Study on Component Characteristic Correction Algorithm of Turbofan Engine

The optimization of component characteristics is an important part of the research on model modification of turbofan engine. This paper studied three kinds of modification and optimization algorithms: genetic algorithm (GA), differential evolution algorithm (DE) and particle swarm optimization algorithm (PSO), and made a comparative analysis on the principles of the algorithms. In addition, the DGEN380 turbofan engine component model was taken as the research object to test and compare the modification performance of each algorithm. The results show that ： compared with GA, DE algorithm and PSO algorithm have higher adaptability to multi-objective and multi-parameter correction, the success rate can reach 100%, the error is kept within 1%, and the convergence ability is strong. Compared with the other two algorithms, the solving speed of PSO algorithm is more influenced by the initial parameters. This conclusion can provide an important reference for selecting and improving the algorithm.


Introduction
Aeroengine model is an important tool for the development of advanced aerodynamic power plants, and the accuracy of the model determines the scope of application of the model. Due to the theoretical and random errors of the components, the deviation of installation of each component and the variety in the working environment, the actual characteristics of the engine components have changed. In order to make the output parameters of the engine model more accurate, it is necessary to study a method to further modify the characteristics of the engine components through the engine test data, and to modify the engine model to meet the accuracy requirements and match the actual engine conditions as much as possible.
In the field of aero-engine model correction. Stamatis A 1 proposed a method that can adaptively correct the characteristics of engine components according to the actual measurement data, and use the nonlinear generalized minimum residual method to obtain the guess vector and the correction factor. C Kong 2 proposed a method of scaling the characteristics map of components based on system identification, which can obtain high-precision simulation results. Later, he used genetic algorithm to solve the compressor component characteristic diagram with higher accuracy, and proposed a method to improve the design of fitness function, which reduced the dependence on experimental data and shortened the calculation time, and improved the correction efficiency 3 , 4 . G Li 5 proposed that in order to solve the problem of inaccurate calculation of the model design point, the unmeasurable parameters such as the work efficiency of the components were regarded as unknown quantities, and the Newton-Raphson algorithm was used to solve the correction coefficients, thereby obtaining the accurate true values of the unmeasurable parameters. On this basis, G Li 6 put forward the concept of component characteristic correction factor function. This method can reduce the amount of calculation and is suitable for correction of multiple operating points.
Related research focused on the matching of component characteristics. Choosing an appropriate algorithm is the key to correcting component characteristics. There is no relevant research on how to select component characteristics correction algorithms and the differences in correction effects of each algorithm. Aiming at the problem of the selection of correcting algorithm, this paper takes DGEN380 engine compressor components as the correcting object. The three different types of optimization algorithms are systematically compared and analyzed in theory, and examples and model simulations are used to more objectively describe the effect and difference characteristics of model correction of each algorithm. This paper provides reliable theoretical support for the research on the components characteristics correction algorithm of turbofan engine in the future.

Component-level Model
The model proposed in this paper is based on the component method modeling 7,8 using the DGEN380 engine as the object of modeling. This engine is a dual-rotor, separated exhaust turbofan engine. The core uses a single-stage centrifugal compressor, an annular reversed combustion chamber and an axial-flow high pressure turbine. Low pressure rotor is composed of high bypass ratio fan, gear box (low pressure speed/fan speed=3.32) and axial-flow low pressure turbine.
Each parameter of the model is not independent, but there is a unique solution that simultaneously satisfies the dynamic co-working equation. Therefore, an iterative process is required to find the steady-state operating point of the engine 9 . The dynamic co-working equation can be expressed as the following 4 flow balance equations and 2 power balance equations, as shown in Eq.(1) and Eq.(2).
(1) According to the continuous flow, the flow balance relationship of fan outlet, combustion chamber outlet, high pressure turbine outlet, and low pressure turbine outlet can be obtained.
(2) High and low pressure rotor power balance equation In Eq.(2), ht  and lt  are the mechanical efficiency of the high-pressure rotor shaft and the low-pressure rotor shaft, respectively.
High pressure speed nht, low pressure speed nlt, and fan, compressor, high and low pressure turbine pressure ratio πf, πc, πht, πlt are selected as the initial guess before solving. In particular, the main function of the high and low pressure speeds nht and nlt is to determine the component characteristic value on the component characteristic map. Taking the compressor component for instance, πc is known and calculate the corrected spool speed by using nht, and then use the interpolation method to calculate the flow rate and efficiency under the corrected spool speed and pressure ratio from the component characteristic map. Therefore, the deviation of the component characteristics will cause large errors in model calculation results.

Component Characteristic Correction Algorithm
Because the difference in component characteristics will reduce the accuracy of the model, it is necessary to modify the component characteristics to match the actual engine state. The correction problem is essentially a multi-objective parameter optimization problem. At present, genetic algorithm (GA), differential evolution algorithm (DE) and particle swarm optimization algorithm (PSO) are widely used in component characteristics correction. This paper compares and analyzes the three algorithms in component characteristics correction application.

