# Estimation of Performance Parameters of Turbine Engine Components Using Experimental Data in Parametric Uncertainty Conditions

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Basic Identification Procedure

#### 2.2. Regularized Identification Procedure

_{i}is the weight coefficient that describes measurement error (${\mathbf{g}}_{\mathbf{i}}=\frac{1}{{\mathsf{\sigma}}_{\mathbf{i}}^{2}}$), y

_{i}is a component of vector $\overrightarrow{\mathbf{Y}}$ and ${\mathsf{\theta}}_{\mathbf{k}}$ is a component of vector $\overrightarrow{\mathsf{\Theta}}$.

#### 2.3. Numerical Simulation

- (1)
- Without measuring noise, for a wide range of α values;
- (2)
- With measuring noise, for selected α values.

- (1)
- Compressor discharge pressure p
_{2}; - (2)
- Turbine inlet temperature TIT;
- (3)
- Power turbine discharge temperature;
- (4)
- Rotational speeds of rotors N
_{2}and N_{1}.

- Deviation of mass flow rate: ${\mathsf{\Theta}}_{1\text{}\mathrm{sim}}$= ΔW
_{C}= −0.03; - Deviation of compressor efficiency: ${\mathsf{\Theta}}_{2\text{}\mathrm{sim}}$ = Δη
_{C}= −0.04.

- Deviation of flow rate: ${\mathsf{\Theta}}_{1}$ = ΔW
_{C}; - Deviation of compressor efficiency: ${\mathsf{\Theta}}_{2}$ = Δη
_{C}; - Deviation of HPT mass flow: ${\mathsf{\Theta}}_{3}$ = ΔW
_{HPT}; - Deviation of HPT efficiency: ${\mathsf{\Theta}}_{4}$ = Δη
_{HPT}.

_{av}is the average deviation between initial and estimated models:

#### 2.4. Regularized Identification Procedure Development Using a Priori Information

- Exact statement (for example, a part-load performance in a determined area is smooth);
- Statement in the form of limitations of the area of acceptable solutions (for example, the efficiency of the individual compressor cannot differ more than 3% from the efficiency of an “average” compressor, the performance of which is used in the initial model);
- Statement in the form of fuzzy information (for example, the gas temperature in the turbine will grow with the engine’s life);
- Statistical form (for example, probability density functions of parameters).

_{min}, x

_{max}) and $0\le \mathsf{\mu}(\mathrm{x})\le 1$. We will express each a priori information as a particular functional of a priori risk ${\mathsf{\Phi}}_{\mathrm{a}\text{}\mathrm{q}}(\overrightarrow{\mathsf{\theta}})$ and will determine the general functional of a priori risk as a linear composition of particular functionals:

**Case 1.**Limitations of some parameters are known. For example, it is known that 0 < η < 1 and 0 < σ < 1, etc. Using experience and calculation results, these limits can be significantly reduced; for example, 0.5 < η < 0.9 and 0.9 < σ < 0.99 (Figure 6).

**Case 2.**The a priori mathematical model is known. This can be the model with design maps of components or the model of the average engine, which is matched with previous testing results.

_{a}or Φ

_{a}and the difference between parameters that correspond to the matched and a priori models. These parameters may be the model parameters $\overrightarrow{\mathsf{\Theta}}$ as measured (for example fuel flow) or non-measured calculated parameters (for example thrust). The membership function, in this case, can be of symmetrical triangular shape and the functional of a priori risk ${\mathsf{\Phi}}_{\mathrm{a}\text{}\mathrm{q}}(\overrightarrow{\mathsf{\theta}})$ that characterizes the similarity between values of the parameter ${\mathrm{x}}_{\mathrm{q}}$ of matched and a priori models can be formed as

_{q}is a calculated parameter of the engine; ${\mathrm{x}}_{\mathrm{q}}({\overrightarrow{\mathrm{U}}}_{\mathrm{j}},\text{}\widehat{\overrightarrow{\mathsf{\Theta}}})$ is the value calculated by the model to be matched; ${\mathrm{x}}_{\mathrm{q}\text{}0}({\overrightarrow{\mathrm{U}}}_{\mathrm{j}},{\overrightarrow{\mathsf{\Theta}}}_{0})$—the value calculated by the a priori model; ${\overrightarrow{\mathsf{\Theta}}}_{0}$ are the parameters of the a priori model.

**Case 3.**Information about the confidence in different sets of experimental data obtained in different conditions with different precision.

**Case 4.**Confidence in the available maps of the engine components. For example, we know that the compressor map used in the model corresponds to the old version of the engine and is far from the actual map. Thus, we can express this knowledge in view of the confidence functions that are in this case the membership functions.

## 3. Results

_{i}, which were used to form the diagonal weight matrix G (Equation (8)).

