CN113688565B - Supercritical CO based on optimal trade-offs 2 Bayes optimizing method for centrifugal compressor - Google Patents
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Abstract
Supercritical CO based on optimal trade-offs 2 The Bayesian optimization method of the centrifugal compressor comprises the steps of S1, sample data construction, S2, model construction, S3 and sample data set calculationFour steps of (1) a potential optimal balance coefficient set vector, S4, and outputting an optimal sample point vector and a response value thereof. The invention is directed to supercritical CO 2 The working medium near the critical point of the centrifugal compressor brings design and optimization problems, and the defects of the existing Bayesian optimization method based on confidence upper bound can dynamically balance local development and global exploration in the optimization process, improve the optimization efficiency and global optimization capacity, and improve supercritical CO 2 And (5) pneumatic optimization design result of the centrifugal compressor.
Description
Technical Field
The invention relates to supercritical CO 2 Centrifugal compressor design and optimization field, in particular supercritical CO based on optimal trade-offs 2 A Bayesian optimization method for centrifugal compressors.
Background
Fossil energy is used as the most important component of energy structures in China, and the problems of reduced reserves, environmental limitation and the like are faced. In recent years, supercritical CO 2 The Brayton power cycle is a working medium, and the advantages of high efficiency potential, high energy density and the like are widely paid attention to. The centrifugal compressor is used as one of the core devices of the cycle, and has the advantages of wide operation range, compact structure, high efficiency, large single-stage pressure ratio and the like. However, the physical property of the special working medium near the critical point is supercritical CO 2 Centrifugal compressor design and optimization presents challenges.
Along with the development of computers, high-precision numerical simulation calculation becomes an important tool in the field of engineering design optimization. With the continuous improvement of numerical calculation accuracy and model complexity, the amount of calculation required for high-accuracy numerical simulation is also drastically increasing. Therefore, optimization problems in the modern engineering field often require fast and accurate acquisition of globally optimal solutions of the problem by means of mathematics, helping designers to reduce unnecessary time and cost overhead. For design optimization based on numerical calculation, besides the high calculation cost, the problem that an expression in a closed form is lacked, derivative information is difficult to obtain and the like exists. Bayesian optimization is one of the most suitable approaches to solve such problems. Bayesian optimization builds a statistical approximation model of the problem through a small amount of data and priori knowledge, replaces a complex and expensive real physical problem to a certain extent, and then actively iterates through an acquisition function to explore a design area of interest, so as to quickly approximate a global optimal solution of the problem.
The acquisition function is used as an active learning strategy in Bayesian optimization, and the trade-off between global exploration and local development needs to be fully considered, so that the efficiency and global optimization capacity of the algorithm are ensured. The confidence upper bound strategy is used as one of the most commonly used acquisition functions by an explicit trade-off coefficientThe weight between the posterior mean and the standard deviation is adjusted, so that the purpose of weighing local development and global exploration is achieved. The use of too large a trade-off factor results in a slower optimization efficiency, while the use of too small a trade-off factor runs the risk of sinking into a locally optimal solution. At present, the common local and global trade-off strategies based on the confidence upper bound acquisition function mainly comprise: adopting a fixed weighing coefficient; gaussian process confidence upper bound policy (GP-UCB); random gaussian process confidence upper bound strategy (RGP-UCB). Existing trade-off strategies often employ a relatively fixed trade-off factor or determine the trade-off factor from the standpoint of ensuring algorithm convergence. However, these trade-off strategies are inadequate and adaptive balance optimization is in progressLocal development and global exploration are difficult to adapt to complex problem backgrounds.
Disclosure of Invention
The invention aims to provide a method for supercritical CO 2 The working medium near the critical point of the centrifugal compressor brings design and optimization problems, and the defects of the existing Bayesian optimization method based on confidence upper bound can dynamically balance local development and global exploration in the optimization process, improve the optimization efficiency and global optimization capacity, and improve supercritical CO 2 Supercritical CO based on optimal balance of pneumatic optimization design result of centrifugal compressor 2 A Bayesian optimization method for centrifugal compressors.