Component Characteristic Correction
For engines, because it is difficult to obtain accurate component characteristics, general characteristics are usually used in the process of modeling, which will directly affect the accuracy of the model. Therefore, this article uses component characteristic correction factors to correct component characteristics. Taking the compressor component for instance, let the characteristics given by its design point (π, w, η), which represent pressure ratio, flow rate and efficiency respectively. (△C π , △Cw, △Cη) are selected as the correction factors of the corresponding component characteristic, and are used to calculate the new component characteristic, as shown in Eq.(3).
In the Eq.3, the subscript "sim" is the simulation value, "real" represents the test value, and "amend" represents the new component characteristic value.
Combined with the engine test data, the relative error between it and the output of corrected model is used as the criterion for judging the accuracy of model. The fitness function F is established, , , Which k=1, i p represents the target parameter, i w represents the weight coefficient of the above-mentioned target parameter error, and all i w take 1 here. The larger fitness function, the higher agreement between the model simulation output data and the experimental measurement data, and the smaller model error.
The specific component characteristic correction process is shown in the Fig.1.  Choosing an appropriate algorithm is the key to correct the component characteristic. Therefore, this paper compares and analyzes three commonly used optimization algorithms of model component characteristic.

Mechanism Analysis of 3 Correction Algorithms
The process of solving the component characteristic correction coefficients by three correction algorithms (GA, DE, PSO) is shown schematically in Fig.2. The basic processes of the three algorithms and the establishment of fitness functions are consistent, but there are certain differences in the implementation of specific steps. Compare and describe the differences between the three algorithms in the following two parts.

Initialize the population
Genetic algorithm (GA) and differential evolution algorithm (DE) randomly generate N-digit M binary individuals in the solution domain Ω to form the initial population. However, DE forms the initial population in which each individual is a real number and does not need to be binary coded. PSO does not use evolutionary operators for individuals like other evolutionary algorithms, but treats each individual as a particle with no volume and weight in the n-dimensional search space. Each particle will move in the solution space and its direction and distance will be determined by a speed. It is necessary to randomly generate the position P and velocity V of N particles in the solution domain Ω, and set the initial parameter learning factors C1, C2 and inertia weight  .
2. Update the population GA uses selection, crossover, and mutation operations to update the population to generate the next generation. DE makes the population evolve toward a larger population fitness value through three operations of mutation, crossover, and selection. Both algorithms use these three operations to update the population, but their order is different, and the specific implementation of each operation is also different. Then use the row-in-themiddle method for random sampling with replacement to select the next generation 1 1 Which r is a random number in the range of [0,fitnessXN], Xnew is a new generation population.
In this paper, GA adopts single-point crossover. It means that only one crossover point is randomly set in the individual coding string, and then part of the chromosomes of two paired individuals are exchanged at this point. , : , : , : Which r is a random number in the range of Which ris a random number in the range of Which U is the cross vector and X is the parent population. DE generates the cross vector Ui according to the following equation, Which k is a random number in the range of

PSO
The particle swarm algorithm does not update the next generation through selection, crossover and mutation operations, but determines the next generation population by tracking two extreme values of particles. One is the optimal solution pbest found by the particle itself so far, and the other is the optimal solution gbest found so far for the entire population. The specific update method is as follows, Speed update: , and Vmax, Vmin are the maximum and minimum set speed.

Example Simulation
Construct the following two nonlinear sets of equations as simulation research objects.
(1) The set of equations 1 (simplified equation of compressor) This paper takes the compressor components as the correction object. The overall component model of the compressor is integrated and packaged into a mathematical function model. In this model, the total temperature of the gas at the compressor inlet T 25 , total pressure at the compressor inlet P 25 , and compressor compression ratio π c are as input, and the total gas temperature at the compressor outlet T 3 , total pressure at the compressor outlet P 3 , actual compressor flow W 3 and compressor power N c are as output. According to the principle of compressor, it can be simplified into the following form, The fitness function is the above two sets of equations. The smaller fitness value, the smaller error. Set the average fitness value to be less than 0.1 to meet the solution requirements. The test results of each algorithm running 20 times are as follows.   Table 3a The solution result of the system of equations 2 for each algorithm  Combining Fig.4 and Fig.5, the optimal algebra of GA algorithm for solving the two equations is relatively small, but the average fitness value will be stable in a certain range around 50 generations, fluctuate and no longer continue to look for better solution. The optimal solution of DE and PSO algorithm is slower than that of GA algorithm, and its average fitness value can be stabilized in the ideal convergence area within a few generations, and continue to search for better solutions.
According to the simulation results of the above examples, it can be inferred that the traditional GA algorithm for multi-objective multi-parameter solving problems is slower than the DE and PSO algorithms, and the solution accuracy is not high. The PSO algorithm is slower to optimize the problem with multiple solutions.

Component Model Simulation Analysis
In order to verify the correction performance of each algorithm, the actual compressor component model is used as the object to compare and analyze the correction effect, and analyze the results from the principle of the algorithm.

