An engine model solving method combining newton-raphson and differential evolution method

Abstract

The application discloses a kind of engine model solving method combined Newton-Raphson and differential evolution method, in the initial stage of solving engine simulation model balance equation, using Newton-Raphson method to calculate, and whether it is judged to tend to diverge by divergence coefficient.If iterative process tends to diverge, then hybrid algorithm is switched to differential evolution algorithm to solve initial value again using Newton-Raphson method iteration, and divergence trend is judged.When iterative process tends to converge again, hybrid algorithm is switched to Newton-Raphson method.The above process is repeatedly carried out until the iterative error reaches the required accuracy to complete the calculation process.The method of the application combines the better local convergence and solving accuracy of Newton-Raphson method and the strong global search ability and robustness of differential evolution algorithm, and the convergence trend of iterative process is judged by introducing divergence coefficient reflecting the uniformity of independent variable error and error change trend to improve the convergence and convergence consistency of engine simulation model solving.

Description

Technical Field

[0001] This invention belongs to the field of aero-engine technology, specifically relating to a method for solving engine models. Background Technology

[0002] For the overall performance simulation of aero-engines, developing robust and fast-converging model solution methods is of great significance. Currently, widely used numerical methods for solving the equilibrium equations of engine simulation models include the Newton-Raphson method, the Broyden rank-1 method, and the N+1 point residual method. These methods are all based on function gradients for solving engine equilibrium equations. While they offer high accuracy and efficiency, they require continuous, differentiable, and locally convergent nonlinear equations. Furthermore, with the deepening research into high-performance engine technologies such as variable-cycle engines, the equilibrium equations of engine simulation models are becoming increasingly complex. The diverse operating modes, wide operating range, numerous and highly variable parameters, and lack of explicit expression in the nonlinear mathematical model of variable-cycle engines make traditional numerical iterative methods such as the Newton-Raphson method and the N+1 residual method extremely difficult to solve. In addition, while modern optimization algorithms such as particle swarm optimization, genetic algorithms, and differential evolution algorithms have high probabilities of global convergence, their solution efficiency and accuracy are relatively low.

[0003] Therefore, it is necessary to develop new methods for solving aero-engine simulation models that can retain the high solution efficiency and accuracy of traditional numerical iterative methods while also taking into account the better global convergence of modern optimization algorithms, so as to obtain more accurate results when solving more complex engine balance equations. Summary of the Invention

[0004] To overcome the shortcomings of existing technologies, this invention provides a method for solving engine models that combines the Newton-Raphson method and the differential evolution method. This method is a hybrid algorithm combining the Newton-Raphson method and the differential evolution algorithm. In the initial stage of solving the equilibrium equations of the engine simulation model, the Newton-Raphson method is used for calculation, and the divergence coefficient is used to determine whether it tends towards divergence. If the iteration process tends towards divergence, the hybrid algorithm switches to the differential evolution algorithm to find initial values, and then the Newton-Raphson method is used again for iteration, and the divergence trend is judged. When the iteration process tends towards convergence again, the hybrid algorithm switches back to the Newton-Raphson method. This process is repeated until the iteration error reaches the required accuracy, completing the calculation. This invention combines the good local convergence and solution accuracy of the Newton-Raphson method with the strong global search capability and robustness of the differential evolution algorithm. By introducing a divergence coefficient that reflects the uniformity and trend of error change of the independent variable, the convergence trend of the iteration process is judged, thereby improving the convergence and consistency of the engine simulation model solution.

[0005] The technical solution adopted by this invention to solve its technical problem includes the following steps:

[0006] Step 1: Perform engine design point performance calculations to obtain initial values ​​for iterative variables. The iterative variables include at least the compressor boost ratio, high and low pressure rotor speeds, turbine inlet temperature, turbine inlet converted flow rate, exhaust nozzle throat cross-sectional area, compressor component inlet guide vane angle, turbine component inlet guide vane cross-sectional area, and combustion chamber fuel supply.

[0007] Step 2: Establish a common working equation set for the engine based on the input control law, initial values ​​of iterative variables, and the principle of flow, pressure, and power balance between components, and calculate the initial residual value of the balance equation; the problem of solving the common working point of the engine is transformed into a problem of solving a nonlinear equation set under a given regulation law;

[0008] Step 3: Solve the nonlinear equation system iteratively using the Newton-Raphson method;

[0009] Step 4: Calculate the residual values ​​of the equilibrium equations and determine whether the residuals meet the convergence accuracy. If the convergence accuracy is met, output the calculation results for non-design points; otherwise, proceed to step 5.