The invention solves the problems in the prior art by adopting the technical scheme that: supercritical CO based on optimal trade-offs 2 The Bayesian optimization method of the centrifugal compressor comprises the following steps:
s1, sample data construction: from true gas based multi-stage supercritical CO 2 Determining d design variables z in one-dimensional aerodynamic calculation model of centrifugal compressor j Wherein z is j Representing the j-th design variable, wherein j is more than or equal to 1 and less than or equal to d, and d is the number of the design variables; the d design variables z j Saved as design variable vectorAnd presets each design variable z j Is a design range of (a);
in the design variable z j For d design variables z within the design range of (2) j Performing assignment, and taking the design variable vector after each assignment as a sample point; selecting t groups to be assigned according to an experimental design method to obtain t sample pointsI is more than or equal to 1 and less than or equal to t, and t is more than or equal to 1; wherein z is ji Representing the jth design variable value in the ith sample point; and forming the t sample points into a sample setThe sample set X is then input to a model and energy containing total loss of pressureSupercritical CO for a model of quantity loss 2 In a one-dimensional pneumatic calculation program of the centrifugal compressor, each sample point x is calculated i Is a sample response value y (x i ) And outputs a response value y (x i ) The sample response vector Y is formed, and the sample set X and the sample response vector Y form a sample data set D= { X, Y };
s2, constructing a model: constructing a Gaussian process proxy model between the design variable vector and the sample response value by using the sample data set D; to obtain a predictive distribution of any new sample points outside the sample dataset D within the design envelope;
s3, calculating a potential optimal balance coefficient set vector of the sample data set Dn is the number of potential optimal sample points; the method comprises the following steps:
s3-1, calculating each sample point x in the sample data set D i Near uncertainty vector
Uncertainty vectors around individual sample pointsBy leaving a cross-validation standard deviationMeasuring, wherein->Is the sample data set D with the sample point x removed i After that, the sample point x obtained in step S2 i Prediction standard deviation at location;
s3-2, according to uncertainty vectorAnd sample response vector Y, selecting a potential optimal sample point vectorn is the number of potential optimal sample points, +.>Representing the mth potential optimal sample point; selecting a potentially optimal sample point by solving the following inequality>m＝1，...，n：
Wherein,to balance the coefficients, y (x i ) And->Respectively represent sample points x i The response value of (c) and leave a cross-validation standard deviation, i=1.. t is; />And->Respectively represent potential optimal sample points->A cross validation standard deviation is reserved; y is max Epsilon is a small non-negative number for the optimal response value in the sample dataset D;
s3-3, arranging the potential optimal sample points according to the numerical value of the sample uncertainty from large to small;
s3-4, calculating the slope of adjacent potential optimal sample pointsIs the absolute value of the potentially optimal trade-off coefficient, wherein +.>m＝1，...，n-1；/>And->Respectively represent potential optimal sample pointsA cross validation standard deviation is reserved;
s3-5, increasing two trade-off coefficient values with the size of 0 and approaching infinityAnd->And the balance coefficient values are arranged in reverse order to obtain a potential optimal balance coefficient set vector +.>
S4, outputting an optimal sample point vector and a response value thereof; selecting point iterative optimization by utilizing a self-adaptive confidence upper bound strategy based on optimal balance to obtain an optimal sample point vector and a response value thereof; obtaining the geometric parameters and aerodynamic index values of the corresponding supercritical carbon dioxide centrifugal compressor according to the obtained optimal sample point vector and the response value thereof; the method comprises the following steps:
s4-1, selecting a weighing coefficient by using a pointer circulation methodSelecting optimal weighting coefficient set vector +/according to pointer theta loop with initial value of 1>Trade-off coefficient +.>
S4-2 based on trade-off coefficient valuesConstruction of an adaptive confidence upper bound acquisition function α (x * The method comprises the steps of carrying out a first treatment on the surface of the D) The method comprises the following steps:wherein μ (x * ) Sum sigma (x) * ) Respectively new sample points x * Is a prediction mean and a prediction standard deviation of (1); the objective function of constructing the acquisition function according to the constraint condition is as follows:
Maximizeα(x * ；D)