Simulation Results of Compressor Components
DGEN380 engine compressor components is as the correction object. At the design point (maximum continuous climb point) of DGEN380 (height H=3048m, Mach number Ma=0.338, fuel flow rate W f =0.027782kg/s), the optimization algorithm is tested. The section parameters of the compressor components in Table 4 are selected as the target parameters. The results of each algorithm running 5 times are as follows.  Fig.6 The correction error of each algorithm Fig.7 The average fitness of each algorithm General engineering requires the correction accuracy to be within 1%. According to the calculation results shown in Table 5, it can be seen that the average error of a single performance parameter of the compressor obtained by the three correction algorithms meets the accuracy requirements. However, relatively, the GA algorithm correction model results have a larger deviation. From Fig.6, it can be seen that the single correction result fluctuates greatly, and there are several performance parameters exceeding 1%. For example, the N c error of the fifth time is 1.24%. By contrast, the DE and PSO algorithm have better correction effects, the accuracy of each performance parameter can be kept within 10 -5 , and the results of each time are basically the same. According to Fig.7, the GA algorithm tends to be stable around the 80th generation, but fluctuates greatly, the DE algorithm converges faster, and the PSO algorithm is relatively slow to converge. The result obtained above is similar to the result of solving the set of equations 1 simplified by the compressor model.

Analysis of Result
The GA algorithm selection operation is carried out according to Eq.(5) and Eq.(6). The probability of each individual being selected is proportional to the value of its fitness function. Since r is a random number in the range of [0,fitnessX N ], this randomness may slow down the speed of convergence. The DE algorithm selects the next generation according to Eq.(9), and directly selects individuals with high fitness by comparing the previous generation and the population fitness value after crossover and mutation operation. The PSO algorithm selects the next generation according to Eq.(13). V determines the direction of update for it and make the particles move closer to the optimal solution. This deterministic selection of DE and PSO algorithms accelerates the convergence speed.
The GA algorithm population individuals use binary coding. During the crossover and mutation operation, the individual coding string is first transformed according to the rules of Eq.(7) and Eq. (8). However, when the best point is at or close to the boundary of the feasible solution, the randomness of this method may prevent it from effectively continuing to optimize. For example, the theoretical solution is 1 in Eq.2. When it is close to the theoretical solution, the possible results of crossover and mutation are shown in the following table. It can be seen from Table 6 that the original individual is already close to the theoretical solution, but is far away from the theoretical solution after crossover and mutation. The randomness of GA algorithm crossover-mutation can maintain the diversity of the population and avoid local optima. However, when it is close to the theoretical solution, it may make it impossible to continue to search for optimization, and the phenomenon of average fitness value will fluctuate.
The DE algorithm performs crossover-mutation operations based on the optimal solution gbest found so far in the entire population. The PSO algorithm performs the next-generation update based on the optimal solutions pbest and gbest found so far by the particle itself. In the early and late stages of the solution process, both two algorithms can make it continuously move closer to the optimal solution.
The PSO algorithm selects the next generation according to Eq.(12). Compared with the DE algorithm, the inertia weight  is introduced here. Its size affects the algorithm's global and local optimization capabilities. When  is larger, the global optimization ability will be stronger, and the local optimization ability will be weaker. According to the above example, the PSO algorithm has a slower optimization speed for problems with multiple solutions, which may cause the local optimization ability to be too weak due to the value of  . Therefore, the simulation is performed under different values of  , and the results are as follows,  Fig.8 and Fig.9, with the change of  , the correction convergence speed also changes. The larger  , the slower convergence. The error of each performance parameter is basically the same at different values of  , because both  =0.7 and  =0.8 do not reach optimal convergence effect, so the relative error of performance parameters has increased.

Conclusion
In this paper, the mechanism analysis and algorithm realization of the three main component characteristic correction optimization algorithms are carried out, and the three optimization algorithms are simulated and compared using the turbofan engine compressor component model. The algorithm performance are analyzed through the algorithm mechanism and simulation results. Specifically, the performance of each algorithm is tested in terms of correction accuracy, convergence speed and so on. It provides a reference for the selection and improvement of the component characteristic correction algorithm of turbofan engine in the future.
(1) When using the traditional GA algorithm solves multi-objective and multiparameter problems such as model correction, due to the randomness of the algorithm, the convergence speed may be slow, the accuracy is low, and the solution is unstable. If you want to improve the performance of the correction algorithm, you need to further improve the selection, crossover and mutation operations of algorithm, and then the loss of performance caused by randomness can be offset.
(2) The traditional DE algorithm has high correction accuracy and does not need to adjust the parameters too much, but the convergence speed needs to be improved, N c which can be improved from the local convergence ability.
(3) When using the traditional PSO algorithm solves multi-objective and multiparameter problems such as model correction, the correction accuracy is not much different from the DE algorithm. The convergence speed is affected by the value of the inertia weight  . Therefore, the algorithm parameters need to be adjusted when the PSO algorithm is used for model correction.