[0010] Step 5: Determine if the maximum number of iterations has been reached. If the number of iterations has reached the upper limit, output the calculation results for non-design points; otherwise, go to step 6.

[0011] Step 6: Calculate the divergence coefficient based on the residuals to determine whether the iteration process has a divergence trend. If the divergence coefficient is less than the threshold, that is, the iteration process has a convergence trend, then go to step 3 to continue iterating using the Newton-Raphson method; otherwise, go to step 7.

[0012] Step 7: Use the differential evolution algorithm to iteratively solve the nonlinear equation system, and use the calculated result as the initial value to proceed to step 8;

[0013] Step 8: Using the optimal individual obtained from one iteration of the differential evolution algorithm as the initial value, perform one Newton-Raphson iteration to obtain a new solution vector;

[0014] Step 9: Substitute the new solution vector into the equilibrium equation, calculate the residual value of the equilibrium equation at this time, and go to step 6;

[0015] Step 10: When the residual in step 4 meets the convergence accuracy and the non-design point calculation result is output, the solution of the aero-engine simulation model is completed. If the non-design point calculation result is output because the number of iterations has reached the upper limit, it proves that a convergent solution has not been found.

[0016] Furthermore, step 3 is specifically as follows:

[0017] Step 3-1: Use the difference operator to replace the differential operator to establish the Jacobian coefficient matrix;

[0018] Step 3-2: Determine whether the Jacobian coefficient matrix is ​​singular. If the coefficient matrix is ​​not singular, proceed to step 3-3; otherwise, apply a small perturbation before proceeding to step 3-3.

[0019] Step 3-3: Solve the linear equation system using Gaussian elimination to obtain the changes in the iterative variables;

[0020] Steps 3-4: Add the change in the iteration variable to the old iteration variable to obtain the new iteration variable.

[0021] Furthermore, step 7 specifically includes:

[0022] Step 7-1: Use the solution calculated in step 3 as the initial value for the differential evolution algorithm, and make changes based on the initial value to generate the initial population;

[0023] Step 7-2: Set the scaling factor F, select the mutation strategy and perform the mutation operation to generate mutated individuals;

[0024] Step 7-3: Perform element-wise crossover on the mutated individuals and the initial individuals generated in sub-step 1 according to the crossover probability CR to generate crossover individuals;

[0025] Step 7-4: Calculate the fitness values ​​of the initial individuals and crossover individuals, i.e., the residual values ​​of the equilibrium equation;

[0026] Step 7-5: Compare the fitness values ​​of the crossover individuals with the fitness values ​​of the initial individuals, and select the individual with the smaller fitness value to enter the new population;

[0027] Steps 7-6: Select the best individual from the new population to proceed to step 8.

[0028] The beneficial effects of this invention are as follows:

[0029] The present invention applies an engine simulation model solution method that combines the Newton-Raphson method and the differential evolution algorithm. The differential evolution algorithm is used to modify the Newton-Raphson method. By introducing a divergence coefficient that reflects the uniformity of error and the trend of error change of the independent variable, the convergence trend of the iteration process is judged, thereby improving the convergence and convergence consistency of the engine simulation model solution. This can effectively solve the problem of the contradiction between computational efficiency and convergence caused by a single algorithm. Attached Figure Description

[0030] Figure 1 This is a schematic diagram of the method flow of the present invention.

[0031] Figure 2 This is a schematic diagram of the Newton-Raphson method in the method of this invention.

[0032] Figure 3 This is a schematic diagram of the differential evolution algorithm in the method of this invention. Detailed Implementation

[0033] The present invention will be further described below with reference to the accompanying drawings and embodiments.

[0034] This invention provides a method for solving engine models that combines the Newton-Raphson method and the differential evolution method, integrating the traditional Newton-Raphson iterative method with the differential evolution algorithm. This method leverages the high efficiency and accuracy of the Newton-Raphson method and the good global convergence of the differential evolution algorithm to improve the overall performance of the solution method. It addresses at least the problems of existing engine simulation model solution methods that cannot guarantee global convergence and have a strong dependence on initial values.

[0035] The technical solution of this invention is: a method for solving engine models combining Newton-Raphson and differential evolution methods, comprising the following steps:

[0036] Step 1: Perform engine design point performance calculations to obtain initial values ​​for iterative variables, including but not limited to compressor boost ratio, high and low pressure rotor speeds, turbine inlet temperature, turbine inlet converted flow rate, exhaust nozzle throat cross-sectional area, compressor component inlet guide vane angle, turbine component inlet guide vane cross-sectional area, combustion chamber fuel supply, etc.