wherein,respectively the target flow working condition pressure ratio and the design flow working condition efficiency of the initial compressor model, wherein M is the designated flow, tt, is represents the total-to-total isentropic condition, a represents the target working condition, b represents the design working condition, and x represents the design working condition * Representing a new sample point outside of any one sample dataset D,/for>Representing a compressor model at a new sample point x * Target flow operating mode pressure ratio at +.>Representing a compressor model at a new sample point x * The design flow operating mode efficiency is provided,representing a compressor model at a new sample point x * A maximum flow value at;
s4-3, selecting an expansion sample point x by utilizing an objective function of a particle swarm optimization maximization acquisition function t+1 By supercritical CO 2 One-dimensional pneumatic calculation program solving response value y (x t+1 )；
S4-4, response value y (x t+1 ) Performing stop judgment, and expanding the sample point x if the iteration termination condition is met t+1 And its response value y (x t+1 ) Outputting the optimized centrifugal compressor structure and the aerodynamic performance thereof as an optimal result; if the iteration termination condition is not satisfied, executing the next step;
s4-5, sample Point x will be expanded t+1 And its response value y (x t+1 ) Adding the data into the current sample data to expand the data sample set D, wherein the obtained updated sample data is D=DU { x } t+1 ，y(x t+1 ) And executing the steps S2-S3, carrying out reset judgment on the pointer, if the numerical value of the pointer is smaller than the number of the weighing coefficient values, adding 1 to the numerical value of the pointer theta, otherwise, resetting the pointer theta to be 1, and executing the step S4.
The design variables include: two-stage supercritical CO 2 The centrifugal compressor comprises a first stage rim radius, an outlet width, a blade root height, a blade top height, a diffuser outlet height and a second stage rim radius.
The proxy model in the step S2 is a gaussian process model, where the gaussian process is described as follows:
y(x)～GP(m(x)，k(x，x′))，
wherein x and y (x) represent sample points and corresponding response values, and the mean function m (x) is a zero-order polynomial regression function, i.e., m (x) =1; the covariance function k (x, x ') is a radial basis function representing the correlation between any two sample points x and x' in the sample data set D, i.eWherein (1)>z j ，z′ j Is a design variable; delta and l j The kernel hyper-parameters determined for optimizing the log-marginal likelihood function, delta representing the output ratio, l j Representing the input ratio of the j-th dimension of the design variable, exp being an exponential function with a base number e, d being the number of design variables, j=1, 2,..d represents the individual dimensions of the design variable;
combining the sample data set D= { X, Y }, obtaining a new sample point X at any position except the sample data set D * Is a predictive distribution of:
p(y * |D)＝Normal(μ(x * )，σ 2 (x * ))，
wherein μ (x * )＝m(x * )+k(x * ，X)(k(X，X)) -1 (Y-m (X)) is the new sample point X * Mean value of the position prediction, sigma 2 (x * )＝k(x * ，x * )-k(x * ，X)(k(X，X)) -1 k(x * ，X) T For a new sample point x * The variance is predicted.
T groups of assignment are selected by Latin hypercube sampling experimental design method, and t sample points are obtained
In step S3-2, ε is 0 or 0.01.
The iteration termination condition in the step S4-4 is the maximum function evaluation times N max And relative error E:
wherein y is max Is the maximum value in the sample response value vector Y, Y global Is a known global maximum.
Maximum number of function evaluations N max =20d, d represents the number of design variables. Maximum number of function evaluations N max Can be adjusted according to the complexity of the particular problem.
The invention has the beneficial effects that: the invention is directed to supercritical CO 2 The design and optimization problems brought by the working medium with the near critical point of the centrifugal compressor and the defects of the existing Bayesian optimization method based on the confidence upper bound provide a Bayesian optimization processing method based on optimal balance, so that the local development and global exploration in the optimization process can be dynamically balanced, the optimization efficiency and the global optimization capacity of an algorithm are improved, and the supercritical CO is improved 2 And (3) pneumatic optimization design result of the centrifugal compressor. The main aspects are as follows:
1. different from a general Bayesian optimization method based on confidence upper bound, the invention designs a self-adaptive confidence upper bound Bayesian optimization processing method based on optimal trade-off, and a group of optimal trade-off coefficients are adaptively constructed by searching potential optimal points in a sample, so that the dynamic adjustment method can perform local and global search in the active learning process, thereby improving the search efficiency and avoiding trapping in a local optimal solution.
