ast}\left({x}_{p(nx+1)}\right)\end{array}\right]}^{T}$ and ${U}_{i}\stackrel{\Delta}{\phantom{\rule{0.0pt}{0ex}}=}{\left[\begin{array}{ccc}{u}^{\ast}\left({x}_{p1}\right)& \cdots & {u}^{\ast}\left({x}_{p(nx+1)}\right)\end{array}\right]}^{T}$ at simplexes vertexes state points via (10), build ${M}_{i}$, preset threshold $\sigma $, $\alpha $ and $\beta $, let $i=1$;
- If $i\le l$, select the current simplex ${S}_{i}$ and turn to Step 3; if $i>l$, algorithm ends, return $U\stackrel{\Delta}{=}\begin{array}{ccc}[{U}_{1}& \cdots & {U}_{h}\end{array}]$ and $M\stackrel{\Delta}{=}\left[\begin{array}{ccc}{M}_{1}& \cdots & {M}_{h}\end{array}\right]$, h is size of well-partitioned space ${\Phi}_{0}$;
- Compare the size of $\alpha $ and conditional number of ${M}_{i}$, if $cond\left({M}_{i}\right)>\alpha $, delete ${S}_{i}$, $l=l-1$, and turn to Step 2; if $cond\left({M}_{i}\right)\le \alpha $, turn to Step 4;
- Obtain $\underset{{x}_{pmax}\in S}{max}{\xi}_{a}\left({x}_{pmax}\right)$, ${x}_{pmax}$, ${V}^{\ast}\left({x}_{pmax}\right)$ and ${u}^{\ast}\left({x}_{pmax}\right)$ via (19), if $\underset{{x}_{pmax}\in S}{max}{\xi}_{a}\left({x}_{pmax}\right)\le \sigma $, $i=i+1$, and turn to Step 2; if $\underset{{x}_{pmax}\in S}{max}{\xi}_{a}\left({x}_{pmax}\right)>\sigma $, replace the $nx+1$ vertexes of ${S}_{i}$ with ${x}_{pmax}$ in sequence yielding $nx+1$ new simplex, add them to ${\Phi}_{0}$, and delete ${S}_{i}$, $l=l+nx$, turn to Step 5;
- Determine whether the longest side is $\beta $ times longer than the shortest side, if not, turn to Step 2; if yes, take the midpoint of the longest side as a new point ${x}_{pz}$, calculate ${V}^{\ast}\left({x}_{pz}\right)$ and ${u}^{\ast}\left({x}_{pz}\right)$, two end points of the longest side are replaced with ${x}_{pz}$ in sequence, then two simplexes yield, delete ${S}_{i}$, $l=l+1$, turn to Step 2.

**Remark**

**1.**

**Remark**

**2.**

#### 4.2. The Online Implementation for Offline Designed RMPC Control Law

**Remark**

**3.**

## 5. Simulation Results

#### 5.1. Establishment of the Zonotope-Type Uncertain Model for MST System

#### 5.2. Control Simulation for MST System

**Case 1. Power plant unit load varies**

- the proposed OFAERMPC;
- incremental model predictive controller (IMPC) based on the nominal model of the identified zonotope (29) with weighting coefficients ${L}_{x}=1$ and ${L}_{u}=0.01$, control horizon 5, prediction horizon 500 and sampling interval $Ts$ = 5 s;
- digital PI controller with proportional coefficient 5.26 and integral time 292.22 (design by matlab PID controller tuning modular);
- standard AEMPC.

**Case 2. Unpredictable disturbance occurs**

**Case 3. Plant behavior changes**

## 6. Conclusions

- In the offline design stage, an explicit RMPC control law design method with improved convergence is proposed by introducing two extra parameters;
- Based on the nominal model of zonotope, a manipulated variable target observer is developed to make control results no offset exists.

## Author Contributions

## Funding

## Conflicts of Interest

## Abbreviations

AEMPC | Approximated explicit model predictive control |

CFPP | Coal-fired power plant |

MIMO | Multiple inputs multiple outputs |

MPC | Model predictive control |

MST | Main steam temperature |

NN | Neural network |

OFAERMPC | Offset-free approximated explicit robust model predictive control |

PWA | Piecewise affine |

PID | Proportion integration differentiation |

RMPC | Robust model predictive control |

SISO | Single input single output |

SMI | Set membership identification |

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**Figure 3.**Operating data of MST system during unit load varying (gray line: steam temperature at attemperator inlet $Ti$; red line: MST; blue line: steam temperature at attemperator outlet).

