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Exploitation of multi-models identification with decoupled states in twin shaft gas turbine variables for its diagnosis based on parity space approach

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Abstract

In practice, model-based fault diagnosis methods are essential to improve availability with reduced operating costs and good operational reliability of industrial systems. This is based firstly on the choice of the model identification method adapted to the system, depending on the complexity of the system and its interaction with its environment. Then, on the choice of an adequate diagnostic strategy for the generation of system failure indicators. In this work, the identification problem of the model variables of a double-shaft gas turbine is treated, to deal with the dynamics of model nonlinearities of this rotating machine. Hence, the equations which govern this turbine are carried out, using the local multi-models’ techniques with decoupled states, from the input/output measurements collected on the examined turbine. To best characterize their dynamic behavior in diverse operating areas. Subsequently, the resulting multi-model decoupled states are used to develop a fault diagnosis approach for this turbine. This makes it possible to generate symptoms of turbine failure from consistency tests between the measurements extracted on its real behavior, and the estimated signals which translate the reference behavior, given by the obtained multi-models. The obtained results in this work show the implementation efficiency of the proposed techniques of modeling and estimation of the examined decoupled turbine states, up to the phase of its implementation in the diagnostic strategy of the examined turbine based on the parity space approach.

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Abbreviations

\(L\) :

Partition blocks

\(\chi\) :

Coefficient of constriction

\(\sigma\) :

Coefficient of dispersion

\(c_{1}\) and \(c_{2}\) :

Acceleration coefficients

\(w_{f}\) :

Fuel flow

\(D\left( h \right)\) :

Research direction in parametric space

\(\Delta\) :

Adjustment factor

\(\mu_{i}\) :

Weighting function (activation)

\(J\left( \theta \right)\) :

Objective cost minimization function

\(\omega_{i}\) :

Weight functions

\(G\) :

Gradient

GT:

Gas Turbine

h:

Iteration index

\(H\left( \theta \right)\) :

Hessian

HP:

High pressure

\(\xi\) :

Function index

\(\mu_{i}\) :

Index of weighting functions

\(k\) :

Discrete time iteration index

\(i\) :

Local model index

\(j\) :

Multi-model output index

Li:

Sub-models i (i = 2, 3, 4)

LP:

Low pressure

A, B, C & D:

State model matrix

\(I\) :

Identity matrix

\(A_{i}\), \(B_{i}\), \(C_{i,j}\) and \(D_{i,j}\) :

Matrices of decoupled state multi-models

\(g(t)\) :

Best overall position

MMA:

Multi model approach

MOO:

Multi objective optimization

N:

Number of measurements

NGP:

High pressure turbine

\(p\) :

Number of multi-model outputs

NPT:

Low pressure turbine

NSGA:

Non dominated sorting genetic algorithm

\(L\) :

Parity space order

\(\lambda\) :

Regularization parameter

\(w\) :

Inertia weight

\(x_{i} (t)\) :

Position of the particle

PSO:

Particle Swarm Optimization

\(y\) :

Model output

\(\hat{y}\left( k \right)\) :

Estimated multi-model output

\(y\left( k \right)\) :

Measured system output

\(T_{7}\) :

Exhaust temperature

\(T_{5}\) :

HP turbine temperature

\(c_{i}\) :

Center variable

\(r_{1}\), \(r_{2}\) :

Random variables

\(u\) :

Control vector

\(f(k)\) :

Fault vector

\(\theta\) :

Parameter vector

\(V_{s}\) :

Parity vector

\(r(k)\) :

Residuals vector

\(x\) :

State vector

\(v_{i} (t)\) :

Particle speed

\(N_{npt}\) :

LP turbine rotational speed

\(N_{ngp}\) :

HP turbine rotation speed

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Acknowledgements

This article is the result of a work on the identification of model variables of a twin-shaft gas turbine using the techniques of local decoupled state multi-models, for their use in the diagnosis of faults of this rotating machine with a parity space approach, based on the input/output measurements collected on the turbine under examination. This work is supported and carried out by the joint research team on gas turbines with the applied automation and industrial diagnosis laboratory, University of Djelfa, Algeria. However, the authors express their sincere thanks to the General Directorate of Scientific Research and Technological Development (DGRSDT), Algeria.

Author information

Authors and Affiliations

  1. Gas Turbine Joint Research Team, University of Djelfa, 17000, Djelfa, DZ, Algeria

    Sidali Aissat & Ahmed Hafaifa

  2. Applied Automation and Industrial Diagnostics Laboratory, Faculty of Science and Technology, University of Djelfa, 17000, Djelfa, DZ, Algeria

    Sidali Aissat, Ahmed Hafaifa & Mouloud Guemana

  3. Faculty of Science and Technology, University of Bordj Bou Arreridj, 34000, Bordj Bou Arreridj, DZ, Algeria

    Abdelhamid Iratni

  4. Manufacturing Futures Research Institute, Faculty of Science, Engineering and Technology, Swinburne University of Technology, Melbourne, Australia

    XiaoQi Chen

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  1. Sidali Aissat
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  2. Ahmed Hafaifa
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  4. Mouloud Guemana
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Corresponding author

Correspondence to Ahmed Hafaifa.