[0037] Step 2: Establish a common operating equation set for the engine based on the input control law, initial values ​​of iterative variables, and the principles of flow, pressure, and power balance between components, and calculate the initial residual values ​​of the balance equations. In this way, the problem of solving the common operating point of the engine is transformed into a problem of solving a nonlinear equation set under a given regulation law.

[0038] Step 3: Solve the nonlinear equation system iteratively using the Newton-Raphson method;

[0039] Sub-step 1: Use the difference operator to replace the differential operator to establish the Jacobian coefficient matrix;

[0040] Sub-step 2: Determine whether the Jacobian coefficient matrix is ​​singular. If the coefficient matrix is ​​not singular, proceed to sub-step 3; otherwise, apply a small perturbation before proceeding to sub-step 3.

[0041] Sub-step 3: Solve the system of linear equations using Gaussian elimination to obtain the changes in the iterative variables;

[0042] Sub-step four: Add the change in the iteration variable to the old iteration variable to obtain the new iteration variable;

[0043] Step 4: Calculate the residual values ​​of the equilibrium equations and determine whether the residuals meet the convergence accuracy. If the convergence accuracy is met, output the calculation results for non-design points; otherwise, proceed to step 5.

[0044] Step 5: Determine if the maximum number of iterations has been reached. If the maximum number of iterations has been reached, output the calculation results for non-design points; otherwise, proceed to Step 6.

[0045] Step 6: Calculate the divergence coefficient based on the residuals to determine whether the iteration process has a divergence trend. If the divergence coefficient is less than the threshold, that is, the iteration process has a convergence trend, then go to step 3 to continue iterating using the Newton-Raphson method; otherwise, go to step 7.

[0046] Step 7: Use the differential evolution algorithm to iteratively solve the nonlinear equation system, and use the calculated result as the initial value to proceed to step 8;

[0047] Sub-step 1: Use the solution calculated in step 3 as the initial value for the differential evolution algorithm, and make changes based on the initial value to generate the initial population;

[0048] Sub-step 2: Set the scaling factor F, select the mutation strategy and perform the mutation operation to generate mutated individuals;

[0049] Sub-step 3: Perform element-wise crossover on the mutated individuals generated in sub-step 1 and the initial individuals according to the crossover probability CR to generate crossover individuals;

[0050] Sub-step four: Calculate the fitness values ​​of the initial individuals and crossover individuals, i.e., the residual values ​​of the equilibrium equation;

[0051] Sub-step 5: Compare the fitness values ​​of the crossover individuals with the fitness values ​​of the initial individuals, and select the individual with the smaller fitness value to enter the new population;

[0052] Sub-step six: Select the best individual from the new population to proceed to step eight;

[0053] Step 8: Using the optimal individual obtained from one iteration of the differential evolution algorithm as the initial value, perform one Newton-Raphson iteration to obtain a new solution vector;

[0054] Step 9: Substitute the new solution vector into the equilibrium equation, calculate the residual value of the equilibrium equation at this time, and go to step 6;

[0055] Step 10: When the residual in Step 4 meets the convergence accuracy and outputs the non-design point calculation results, the solution of the aero-engine simulation model is completed. If the non-design point calculation results are output because the number of iterations has reached the upper limit, it proves that a convergent solution has not been found. Specific implementation examples:

[0057] See Figure 1 — Figure 3To enable those skilled in the art to better understand the present invention, the technical solutions of the present invention will be clearly and completely described below with reference to the accompanying drawings of the embodiments of the present invention. Obviously, the described embodiments are only some embodiments of the present invention, and not all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort should fall within the scope of protection of the present invention.

[0058] A specific embodiment of the present invention includes the following steps.

[0059] Step 1: The solution object of this method is an overall performance simulation model of an aero-engine with arbitrary configuration. First, the engine design point performance needs to be calculated to obtain the initial values ​​of the iterative variables. Taking a twin-shaft hybrid exhaust turbofan engine as an example, the iterative variables include the high-pressure compressor boost ratio, fan boost ratio, high-pressure rotor speed, turbine inlet temperature, low-pressure turbine inlet equivalent flow rate, and high-pressure turbine inlet equivalent flow rate.

