Nonlinear Dynamic Model Identification Of Gas Turbine Engine.

Design of gas turbine engine itself and its diagnosing systems is deeply related with calculations on the base of gaspath mathematical models of different complexity, and the identification procedure represents not only an effective instrument to raise an accuracy of the models and related calculations but also an important component of diagnosing algorithms. Practical application of such instruments as nonlinear mathematical model and its identification procedure becomes a common practice [1,2]. Last computer progress stimulates an elaboration and application of more sophisticated instruments.

Numerous developments of described models for gas turbines of basic schemes and static model identification researches were fulfilled using the universal gas turbine model approach [3-5]. The procedure of dynamic model identification of gas turbine for gas pumping unit was elaborated later on a base of this experience.

Numerous gas turbine gaspath models widely used for the diagnosing may be classified as follows: a) linear and nonlinear; b) static and dynamic; c) according to a depth of description.

Two types of mentioned models are principal and most complex: nonlinear static and dynamic models. Another types may be generated from these ones.

Every component (compressor, turbine, combustion chamber etc.) of the multi-regime nonlinear static model is presented by its performance. In common form this model may be described as follows

                              (1)

where the vector  of gaspath parameters depends on the vector  of engine regime parameters and the vector  of gaspath state parameters of the dimension r.

In the capacity of state parameters the parameters of gas turbine component performances (performance parameters) are chosen. These parameters are able to move and deform the component performances and, as a result, to adjust the model performance to the real one or simulate engine failures.

Mathematically the relation (1) is formed after the solution of the system of algebraic equations reflecting the conditions of components combined work on steady-state regimes.

Modification of these equations to the conditions of transient regimes permits in the base of the static model to form a dynamic model of the following view:

                       (2)

where the vector of regime parameters, in contrast to the static model, is given as a function of time , and a separate influence of time variable t is explained by an inertia nature of gas turbine dynamic processes.

In the model (2) the influence of main inertia factors, the inertia moments JGG and JFT of gas-generator and free turbine rotors correspondingly, is taken into account. The appropriate parameters δJGG and δJFT were introduced in the vector  for correcting the initial inertia moments JGG and JFT in identification aims.

The real measured values  differs from the model values  due to the model and measurement errors, and the object of dynamic model identification is to find such state parameter estimations , which minimize the error level of model. That is why the task of dynamic model identification may be classified as the optimization problem, and the estimations may be found according to the following expression written in common form

.            (4)

Due to model non-linearity any exact analytic solution for the estimations  does no exist, and following numeric iteration procedure is applied.

For any iteration of number n+1 the current solution may be written as a sum of previous solution and current correction

,                   (5)

where the correction  presents a regularized solution

      (6)

of linear system

,                      (7)

where  - Fisher information matrix;    W – weight matrix; I – single matrix;

 - generalized vector of deviations of the model values from measured ones formed in  calculation points of a transient process interval selected for the identification;

 - generalized matrix of influence of state parameters  on gaspath measured parameters  calculated in these same points.

The iterations are repeated until the moment when the estimations reach their stable level.

During the iteration calculations dynamic model is called one time to calculate the vector  and r+1 times to form the matrix .

Developed software of the nonlinear dynamic model identification presents a complex program package and approximately consists of 70 program modules of different hierarchical levels. The modules of highest levels are presented in the following list

·   Iteration Procedure of Identification (level 1);

·   Reading Initial Data (level 2);

·   Computing Coefficient Matrix (level 2);

·   Regularized Solution of Linear System (level 2);

·   Calculation of Transient Process (level 3);

·   Static Regime Calculation (level 4);

·   Solution of Differential Equation System of Dynamics (level 4);

·   Real Time Reading the Regime Parameters (level 5);

·   Computing Right Parts of Differential Equations (level 5);

·   Gaspath Parameters Calculation (level 6).

Another modules lie below and form the block of through consecutive calculation of gas path parameters.

The testing of software was divided into two phases: a) testing on simulated data; b) testing on real data. Two objects are pursued: at first, to check the identification procedure functioning; at second, to estimate the possibility of the procedure application in gas turbine control and diagnosing systems.

Testing on simulated data has demonstrated the software correct functioning and a quick convergence of the iteration procedure.

Automatic maintenance parameter registration was used as a source of the real information and the following parameter structure was chosen: temperature tН and pressure PН of atmospheric air, fuel consumption Gf and power turbine rotation speed nPT (vector ); 7 regular measured parameters (vector ); 6 estimated parameters of gas turbine component performance – two fault indices (a gas consumption parameter and an efficiency parameter) for every principal mechanic component: axial compressor, gas generator turbine, free turbine (vector ).

Originally, 4 archives of transient regimes were used for a verification of common properties of the identification procedure. Table 1 contains following average characteristics of the identification: δY1 ,  δY5 – model deviations from measured values in the first and final iterations; . ΔδY5 - gaspath parameter increment in the final iteration; Θ1, Θ5 – estimated performance parameters in the first and final iterations; ΔΘ5 – performance parameter increment in the final iteration.

Reduction of the model deviation δY during the iteration process confirms a correct identification functioning. Low final increments ΔδY5 and ΔΘ5 as well as a short distance between the first and final estimations testify a quick convergence. However, final deviation level remains considerable after the identification and further structure variation of performance parameters does not permit to reduce this level. That is why an effect of inertia parameters was analyzed too.

Table 1

Results of the identification (4 real data archives)

Archives number

Gaspath parameters

Performance parameters

1

δY1

0.0593

Θ1

0.0730

δY5

0.0386

Θ5

0.0752

ΔδY5

0.0008

ΔΘ5

0.0006

2

δY1

0.0815

Θ1

0.0875

δY5

0.0426

Θ5

0.0823

ΔδY5

0.0004

ΔΘ5

0.0006

3

δY1

0.0750

Θ1

0.0884

δY5

0.0417

Θ5

0.0849

ΔδY5

0.0002

ΔΘ5

0.0004

4

δY1

0.0641

Θ1

0.0735

δY5

0.0410

Θ5

0.0722

ΔδY5

0.0008

ΔΘ5

0.0006

For the archive 4 the following structure variants was considered: 1) 6 performance parameters; 2) 6 performance parameters and 2 inertia parameters (δJGG and δJFT); 3) 2 inertia parameters.

The results of new calculations presented in the Table 2 indicate a week influence of parameters δJGG and δJFT on the identification process even if to increase the inertia moments in some times. The possible reason may be related with slow transient regime, which was analyzed (30 percent fuel consumption rise on the 20 second time interval).

Table 2

Results of the identification (structure variation of estimated performance parameters)

Variant number

Gaspath parameters

Performance parameters

1

δY1

0.0580

Θ1

0.0680

δY5

0.0391

Θ5

0.0696

ΔδY5

0.0005

ΔΘ5

0.0004

2

δY1

0.0580

Θ1

0.0618

δY5

0.0391

Θ5

0.0637

ΔδY5

0.0004

ΔΘ5

0.0007

3

δY1

0.0580

Θ1

3.4753

δY5

0.0573

Θ5

6.3518

ΔδY5

0.0006

ΔΘ5

0.6558

The week influence of inertia factors means, that a low accuracy problem is connected with initial static model. The real reason was founded later [5]: a parallel displacement of engine component characteristics used in identification procedure does not provide necessary accuracy; a more complex movement is required.

Described in this paper the development and testing of the identification procedure of gas turbine nonlinear dynamic model may be considered as a first phase of total research work in this direction. The investigations are continued to determine the best way of this instrument practical application for control and diagnosing systems development.