Real-time improvement method of turbo-shaft engine component-level model based on dimensional reduction of residual iterative equation

This paper explores a method to improve the real-time computation performance of turbo-shaft engine component-level model by reducing the dimensionality of residual equations set. For turbo-shaft engine, nozzle typically operates in a subcritical working state, assuming that the gas total temperature and pressure parameters of power turbine inlet are the same, and by assuming the gas flow balance between gas generator outlet and nozzle inlet, the iterative solution of the pressure drop of nozzle can be obtained. Then the pressure ratio of power turbine is naturally obtained, thereby achieving the objective of dimensionality reduction of the residual equations set. The component-level model’s steady-state and transient performance were simulated on a desktop computer with a clock frequency of 2.6 GHz. Simulation results show that, with a 2% margin of error, the steady state computation time for the component-level model was reduced by 54.8%, while the transient performance computation time was reduced by 18.6%. This represents a noticeable improvement in real-time performance. Simulation results on the P2020 embedded processor platform with an 800MHz clock frequency indicate a 22.3% reduction in single-step maximum transient performance computation time of the reduced dimensional model, compared to the original model. This demonstrates the effectiveness of dimensionality reduction of the residual equations set in enhancing the model real-time computation performance.


Introduction
Aircraft engines are complex nonlinear aerothermodynamic systems.High-precision onboard real-time models can be used to describe the internal aerothermodynamic relationships and obtain key parameters at different working state in the flight envelope.These models can be used to perform sensor on-line diagnosis functions, reconstruct fault signals when encountering sensor failures in the engine's electronic control system, so as to improve the safety and reliability of the engine control system [1] .
The application of aircraft engine component-level model is crucial for engine fault diagnosis [2][3][4] , health management [5][6][7] , and performance prediction.High-precision and real-time models are of great significance as it can guide aircraft engines to maximize working performance potential within the flight envelope.However, there is a trade-off between the convergence accuracy and real-time performance of component-level models.The higher the model's refinement level is, the better its accuracy is, and the longer time is needed for single-step transient performance computation.Simplifying component-IOP Publishing doi:10.1088/1742-6596/2764/1/012028 2 level models within a reasonable accuracy range helps to improve the computational real-time performance of engine component-level models.
To improve the real-time computational performance of engine component-level models while maintaining accuracy, the following methods can be utilized.The application of integration methods instead of algebraic equation iteration methods could eliminate iteration processes during the transient performance computation process, for example, the implementation of volumetric dynamics methods [8- 10] .Optimizing iterative computation methods, such as quasi-Newton methods [11][12][13] and improved Broyden methods [14,15] , aim to reduce computation time per flow path calculation and lower the number of flow path calculations per step for transient performance calculations.Simplifying complex aerothermodynamic calculations by using numerous empirical formulas or analytical expressions and optimized interpolation methods [16] is of great importance in reducing model computation time.
Currently, Newton-Raphson method is primarily used for solving residual equations in engine component-level models.This method perturbs residual equations with disturbance variables provided by Newton-Raphson method as initial guess variables for model's corrected variables.During the iterative process, when the initial guess variable is perturbed sequentially, changes in equation residuals is obtained in flow path calculation.Then the Jacobian matrix to correct initial guess variables is updated.Reducing the number of initial guess variables helps to reduce the dimensionality of the Jacobian matrix and the number of flow path calculations per step, which is an effective method for improving model convergence and real-time performance.

Principle of residual equation dimensionality reduction
This passage discusses the application of residual equation dimensionality reduction in a turbo-shaft engine component-level model.All stations of the model are shown in Figure 1.According to the law of flow continuity and power balance when engine components work together, the multivariate nonlinear residual Equation ( 1) is obtained.The residual equations set includes flow balance equation, power balance equation and static pressure balance equation.Equation ( 1) is a four-element nonlinear equation set for turbo-shaft engine component-level models.The independent variable X is determined by the input parameters.The residual equation includes three flow balance equations and one power balance equation.