2. In particular, the uncertainty of the area near the sample is represented by using a cross validation standard deviation, so that the limitation on sample distribution in the feature space and the complex division and distance expression of the feature space are avoided, and the universality and simplicity of an uncertainty-based optimization method are improved.
3. Supercritical CO using improved optimal trade-off Bayesian method 2 The pneumatic performance of the centrifugal compressor is optimized, and the supercritical CO is effectively enhanced 2 Efficiency of the centrifugal compressor under the target flow working condition is reduced simultaneously 2 The computational overhead of the centrifugal compressor design optimization problem.
Drawings
FIG. 1 is a flow chart of the design of the present invention.
FIG. 2 is a flow chart of the adaptive confidence upper bound iterative point selection strategy based on the optimal trade-off of the present invention.
FIG. 3 is a flow chart of the present invention for calculating an optimal set of trade-off coefficients.
Fig. 4 is a schematic diagram of a potentially optimal sample point and optimal trade-off coefficients of the present invention.
FIG. 5 shows the present inventionMing and Ming two-stage supercritical CO 2 The centrifugal compressor is structurally schematic.
FIG. 6 is a supercritical CO according to the present invention 2 Centrifugal compressor design variables schematic.
Detailed Description
The invention is described below with reference to the drawings and the detailed description:
the design thought of the invention is a Bayesian optimization method based on optimal balance, the uncertainty near the sample is represented by using the left cross validation standard deviation of the sample, and a set of optimal balance coefficient values are adaptively constructed by selecting potential optimal points through the uncertainty and response information of the sample, so that local and global searches are balanced in the iterative optimization process, the sample with the highest potential is selected, and the global optimization efficiency of the algorithm is improved. From the design point of view, the design concept of the invention comprises the following treatment processes: as shown in fig. 1:
firstly, establishing a one-dimensional pneumatic calculation model and a pneumatic optimization design target of a supercritical carbon dioxide centrifugal compressor;
secondly, obtaining initial sample data through experimental design;
again, using the sample data set to build a Gaussian Process (GP) proxy model;
then, introducing an optimal weighing coefficient set of the sample data;
then, selecting point iterative optimization through improved optimal balance self-adaptive confidence upper bound strategy;
and finally, outputting the geometric parameters and aerodynamic indexes of the supercritical carbon dioxide centrifugal compressor corresponding to the optimal design.
The invention adopts the Bayesian optimization method based on optimal balance to perform two-stage supercritical CO 2 The centrifugal compressor performs aerodynamic performance optimization. The structure of the centrifugal compressor is shown in fig. 5, and the design variables selected in the optimization process are shown in fig. 6.
Supercritical CO based on optimal trade-offs 2 The Bayesian optimization method of the centrifugal compressor specifically comprises the following steps:
s1, sample data construction: from based on trueMultistage supercritical CO of solid gas 2 Determining d design variables z in one-dimensional aerodynamic calculation model of centrifugal compressor j Wherein z is j The j design variables are represented, j is more than or equal to 1 and less than or equal to d, and d is the number of the design variables. Design variable z j Comprising the following steps: two-stage supercritical CO 2 The centrifugal compressor comprises a first stage rim radius, an outlet width, a blade root height, a blade top height, a diffuser outlet height and a second stage rim radius. The d design variables z j Saved as design variable vectorAnd presets each design variable z j Is a design range of (a);
example selection of two-stage supercritical CO shown in FIG. 5 2 The centrifugal compressor was optimized and the supercritical CO shown in fig. 6 was selected 2 First stage rim radius R 'of centrifugal compressor' 2 Width of outlet B' 2 Blade root height R' 1s Height of leaf top R' 1h Diffuser outlet height R 2 Second stage rim radius R'. 2 Component variable vector as d=6 design variablesAnd given the design ranges of the individual design variables as shown in table 1 below:
table 1 design range table of design variables
| R′ 2 | B′ 2 | R′ 1s | R′ 1h | R″′ 2 | R″′ 2 | |
| Lower bound of variable/m | 0.075 | 0.0075 | 0.05 | 0.016 | 0.12 | 0.085 |
| Variable upper bound/m | 0.095 | 0.0085 | 0.07 | 0.035 | 0.2 | 0.12 |
Within the range of Table 1, for d design variables z j Performing assignment, and taking the design variable vector after each assignment as a sample point; t groups of assignment are selected by using Latin hypercube sampling experimental design method to obtain t sample pointsI is more than or equal to 1 and less than or equal to t, and t is more than or equal to 1. The present embodiment selects t=20 sample points, where z ji Representing the jth design variable value in the ith sample point; and the t sample points are combined into a sample set +.>The sample set X is then input into supercritical CO 2 In a one-dimensional pneumatic calculation program of the centrifugal compressor, each sample point x is calculated i Is a sample response value y (x i ) And outputs a response value y (x i ) The sample response vector Y is formed, and the sample set X and the sample response vector Y form sample data D= { X, Y };
wherein, supercritical CO 2 The centrifugal compressor one-dimensional aerodynamic calculation program mixes the total pressure loss model and the energy loss model.