[0060] Step Two: Establish a set of common operating equations for the engine based on the input control laws, initial values ​​of iterative variables, and the principles of flow, pressure, and power balance between components. A twin-shaft, mixed-exhaust turbofan engine must meet the following six common operating conditions: flow balance between the low-pressure turbine and fan; power balance between the low-pressure turbine and fan; flow balance between the high-pressure turbine and high-pressure compressor; power balance between the high-pressure turbine and high-pressure compressor; static pressure balance of the airflow inside and outside the mixing chamber at the inlet; and flow balance between the afterburner outlet and the exhaust nozzle. After establishing the balance equations based on the above common operating conditions, calculate the initial residual values ​​of the balance equations.

[0061] Step 3: Solve the nonlinear equation system iteratively using the Newton-Raphson method.

[0062] Sub-step 1: Use the difference operator to replace the differential operator to establish the Jacobian coefficient matrix;

[0063] Sub-step 2: Determine whether the Jacobian coefficient matrix is ​​singular. If the coefficient matrix is ​​not singular, proceed to sub-step 3; otherwise, apply a small perturbation before proceeding to sub-step 3.

[0064] Sub-step 3: Solve the system of linear equations using Gaussian elimination to obtain the changes in the iterative variables;

[0065] Sub-step four: Add the change in the iteration variable to the old iteration variable to obtain the new iteration variable;

[0066] Step 4: Calculate the residual values ​​of the equilibrium equations and determine whether the residuals meet the convergence accuracy. If the convergence accuracy is met, output the calculation results for non-design points; otherwise, proceed to Step 5.

[0067] Step 5: Determine if the maximum number of iterations has been reached. If the number of iterations has reached the upper limit, output the calculation results for non-design points; otherwise, proceed to Step 6.

[0068] Step Six: Calculate the divergence coefficient γ(k) based on the residuals to determine if the iteration process has a divergence trend. If the divergence coefficient is less than the threshold, indicating a convergence trend, proceed to Step Three to continue iterating using the Newton-Raphson method; otherwise, proceed to Step Seven. The divergence coefficient γ(k) can be calculated using the following formula:

[0069]

[0070]

[0071] Where e(k) is the iteration error at the k-th step, e(k) = f(x(k)); i (k) represents the error value of the i-th component of the independent variable in the k-th step of the iterative calculation.

[0072] It should be noted that, The components characterize the uniformity of error during the iteration process, and the ||e(k)|| / ||e(k-1)|| components characterize the trend of error change. In the hybrid algorithm, the convergence of the iteration process is judged by the divergence coefficient, and a reasonable divergence coefficient threshold is set to switch the algorithm. This invention suggests that the divergence coefficient threshold be 20% of the initial iteration error.

[0073] Step 7: Use the differential evolution algorithm to iteratively solve the nonlinear equation system, and use the calculated result as the initial value to proceed to step 8.

[0074] Sub-step 1: Use the solution calculated in step 3 as the initial value for the differential evolution algorithm, and make changes to the initial value to generate the initial population. For example, the initial population can be calculated using the following formula:

[0075] x i,j (g)=x j (g-1)+rand(-1,1)×x j (g-1)×q

[0076] Where, x i,j (g) represents the value of the j-th dimension of the i-th individual; x j (g-1) represents the j-th dimension value of the new individual generated in the previous iteration; rand(-1,1) represents a random number between [-1,1]; q is a range constant, which is recommended to be 50% in this invention.

[0077] Sub-step 2: Set the scaling factor F. Different mutation operations can be selected based on different mutation strategies. For example, choosing a random mutation strategy means selecting three random individuals from the initial population to perform the mutation operation, generating mutated individuals V.i (g+1):

[0078] V i (g+1)=X r1 (g)+F(X r2 (g)-X r3 (g))

[0079] Among them, X r1 (g), X r2 (g), X r3 (g) represents three random individuals in the population.

[0080] It should be noted that the scaling factor F is generally taken between [0,2], and in this invention, F = 0.5 is used.

[0081] Sub-step 3: Perform element-wise crossover on the mutated individuals generated in sub-step 1 and the initial individuals based on the crossover probability CR, generating crossover individuals U. i (g+1):

[0082]

[0083] It should be noted that the crossover probability CR is generally taken between [0,1], and in this invention, CR = 0.4 is used.

[0084] Sub-step four: Calculate the fitness values ​​of the initial individuals and crossover individuals, i.e., the residual values ​​of the equilibrium equation;

[0085] Sub-step 5: Compare the fitness values ​​of the crossover individuals with the fitness values ​​of the initial individuals, and select the individual with the lower fitness value to enter the new population.

[0086]

[0087] Sub-step six: Select the best individual from the new population to proceed to step eight.