( )
The model involves four initial guess variables, including gas turbine rotor speed g n , compressor pressure ratio C  , gas turbine pressure ratio gt  , and power turbine pressure ratio pt  .These variables influence different flow path components, as illustrated in Figure 2. Specifically, the gas turbine rotor speed g n and compressor pressure ratio C  primarily affect component calculation region (1), while gas turbine pressure ratio gt  influences component calculation region (2).Power turbine pressure ratio pt  mainly affects component calculation region (3).Additionally, power turbine pressure ratio initial guess variable primarily matches the constraints of the calculation parameters from the power turbine to the nozzle. and pt  .When a turbo-shaft engine operates smoothly and safety, its power turbine inlet is in a critical state and nozzle is in a subcritical state.Given the same power turbine inlet gas flow total temperature and total pressure parameters calculated by the upstream components, it can be concluded that the gas flow entering into the power turbine is consistent through the calculation of the flow formula for the different initial guess variables of power turbine pressure ratio.
The following is the method for obtaining the characteristics of , s pt  ,which is referred to as the totalstatic pressure ratio.The gas flow equation of power turbine inlet is shown in Equation ( 2 In the equation, 45 W is the mass flow rate at the power turbine inlet, 45 K is the amplification factor, which is related to the gas adiabatic 45 k index and gas constant R at power turbine inlet, 45 A denotes the aerodynamic area at power turbine inlet, 45 t P represents the gas flow total pressure at power turbine inlet, 45 t T stands for the gas flow total temperature at power turbine inlet, and   45 q Ma is the gas flow function that depends on the Mach number 45 Ma at power turbine inlet.According to the conversion by similar principles, the inlet corrected gas flow rate 45,cor The gas flow equation at power turbine outlet is shown in Equation ( 4), and the subscript 5 represents the outlet station of power turbine: 1/ 2 t5 ( ) Similarly, the corrected gas flow rate 5,cor W at power turbine outlet is obtained after the conversion of similar principles.
According to Equation ( 5) and Equation ( 3), the gas flow rate relationship between the inlet and outlet of power turbine can be obtained: Under the condition that the inlet and outlet flow rate of power turbine is continuous, that is And the flow rate relationship between inlet and outlet of power turbine can be obtained from the characteristic diagram of power turbine: In the equation, pt  is power turbine drop pressure ratio, and , pt cor n is the inlet corrected speed of power turbine.In the power turbine characteristic diagram, there is a gas conversion flow rate relationship at the inlet station, as shown in Equation ( 8).

45, ,
, Therefore, the corrected gas flow rate of power turbine outlet can be expressed as： According to Equation ( 5), the outlet Mach number According to the relationship between total pressure and static pressure and Equation (10), the totalstatic pressure relation of power turbine outlet can be obtained as Equation (11).
Then, the relationship between inlet total pressure and outlet static pressure of power turbine is obtained as Equation (12).
In the component-level model of a turbo-shaft engine, nozzle outlet static pressure is used instead of power turbine outlet static pressure to describe the characteristics of power turbine.The outlet total pressure parameter of power turbine is calculated by the original total pressure ratio characteristics.Considering the cooling air bleed mixing and total pressure loss loss  after power turbine, the exit static pressure at nozzle is calculated for different turbine pressure ratios.The downstream component characteristics of power turbine are integrated to describe the total-static pressure ratio characteristic of power turbine as described in Equation (13).
Power turbine pressure ratio can be obtained by interpolating Equation (13).Since the flow balance is a prerequisite for obtaining the

Analysis of steady-state calculation error
The original total pressure characteristic map is replaced by the updated total-static pressure characteristic map.The power turbine pressure ratio for different operation states is obtained by interpolating the   According to the simulation results, the errors of all parameters are within 0.81%.Table 1 shows the maximum errors of main parameters.After dimensionality reduction of the residual equations, the influence on the station parameters of the upstream components is small, and the errors are all within 0.5%.Among the key station parameters of the downstream components, the maximum error of 49 P is 0.81%, pt  is 0.71%, and pt E is 0.73%.Overall, after dimensionality reduction of the residual equations, the errors in calculating various stations parameters of the engine model are acceptable.

Analysis of transient performance calculation error
The fuel supply schedule used in transient performance simulation for turbo-shaft engine componentlevel model is showed in Figure 6.The tolerance was set to 0.001 during simulation process.The error curves of the main station parameters are shown in the Figure 7.The errors of the upstream component parameters of power turbine are within 0.25%.The maximum dynamic simulation error of 45 P is basically within 1%.Among the downstream component parameters of power turbine, the maximum error of pt E is 1.7%, and the errors of other parameters are mostly within 1%.

Model for real-time simulation results analysis
To test the convergence of the improved model, the tolerance was set to 0.001, and the design point parameters are used as initial guess variables.For different steady-state operating points, the calculation program is executed repeatedly for 100,000 times.Simulation results show in Figure 8 that the average execution time used for the single-step steady-state simulation of the original model is 0.31ms, while the improved model is 0.14ms.The average number of flow path calculations per step of the original model is 203 times, while the improved model is 88 times.Under the allowable error conditions.compared to the original model, the steady-state simulation calculations of the turbo-shaft engine component-level model reduced the average number of flow path calculations per step by 56.7% and the average computation time per step by 54.8%.Overall, the real-time calculation performance of the turbo-shaft engine model has been significantly improved with minimal loss in accuracy.