Sample response value y (x i ) Is supercritical CO 2 The centrifugal compressor is at sample point x i Total to total isentropic efficiency at target flow conditions of 50kg/sAnd 3 structural design constraint conditions of the centrifugal compressor are included in the optimization process, namely: target flow operating pressure->Flow capacity->Design flow regime efficiency->The purpose of this step is to hope to find the sample point x i Maximum total to total isentropic efficiency in the design range +.>Thus, the objective function of the optimization problem is expressed as:
Maximize
wherein,the target flow working condition pressure ratio and the design flow working condition efficiency of the initial model are respectively, M is the designated flow 100kg/s, tt, is represents the total isentropic condition, a represents the target working condition 50kg/s flow, b represents the design working condition 70kg/s flow, x * Representing new sample points outside of any one sample dataset D.
S2, constructing a model: constructing a Gaussian process proxy model between the design variable vector and the sample response value by using the sample data D to obtain the prediction distribution of any new sample points outside the sample data set D in the design range; wherein the gaussian process is described as follows:
y(x)～GP(m(x)，k(x，x′))，
wherein x and y (x) represent sample points and corresponding response values, and the mean function m (x) is a zero-order polynomial regression function, i.e., m (x) =1; the covariance function k (x, x ') is a radial basis function representing the correlation between any two sample point vectors x and x' in the sample dataset D.
I.e.
Wherein,z j ，z′ j is a design variable; delta and l j The kernel hyper-parameters determined for optimizing the log-marginal likelihood function, delta representing the output ratio, l j Representing the input ratio of the j-th dimension of the design variable, exp being an exponential function with a base number e, d being the number of design variables, j=1, 2,..d represents the individual dimensions of the design variable;
combining the sample data set D= { X, Y }, obtaining a new sample point X at any position outside the sample data set D * Is a predictive distribution of:
p(y * |D)＝Normal(μ(x * )，σ 2 (x * ))，
wherein μ (x * )＝m(x * )+k(x * ，X)(k(X，X)) -1 (Y-m (X)) is the new sample point X * Mean value of the position prediction, sigma 2 (x * )＝k(x * ，x * )-k(x * ，X)(k(X，X)) -1 k(x * ，X) T For a new sample point x * The variance is predicted.
As a proxy model, the step obtains a prediction mean and a prediction standard deviation of a new sample point at any position outside the sample data set D; and then selecting an output result which is better than the sample point in the sample data set D according to the prediction mean value and the prediction standard deviation to carry out real calculation verification. This proxy model is actually composed of the prediction mean and the prediction standard deviation.