Appendix

Appendix

1.1 Obtained gas turbine state model

The sub-models L2:

$$ \left\{ {\begin{array}{*{20}c} {A_{1} = \left[ {\begin{array}{*{20}c} 0 & 1 \\ { - 1.0014} & { - {0}{\text{.2364}}} \\ \end{array} } \right],B_{1} = \left[ {\begin{array}{*{20}c} {0} \\ {1} \\ \end{array} } \right],C_{1} = \left[ {\begin{array}{*{20}c} { - {0}{\text{.0020}}} & {{0}{\text{.0086}}} \\ { - {0}{\text{.0029}}} & {{0}{\text{.0124}}} \\ { - {0}{\text{.0226}}} & {{0}{\text{.0803}}} \\ { - {0}{\text{.0127}}} & {{0}{\text{.0368}}} \\ \end{array} } \right],D_{1} = \left[ {\begin{array}{*{20}c} {{1}{\text{.2763}}} \\ {{1}{\text{.2107}}} \\ {{9}{\text{.4792}}} \\ {{6}{\text{.1181}}} \\ \end{array} } \right]} \\ {A_{2} = \left[ {\begin{array}{*{20}c} {0} & {1} \\ {{0}{\text{.5152}}} & { - {0}{\text{.3829}}} \\ \end{array} } \right],B_{2} = \left[ {\begin{array}{*{20}c} 0 \\ 1 \\ \end{array} } \right],C_{2} = \left[ {\begin{array}{*{20}c} {{0}{\text{.0551}}} & {{0}{\text{.0284}}} \\ {{0}{\text{.0624}}} & {{ 0}{\text{.0676}}} \\ {{ 0}{\text{.6094}}} & {{0}{\text{.6500}}} \\ {{0}{\text{.3075}}} & {{0}{\text{.2969 }}} \\ \end{array} } \right],D_{2} = \left[ {\begin{array}{*{20}c} {{2}{\text{.9163}}} \\ {{1}{\text{.9804}}} \\ {{16}{\text{.1059}}} \\ {{13}{\text{.1460}}} \\ \end{array} } \right]} \\ \end{array} } \right. $$

The sub-models L3:

$$ \left\{ {\begin{array}{*{20}c} {A_{1} = \left[ {\begin{array}{*{20}c} 0 & 1 \\ {{0}{\text{.0181}}} & { - {0}{\text{.1221}}} \\ \end{array} } \right],B_{1} = \left[ {\begin{array}{*{20}c} {0} \\ {1} \\ \end{array} } \right],C_{1} = \left[ {\begin{array}{*{20}c} {{0}{\text{.0184}}} & { - {0}{\text{.0097}}} \\ { - {0}{\text{.0100 }}} & { - {0}{\text{.0055}}} \\ {{0}{\text{.1380}}} & { - {0}{\text{.1303}}} \\ {{0}{\text{.1258}}} & { - {0}{\text{.0317}}} \\ \end{array} } \right],D_{1} = \left[ {\begin{array}{*{20}c} {{1}{\text{.2880}}} \\ {{1}{\text{.2342}}} \\ {{9}{\text{.5753}}} \\ {{6}{\text{.1147}}} \\ \end{array} } \right]} \\ {A_{2} = \left[ {\begin{array}{*{20}c} 0 & 1 \\ {{0}{\text{.1365}}} & { - {0}{\text{.2635}}} \\ \end{array} } \right],B_{2} = \left[ {\begin{array}{*{20}c} {0} \\ {1} \\ \end{array} } \right],C_{2} = \left[ {\begin{array}{*{20}c} { - {0}{\text{.0358}}} & {{0}{\text{.0858}}} \\ {{ 0}{\text{.0581}}} & {{0}{\text{.1546}}} \\ { - {0}{\text{.0088}}} & {{1}{\text{.4640}}} \\ { - {0}{\text{.0772}}} & {{0}{\text{.4979}}} \\ \end{array} } \right],D_{2} = \left[ {\begin{array}{*{20}c} {{2}{\text{.7040}}} \\ {{1}{\text{.8058}}} \\ {{15}{\text{.0291}}} \\ {{12}{\text{.3068}}} \\ \end{array} } \right]} \\ {A_{3} = \left[ {\begin{array}{*{20}c} 0 & 1 \\ {{0}{\text{.2428}}} & { - {0}{\text{.2369}}} \\ \end{array} } \right],B_{3} = \left[ {\begin{array}{*{20}c} {0} \\ {1} \\ \end{array} } \right],C_{3} = \left[ {\begin{array}{*{20}c} {{0}{\text{.4731}}} & {{ 1}{\text{.5000}}} \\ {{0}{\text{.6655}}} & {{1}{\text{.9180}}} \\ {{6}{\text{.9454}}} & {{ 8}{\text{.9683}}} \\ {{3}{\text{.7276}}} & {{3}{\text{.4792}}} \\ \end{array} } \right],D_{3} = \left[ {\begin{array}{*{20}c} {{ 0}{\text{.7236}}} \\ { - {0}{\text{.6001}}} \\ {{0}{\text{.3148}}} \\ {{4}{\text{.0464}}} \\ \end{array} } \right]} \\ \end{array} } \right. $$