[0088] It should be noted that for a dual-shaft hybrid exhaust turbofan engine, the common working equation set contains 6 equilibrium equations. Substituting each individual into the calculation will yield 6 error values. The largest error value is used for comparison, and the individual that minimizes the largest error value in the population is selected as the optimal individual.

[0089] Step 8: Using the optimal individual obtained from one iteration of the differential evolution algorithm as the initial value, perform one Newton-Raphson iteration to obtain a new solution vector.

[0090] Step 9: Substitute the new solution vector into the equilibrium equation, calculate the residual value of the equilibrium equation at this time, and go to Step 6.

[0091] Step 10: When the residual in Step 4 meets the convergence accuracy and outputs the non-design point calculation results, the solution of the aero-engine simulation model is completed. If the non-design point calculation results are output because the number of iterations has reached the upper limit, it proves that a convergent solution has not been found.

Claims

1. A method for solving engine models combining Newton-Raphson and differential evolution methods, characterized in that, Includes the following steps: Step 1: Perform engine design point performance calculations to obtain initial values ​​for iterative variables. The iterative variables include at least the compressor boost ratio, high and low pressure rotor speeds, turbine inlet temperature, turbine inlet converted flow rate, exhaust nozzle throat cross-sectional area, compressor component inlet guide vane angle, turbine component inlet guide vane cross-sectional area, and combustion chamber fuel supply. Step 2: Establish a common working equation set for the engine according to the input control law, the initial values ​​of the iterative variables, and the principle of flow, pressure, and power balance between components, and calculate the initial residual value of the balance equation; The problem of finding the common operating point of the engine is transformed into a problem of solving a system of nonlinear equations under a given regulation law; Step 3: Solve the nonlinear equation system iteratively using the Newton-Raphson method; Step 4: Calculate the residual values ​​of the equilibrium equations and determine whether the residuals meet the convergence accuracy. If the convergence accuracy is met, output the calculation results for non-design points; otherwise, proceed to step 5. Step 5: Determine if the maximum number of iterations has been reached. If the number of iterations has reached the upper limit, output the calculation results for non-design points; otherwise, go to step 6. Step 6: Calculate the divergence coefficient based on the residuals to determine whether the iteration process has a divergence trend. If the divergence coefficient is less than the threshold, that is, the iteration process has a convergence trend, then go to step 3 to continue iterating using the Newton-Raphson method; otherwise, go to step 7. Step 7: Use the differential evolution algorithm to iteratively solve the nonlinear equation system, and use the calculated result as the initial value to proceed to step 8; Step 8: Using the optimal individual obtained from one iteration of the differential evolution algorithm as the initial value, perform one Newton-Raphson iteration to obtain a new solution vector; Step 9: Substitute the new solution vector into the equilibrium equation, calculate the residual value of the equilibrium equation at this time, and go to step 6; Step 10: When the residual in step 4 meets the convergence accuracy and the non-design point calculation result is output, the solution of the aero-engine simulation model is completed. If the non-design point calculation result is output because the number of iterations has reached the upper limit, it proves that a convergent solution has not been found.

2. The method for solving engine models combining Newton-Raphson and differential evolution methods according to claim 1, characterized in that, Step 3 is described in detail below: Step 3-1: Use the difference operator to replace the differential operator to establish the Jacobian coefficient matrix; Step 3-2: Determine whether the Jacobian coefficient matrix is ​​singular. If the coefficient matrix is ​​not singular, proceed to step 3-3; otherwise, apply a small perturbation before proceeding to step 3-3. Step 3-3: Solve the linear equation system using Gaussian elimination to obtain the changes in the iterative variables; Steps 3-4: Add the change in the iteration variable to the old iteration variable to obtain the new iteration variable.

3. The method for solving engine models combining Newton-Raphson and differential evolution methods according to claim 2, characterized in that, Step 7 specifically involves: Step 7-1: Use the solution calculated in step 3 as the initial value for the differential evolution algorithm, and make changes based on the initial value to generate the initial population; Step 7-2: Set the scaling factor F, select the mutation strategy and perform the mutation operation to generate mutated individuals; Step 7-3: Perform element-wise crossover on the mutated individuals and the initial individuals generated in sub-step 1 according to the crossover probability CR to generate crossover individuals; Step 7-4: Calculate the fitness values ​​of the initial individuals and crossover individuals, i.e., the residual values ​​of the equilibrium equation; Step 7-5: Compare the fitness values ​​of the crossover individuals with the fitness values ​​of the initial individuals, and select the individual with the smaller fitness value to enter the new population; Steps 7-6: Select the best individual from the new population to proceed to step 8.

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