S3, calculating a potential optimal balance coefficient set vector of the sample data Dn is the number of potential optimal sample points; as shown in fig. 3, the method comprises the following steps:
s3-1, calculating each sample point x in the sample data set D i Near uncertainty vector
Uncertainty vectors around individual sample pointsBy leaving a cross-validation standard deviationMeasuring, wherein->Is the current sample dataset D with the sample points x removed i After that, the sample point x obtained in step S2 i Prediction standard deviation at location;
s3-2, according to uncertainty vectorAnd sample response vector Y, selecting a potential optimal sample point vectorn is the number of potential optimal sample points, +.>Representing the sample point as a potentially optimal sample point; the potentially optimal sample point requires that there is a certain trade-off coefficient +.>Such that (i) its corresponding Lipschitz upper bound is better than the Lipschitz upper bound of other sample points, and (ii) better than the current optimal solution. Selecting a potentially optimal sample point by solving the following inequality>m＝1，...，n：
Wherein,to weigh the coefficients, i=1,.. i ) And->Respectively represent each sample point x i A cross validation standard deviation is reserved; />And->Respectively represent each potential optimal sample point->A cross validation standard deviation is reserved; y is max For the optimal response value in the current sample data D, epsilon is a smaller non-negative number, and for the unknown optimization problem, 0 or 0.01 can be generally taken, and epsilon=0 is selected in this embodiment;
s3-3, arranging the potential optimal sample points according to the numerical value of the sample uncertainty from large to small;
s3-4, calculating the slope of adjacent potential optimal sample pointsIs the absolute value of the potential optimal trade-off coefficient. As shown in FIG. 4, +.>In the figure, the solid points represent potential optimal sample points, and the potential optimal trade-off coefficient value is the absolute value of the slope of the adjacent potential optimal sample points, and can be expressed as +.>m＝1，...，n-1；/>And->Respectively represent potential optimal sample points->A cross validation standard deviation is reserved;
s3-5, increasing two trade-off coefficient values with the size of 0 and approaching infinityAnd->And inversely ordering the trade-off coefficient valuesColumn, get the potentially optimal trade-off coefficient set vector +.>
S4, outputting an optimal sample point vector and a response value thereof; the optimal sample point vector and the response value thereof are obtained by utilizing the self-adaptive confidence upper bound strategy point selection iterative optimization based on optimal balance (as shown in figure 2); according to the obtained optimal sample point vector and the response value thereof, outputting the geometric parameters and aerodynamic index values of the corresponding supercritical carbon dioxide centrifugal compressor; the method comprises the following steps:
s4-1, selecting a weighing coefficient by using a pointer circulation methodSelecting optimal weighting coefficient set vector +/according to pointer theta loop with initial value of 1>Trade-off coefficient +.>
S4-2 based on trade-off coefficient valuesConstruction of an adaptive confidence upper bound acquisition function α (x * The method comprises the steps of carrying out a first treatment on the surface of the D) The method comprises the following steps:wherein μ (x * ) Sum sigma (x) * ) Respectively new sample points x * And a prediction standard deviation. The objective function of constructing the acquisition function according to the constraint condition is as follows:
Maximizeα(x * ；D)
wherein,respectively the pressure ratio under the working condition of 50kg/s of target flow and the efficiency under the working condition of 70kg/s of design flow of an initial compressor model, wherein M is 100kg/s of designated flow, tt, is represents the total isentropic condition of total pair, a represents the target working condition, b represents the design working condition and x represents the design working condition * Representing a new sample point outside of any one sample dataset D,/for>Representing a compressor model at a new sample point x * Target flow operating mode pressure ratio at +.>Representing a compressor model at a new sample point x * Design flow operating efficiency at->Representing a compressor model at a new sample point x * A maximum flow value at;
s4-3, maximizing an objective function of an acquisition function at any new sample point x by using a particle swarm algorithm * Selecting an expansion sample point x meeting the condition t+1 By supercritical CO 2 One-dimensional pneumatic calculation program solving response value y (x t+1 )；
Selecting an extended sample point x at an objective function that maximizes an acquisition function using a particle swarm algorithm t+1 In this case, the Matlab-based particle swarm optimization algorithm can be used to maximize the acquisition function α (x) * ；D)。
The method comprises the following steps:
s4-3-1 randomly initializes the population of particles, for the embodiment of the invention, the population is set to l=100, and the input dimension is d=6;
s4-3-2, calculating an adaptation value of each particle according to the adaptation function;
for each constraint, data set D is utilized ce Establishing a gaussian process proxy model of the respective constraint by step S2, wherein the dataset D ce ＝{X，f ce E=1, 2,3, c represents a constraint, f ce For passing supercritical CO 2 Each constraint condition obtained by solving one-dimensional calculation program of centrifugal compressorIs a response value vector of (a); if the predicted mean mu of each particle confinement proxy model C (x * ) If the constraint condition is not satisfied, the corresponding particle adaptation value, namely the response value of the objective function, is 0; if the constraint condition is satisfied, the adaptation value of the particle is the predicted mean μ (x * )；
S4-3-3 compares the particle adaptation value solved in S4-3-2 with the particle history optimal solution to solve the individual optimal adaptation value;
s4-3-4 compares the optimal adaptation values in all particles of S4-3-3 with a historical optimal solution to obtain an optimal adaptation value of the population;
s4-3-5, updating the position and speed of the particles by using a position and speed formula defaulted by a particle swarm algorithm in Matlab;
s4-3-6 according to the maximum iteration number n max And (1200) judging whether the particle swarm algorithm is finished or not, if the maximum iteration number is not reached, starting to circulate from 4-4-2, and if not, outputting a result.