_{C}

^{*}(c)—Scaling factor of CPR (Compressor Pressure Ratio);

_{C}

^{*}(a)—Factor of rotation in CPR-W

_{Ccor}plane of the compressor pressure map;

_{C}

^{*}(b)—Factor of rotation in CPR-N

_{cor}plane;

_{HPT}(c)—HPT flow coefficient;

_{PT}(c)—Power turbine flow coefficient;

_{C}

^{*}(a)—Factor of rotation in the η

_{C}

^{*}–W

_{Ccor}plane of the compressor efficiency map;

_{C}

^{*}(c)—Scaling factor of the compressor efficiency map;

_{CC}—Scaling factor of combustion efficiency;

#### 3.1. Least Squares Identification

#### 3.2. Regularized Identification

#### 3.3. Regularized Multi-Criteria Identification

_{net}= const, N

_{1}= const follows the normal distribution with the following mean squared error (Table 5).

_{a}(Equation (13)), and taking into account that in this case, the functionals Φ

_{e}and Φ

_{a}are homogeneous as they contain residuals by parameters of the same names. This facilitated the setting of weight coefficients in Equation (14). They had to relate to a scatter of measurements and a scatter of the engine parameters by series. This provided the composition of the functionals:

## 4. Discussion

## 5. Conclusions

- (1)
- LSM has low stability that can cause the engine simulation software to crash.
- (2)
- Conventional regularization improves the stability, but the intensive bias of the estimates needs to be excluded in advance.
- (3)
- A priori information has a physical sense, so it improves the stability and precision of the estimation. The introduced identification method is effective in ill-conditioned configurations and provides small bias of estimates.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Nomenclature

AI | Artificial intelligence |

av | Average value |

C | Generalized matrix of influence coefficients compressor |

CC | Combustion Chamber |

CPR | Compressor Pressure Ratio |

G | Weight diagonal matrix |

GPA | Gas Path Analysis |

H | Influence coefficient matrix |

HPC | High Pressure Compressor |

HPT | High Pressure Turbine |

I | Identity matrix |

ICM | Influence coefficient matrix |

N | Number of operating points, rotational speed |

LPT | Low Pressure Turbine |

LSM | Least Squares Method |

m | Number of measured parameters |

OP | Operating Point |

P | Pressure |

P_{net} | Net thrust |

r | Number of estimated parameters |

sim | Simulated value |

TIT | Turbine Inlet Temperature |

U | Parameters that determine the engine operating conditions |

W | Mass flow rate of air/gas |

W_{f} | Fuel flow |

Y | Measured parameters |

Z | Generalized vector of measured parameters |

α | Regularization (weighting) factor |

Δ | Parameter’s correction |

δ | Relative deviation |

η | Efficiency |

σ^{2} | Variance |

π | Pressure ratio |

σ | Pressure loss coefficient |

θ | Component’s map parameter |

→ | Vector |

^ | Estimated parameter |

* | Measured value |

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**Figure 8.**Experimental data vs models. (

**a**) fuel flow; (

**b**) high pressure rotor rotation speed; (

**c**) turbine inlet temperature; (

**d**) compressor pressure ratio.

Operating Mode | t_{H}, °C | P*_{H}, Pa | N_{2}, % | N_{1}, % | P_{net}, kW | W_{f}, kg/h | TIT, K | CPR |
---|---|---|---|---|---|---|---|---|

1st cruise | 12.0 | 100,192 | 94.96 | 94.58 | 1223 | 372.5 | 1017 | 8.05 |

2nd cruise | 11.6 | 100,192 | 97.05 | 96.87 | 1521 | 439.0 | 1073 | 8.91 |

Nominal | 12.3 | 100,192 | 98.67 | 98.51 | 1729 | 482.1 | 1106.5 | 9.41 |

Maximum | 11.9 | 100,192 | 102.34 | 102.02 | 2203 | 596.4 | 1197 | 10.54 |

σ_{i}, % | 0.4 | 0.3 | 0.085 | 0.085 | 0.4 | 0.3 | 0.4 | 0.3 |

${\mathsf{\pi}}_{\mathbf{H}\mathbf{P}\mathbf{C}}^{*}(\mathbf{a})$ | ${\mathsf{\pi}}_{\mathbf{H}\mathbf{P}\mathbf{C}}^{*}(\mathbf{b})$ | ${\mathsf{\pi}}_{\mathbf{H}\mathbf{P}\mathbf{C}}^{*}(\mathbf{c})$ | W_{HPT}(c) | ${\mathsf{\eta}}_{\mathbf{H}\mathbf{P}\mathbf{C}}^{*}(\mathbf{a})$ | ${\mathsf{\eta}}_{\mathbf{H}\mathbf{P}\mathbf{C}}^{*}(\mathbf{c})$ | W_{HPT}(c) | W_{PT}(c) | ${\mathsf{\eta}}_{\mathbf{P}\mathbf{T}}^{*}(\mathbf{c})$ | η_{CC} |
---|---|---|---|---|---|---|---|---|---|