Conclusion
Based on a component-level model of turbo-shaft engine, this paper studied the method of dimensionality reduction of the residual equations set, in order to improve the real-time computation.Simulation results on an Intel 2.6GHz CPU computing platform, show that the steady-state singlestep computation time is reduced by 54% on average, with no more than 1% loss in steady-state computation accuracy.During the transient performance computation process, with no more than 2% loss in power turbine power accuracy and no more than 1% loss in transient accuracy for the other parameters, the acceleration and deceleration process model single-step transient performance computation time consumed is reduced by 18.6%.The acceleration and deceleration processes were simulated on the P2020 embedded processor platform with a clock frequency of 800MHz.Simulation results show that the original model had a maximum computation time of 74ms for single step transient performance simulation without dimensionally reduced residual equations.When the residual equations were dimensionally reduced, the maximum computation time was reduced by 22.3% to 57ms.Simulation results show that the real-time performance of the component-level model can be improved significantly by using the dimensionality reduction method.
However, this paper assumes that the static pressure at the nozzle outlet is equal to the ambient pressure in the process of dimensionality reduction of the residual equation set for a turbo-shaft engine component-level model, and after dimensionality reduction, the model uses the corrected speed of the power turbine inlet and the ratio of power turbine inlet total pressure to nozzle outlet static pressure to interpolate power turbine pressure ratio.This differs from the conventional method of computing flow paths before dimensionality reduction, which leads to a decrease in the accuracy of the model after reduction.The next step will be to study how to obtain a more precise static pressure at the outlet of nozzle during different operating states.More study work is needed in order to further improve the realtime performance of turbo-shaft engine component-level model, such as combining the residual equation dimensionality reduction method with other iterative methods.  2 shows the meaning of each station of turbo-shaft engine.

Figure 2 .
Figure 2. The initial guess variable affects the calculation area of the flow path.The steady-state performance simulation of a turbo-shaft engine model requires four initial guess variables: g n , C  , gt  and pt  .These variables correspond to the power balance equation, the combustion chamber outlet and gas turbine inlet flow balance equation, the gas turbine outlet and power turbine inlet flow balance equation, and the power turbine outlet and nozzle inlet flow balance equation, On the other hand, transient performance simulation calculations require three initial guess variables: C  , gt


characteristic of power turbine, Simulation results naturally satisfy the flow balance equations at power turbine outlet and nozzle outlet, without the need to add initial guess variables for power turbine.This also reduces the flow balance equations about power turbine outlet and nozzle outlet.As a result, steady-state calculations of the turbo-shaft engine component-level model only require three initial guess variables ( g n , C  , gt  ), while dynamic simulation calculations only require two initial guess variables ( C  , gt  ).When using Newton-Raphson method for iteration, one flow path calculation can be reduced.The updated total pressure ratio characteristic and the , s pt  characteristic of the power turbine are compared in Figure 3, showing minimal differences between the two characteristic curves.

Figure 3 .
Figure 3.Comparison of total-static pressure characteristics and total pressure characteristics.

6 Figure 5 .
Figure 5. Flow balance equation-1 represents combustion chamber outlet and gas turbine inlet flow balance equation, flow balance equation-2 represents gas turbine outlet and power turbine inlet flow balance equation, and the flow balance equation-3 represents power turbine outlet and nozzle inlet flow balance equation.The original model converged after 166 flow path calculations, while the improved model converged after only 69 flow path calculations.The dimensionality-reduced model of the residual equation exhibits better convergence.

Figure 4 .
Figure 4. Steady-state errors of main parameters.

Figure 5 .
Figure 5. Residual convergence.According to the simulation results, the errors of all parameters are within 0.81%.Table1shows the maximum errors of main parameters.After dimensionality reduction of the residual equations, the influence on the station parameters of the upstream components is small, and the errors are all within 0.5%.Among the key station parameters of the downstream components, the maximum error of 49 P is 0.81%, pt

Figure 6 .
Figure 6.Acceleration and deceleration fuel supply schedule.

Figure 7 .
Figure 7. Error curves of main station parameters.

Figure 8 .
Figure 8.The number of flow path calculation and consuming time in steady-state simulation.The single-step transient performance calculation time is shown in Figure 9.At the beginning of the simulation, computation time used for the single-step transient performance simulation of the original model is 7.5 s  , and the improved model is 6.1 s  .The number of flow path calculations per step of the original model is 5 times and the improved model is 4 times.Compared to the original model, the single-step transient performance calculation time was reduced by 18.6%, and the number of flow path calculations was reduced by 20%.The average reduction in single-step transient performance calculation time during acceleration and deceleration was about 38%, and the reduction in average flow path calculation times was about 35.8%.With an accuracy loss of less than 1%, the execution time of transient performance simulation was reduced by 18.6%.The real-time performance of the model after equation dimensionality reduction is significantly improved compared to the original model.After reducing the dimensions of the residual equations, the model residual equation convergence speed is faster and the real-time performance is better.To check the real-time simulation of the improved component-level model on the computing platform of the airborne equipment, the P2020 processor with the clock frequency of 800MHz was used as an example to test the time spent on single-step transient performance calculation during acceleration and deceleration of the component-level model in the microcontroller.Simulation results show that the maximum one-step transient performance computing time of the original model is 74ms, and the maximum one-step transient performance computing time of the model after dimensional reduction of the residual equation is about 57ms, which reduces the transient performance computing time by 22.3%.

9 Figure 9 .
Figure 9.The consuming time and number of flow path calculation in transient performance simulation.

Table 1 .
Maximum error of key parameters.

Table 2 .
Meaning of each station number of turbo-shaft engine.