The sub-models L4:

$$ \left\{ {\begin{array}{*{20}c} {A_{1} = \left[ {\begin{array}{*{20}c} 0 & 1 \\ { - {0}{\text{.1081}}} & { - {0}{\text{.3903}}} \\ \end{array} } \right],B_{1} = \left[ {\begin{array}{*{20}c} {0} \\ {1} \\ \end{array} } \right],C_{1} = \left[ {\begin{array}{*{20}c} {{0}{\text{.1286}}} & {{0}{\text{.3375}}} \\ {{0}{\text{.1062}}} & {{0}{\text{.5097}}} \\ {{0}{\text{.7245}}} & {{2}{\text{.6477 }}} \\ {{0}{\text{.9639}}} & {{ 2}{\text{.0252}}} \\ \end{array} } \right],D_{1} = \left[ {\begin{array}{*{20}c} {{3}{\text{.3042}}} \\ {{1}{\text{.8189 }}} \\ {{18}{\text{.1980}}} \\ {{13}{\text{.1503}}} \\ \end{array} } \right]} \\ {A_{1} = \left[ {\begin{array}{*{20}c} 0 & 1 \\ { - {0}{\text{.0783 }}} & { - {0}{\text{.3716}}} \\ \end{array} } \right],B_{1} = \left[ {\begin{array}{*{20}c} {0} \\ {1} \\ \end{array} } \right],C_{1} = \left[ {\begin{array}{*{20}c} { - {0}{\text{.3499 }}} & { - {0}{\text{.3726}}} \\ { - {0}{\text{.1752 }}} & {{0}{\text{.0051}}} \\ { - {2}{\text{.5110}}} & {{0}{\text{.0391}}} \\ { - {0}{\text{.1959}}} & {{2}{\text{.1260 }}} \\ \end{array} } \right],D_{1} = \left[ {\begin{array}{*{20}c} {{1}{\text{.4521}}} \\ {{1}{\text{.7159}}} \\ {{9}{\text{.3378}}} \\ {{6}{\text{.8235}}} \\ \end{array} } \right]} \\ {A_{1} = \left[ {\begin{array}{*{20}c} 0 & 1 \\ { - {0}{\text{.0801}}} & { - {0}{\text{.2980}}} \\ \end{array} } \right],B_{1} = \left[ {\begin{array}{*{20}c} {0} \\ {1} \\ \end{array} } \right],C_{1} = \left[ {\begin{array}{*{20}c} {{0}{\text{.1892}}} & {{0}{\text{.2152}}} \\ {{ 0}{\text{.0961}}} & { - {0}{\text{.0819 }}} \\ {{2}{\text{.0746}}} & { - {0}{\text{.4899 }}} \\ {{0}{\text{.2079}}} & { - {1}{\text{.6416}}} \\ \end{array} } \right],D_{1} = \left[ {\begin{array}{*{20}c} {{1}{\text{.0837}}} \\ {{0}{\text{.9147}}} \\ {{ 9}{\text{.0677}}} \\ {{5}{\text{.4967 }}} \\ \end{array} } \right]} \\ {A_{1} = \left[ {\begin{array}{*{20}c} 0 & 1 \\ { - {0}{\text{.0525}}} & {{0}{\text{.0023}}} \\ \end{array} } \right],B_{1} = \left[ {\begin{array}{*{20}c} {0} \\ {1} \\ \end{array} } \right],C_{1} = \left[ {\begin{array}{*{20}c} {{0}{\text{.5542}}} & { - {0}{\text{.1214}}} \\ {{0}{\text{.5514}}} & {{ 0}{\text{.0574}}} \\ {{4}{\text{.6213 }}} & {{1}{\text{.5489}}} \\ {{ 3}{\text{.2510 }}} & { - {0}{\text{.1525}}} \\ \end{array} } \right],D_{1} = \left[ {\begin{array}{*{20}c} {{1}{\text{.9044}}} \\ {{ 1}{\text{.2023}}} \\ {{ 8}{\text{.3144}}} \\ {{7}{\text{.6873}}} \\ \end{array} } \right]} \\ \end{array} } \right. $$

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Aissat, S., Hafaifa, A., Iratni, A. et al. Exploitation of multi-models identification with decoupled states in twin shaft gas turbine variables for its diagnosis based on parity space approach. Int. J. Dynam. Control 10, 25–48 (2022). https://doi.org/10.1007/s40435-021-00804-5

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  • DOI: https://doi.org/10.1007/s40435-021-00804-5

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