−0.0788 | 0.0125 | 0.023 | 0.0464 | 0.0383 | 0.0548 | 0.0265 | −0.0938 | 0.01 | 0.02 |

Iteration | ${\mathsf{\delta}\mathsf{\pi}}_{\mathbf{C}}^{*}\left(\mathbf{a}\right)$ | $\mathsf{\delta}{\mathbf{W}}_{\mathbf{H}\mathbf{P}\mathbf{T}}$ | $\mathsf{\delta}{\mathbf{W}}_{\mathbf{P}\mathbf{T}}$ | ${\mathsf{\delta}\mathsf{\eta}}_{\mathbf{C}}^{*}\left(\mathbf{a}\right)$ | ${\mathsf{\delta}\mathsf{\eta}}_{\mathbf{C}}^{*}\left(\mathbf{a}\right)$ |
---|---|---|---|---|---|

1 | 0.004 | 0.17 | 0.39 | −0.033 | 0.33 |

2 | 0.003 | 0.18 | 0.15 | −0.018 | 0.08 |

3 | 0.003 | 0.09 | 0.08 | −0.016 | 0.02 |

Operating Point | t_{H}, °C | P*_{H}, Pa | N_{2}, % | N_{1}, % | P_{net}, kW | W_{f}, kg/h | TIT, K | CPR |
---|---|---|---|---|---|---|---|---|

1st cruise | 12.0 | 100,192 | 92.38 | 98.42 | 1223 | 374.2 | 1050.3 | 7.48 |

2nd cruise | 11.6 | 100,192 | 94.44 | 98.20 | 1521 | 439.8 | 1105.7 | 8.30 |

Nominal | 12.3 | 100,192 | 96.00 | 98.10 | 1729 | 490.1 | 1146.5 | 8.84 |

Maximum | 11.9 | 100,192 | 98.86 | 98.12 | 2203 | 601.0 | 1224.2 | 10.18 |

Parameter | N_{2} | W_{f} | T_{3} | π_{c} |
---|---|---|---|---|

Mean squared error | 0.85% | 0.7% | 1.1% | 0.6% |

Iteration | ${\mathsf{\delta}\mathsf{\pi}}_{\mathbf{C}}^{*}\left(\mathbf{a}\right)$ | $\mathsf{\delta}{\mathbf{W}}_{\mathbf{H}\mathbf{P}\mathbf{T}}$ | $\mathsf{\delta}{\mathbf{W}}_{\mathbf{P}\mathbf{T}}$ | ${\mathsf{\delta}\mathsf{\eta}}_{\mathbf{C}}^{*}\left(\mathbf{a}\right)$ | ${\mathsf{\delta}\mathsf{\eta}}_{\mathbf{C}}^{*}\left(\mathbf{a}\right)$ |
---|---|---|---|---|---|

1 | 0.005 | 0.15 | 0.27 | −0.04 | 0.28 |

2 | 0.003 | 0.12 | 0.13 | −0.02 | 0.06 |

3 | 0.002 | 0.07 | 0.06 | −0.015 | 0.02 |

Operating Point | t_{H}, °C | P*_{H}, Pa | N_{2}, % | N_{1}, % | P_{net}, kW | W_{f}, kg/h | TIT, K | CPR |
---|---|---|---|---|---|---|---|---|

1st cruise | 12.0 | 100,192 | 94.58 | 94.58 | 1223 | 371.7 | 1013 | 7.98 |

2nd cruise | 11.6 | 100,192 | 96.87 | 96.87 | 1521 | 438.2 | 1066 | 8.85 |

Nominal | 12.3 | 100,192 | 98.51 | 98.51 | 1729 | 484.0 | 1100.5 | 9.43 |

Maximum | 11.9 | 100,192 | 102.02 | 102.02 | 2203 | 600.2 | 1186 | 10.65 |

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## Share and Cite

**MDPI and ACS Style**

Khustochka, O.; Yepifanov, S.; Zelenskyi, R.; Przysowa, R.
Estimation of Performance Parameters of Turbine Engine Components Using Experimental Data in Parametric Uncertainty Conditions. *Aerospace* **2020**, *7*, 6.
https://doi.org/10.3390/aerospace7010006

**AMA Style**

Khustochka O, Yepifanov S, Zelenskyi R, Przysowa R.
Estimation of Performance Parameters of Turbine Engine Components Using Experimental Data in Parametric Uncertainty Conditions. *Aerospace*. 2020; 7(1):6.
https://doi.org/10.3390/aerospace7010006

**Chicago/Turabian Style**

Khustochka, Olexandr, Sergiy Yepifanov, Roman Zelenskyi, and Radoslaw Przysowa.
2020. "Estimation of Performance Parameters of Turbine Engine Components Using Experimental Data in Parametric Uncertainty Conditions" *Aerospace* 7, no. 1: 6.
https://doi.org/10.3390/aerospace7010006