**Figure 5.**Routine operating data for main steam temperature (MST) system identification and verification.

**Figure 6.**Verification of identified uncertain model (blue thin line: identified nominal model output; blue heavy line: identified uncertain model output bounds; black line: real output).

**Figure 7.**Step test of identified uncertain model (blue thin line: identified nominal model output; blue heavy line: identified uncertain model output bounds).

**Figure 9.**MST for a 500 MW–950 MW unit load variation (blue line: OFAERMPC; green line: incremental model predictive controller (IMPC); red line: PI; gray line: standard AEMPC; black line: set point of MST).

**Figure 10.**Valve position of MST for a 500 MW–950 MW unit load variation (blue line: OFAERMPC; green line: IMPC; red line: PI; gray line: standard AEMPC).

**Figure 12.**MST when unpredictable disturbance occurs (blue line: OFAERMPC; green line: IMPC; red line: PI; black line: set point of MST).

**Figure 13.**MST when unpredictable disturbance occurs (blue line: OFAERMPC; green line: IMPC; red line: PI).

**Figure 14.**MST for an unknown change of dynamic property of controlled system (blue line: OFAERMPC; green line: IMPC; red line: PI; black line: set point of MST).

**Figure 15.**Valve position for an unknown change of dynamic property of controlled system (blue line: OFAERMPC; green line: IMPC; red line: PI).

Order of pre-estimation model | 8 |

Error bound to be restrained$\Gamma $ | 1.76 |

Weighting ratio${w}_{H}/{w}_{e}$ | 50 |

${\mathit{v}}_{1}$ | ${\mathit{v}}_{2}$ | ${\mathit{v}}_{3}$ | ${\mathit{v}}_{4}$ | ${\mathit{v}}_{5}$ | ${\mathit{v}}_{6}$ |
---|---|---|---|---|---|

$-0.989561$ | $-0.983721$ | $-0.989562$ | $-0.983721$ | $-0.983722$ | $-0.989562$ |

$-0.002991$ | $-0.000593$ | $-0.002991$ | $-0.000562$ | $-0.000562$ | $-0.002960$ |

**Table 3.**Offline performance of OFAERMPC and standard approximated explicit model predictive control (AEMPC).

Computation Time | Number of Subspaces | |
---|---|---|

Standard AEMPC | 3619 s | 1914 |

OFAERMPC | 170 s | 108 |

PI | IMPC | OLRMPC | OFAERMPC | |
---|---|---|---|---|

Performance index | 10.04 | 9.32 | 8.80 | 8.10 |

Total Simulation time | 0.24 | 11.94 | 8.74 | 0.96 |

PI | IMPC | OFAERMPC | |
---|---|---|---|

Performance index | 5.05 | 0.71 | 0.55 |

Total Simulation time | 0.22 | 0.05 | 0.98 |

PI | IMPC | OFAERMPC | |
---|---|---|---|

Performance index (unchanged behavior) | 5.36 | 0.88 | 0.68 |

Performance index (changed behavior) | 7.17 | 3.32 | 2.28 |

Total Simulation time | 0.23 | 10.36 | 0.98 |

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**MDPI and ACS Style**

Wang, D.; Wu, X.; Shen, J. An Efficient Robust Predictive Control of Main Steam Temperature of Coal-Fired Power Plant. *Energies* **2020**, *13*, 3775.
https://doi.org/10.3390/en13153775

**AMA Style**

Wang D, Wu X, Shen J. An Efficient Robust Predictive Control of Main Steam Temperature of Coal-Fired Power Plant. *Energies*. 2020; 13(15):3775.
https://doi.org/10.3390/en13153775

**Chicago/Turabian Style**

Wang, Di, Xiao Wu, and Jiong Shen. 2020. "An Efficient Robust Predictive Control of Main Steam Temperature of Coal-Fired Power Plant" *Energies* 13, no. 15: 3775.
https://doi.org/10.3390/en13153775