S4-4, response value y (x t+1 ) Stopping judgment is carried out, if the iteration termination condition is met, the current point vector x is calculated t+1 And its response value y (x t+1 ) Outputting the optimized centrifugal compressor structure and the aerodynamic performance thereof as an optimal result; if the iteration termination condition is not satisfied, executing the next step;
the iteration termination condition is the maximum function evaluation times N max =120 and relative error E:
wherein y is max For the maximum value in the current sample response value vector Y, Y global Is a known global maximum.
S4-5, sample Point x will be expanded t+1 And its response value y (x t+1 ) Adding the data into the current sample data to expand the data sample set D, wherein the obtained updated sample data set is D=DU { x } t+1 ，y(x t+1 ) And executing the steps S2-S3, carrying out reset judgment on the pointer, if the numerical value of the pointer is smaller than the number of the weighing coefficient values, adding 1 to the numerical value of the pointer theta, otherwise, resetting the pointer theta to be 1, and executing the step S4.
The result is compared with the optimized result of the algorithm proposed by the invention by using a commonly used Bayesian optimization algorithm based on a confidence upper bound acquisition function, as shown in table 2.
Table 2 optimization average results for 10 replicates of different confidence upper bound trade-off strategies in experiments
The confidence upper bound based acquisition strategies for comparison in table 2 include two strategies employing fixed trade-off values of 1.96 and 10, a gaussian process confidence upper bound strategy (GP-UCB) and a random gaussian process confidence upper bound strategy (RGP-UCB). After 120 steps of iterative optimization, the supercritical C02 centrifugal compressor Bayesian optimization method based on optimal balance provided by the invention uses an improved self-adaptive confidence upper bound strategy (AUCB), and has the best optimization effect compared with other confidence upper bound strategies, so that supercritical CO 2 The efficiency of the centrifugal compressor under the flow working condition of 50kg/s reaches 82.78%, and compared with the initial scheme after experimental design, the efficiency of the centrifugal compressor under the target working condition is improved by 7.52%.
The above is a further detailed description of the invention in connection with specific preferred embodiments, and it is not to be construed as limiting the practice of the invention to these descriptions. It will be apparent to those skilled in the art that several simple deductions or substitutions may be made without departing from the spirit of the invention, and these should be considered to be within the scope of the invention.
Claims (7)
1. Supercritical CO based on optimal trade-offs 2 The Bayesian optimization method for the centrifugal compressor is characterized by comprising the following steps of:
s1, sample data construction: from true gas based multi-stage supercritical CO 2 Determining d design variables z in one-dimensional aerodynamic calculation model of centrifugal compressor j Wherein z is j Representing the j-th design variable, wherein j is more than or equal to 1 and less than or equal to d, and d is the number of the design variables; the d design variables z j Saved as design variable vectorAnd presets each design variable z j Is a design range of (a);
in the design variable z j For d design variables z within the design range of (2) j Performing assignment, and taking the design variable vector after each assignment as a sample point; selecting t groups to be assigned according to an experimental design method to obtain t sample pointsWherein z is ji Representing the jth design variable value in the ith sample point; and the t sample points are combined into a sample set +.>The sample set X is then input into a supercritical CO containing a total pressure loss model and an energy loss model 2 In a one-dimensional pneumatic calculation program of the centrifugal compressor, each sample point x is calculated i Is a sample response value y (x i ) And outputs a response value y (x i ) The sample response vector Y is formed, and the sample set X and the sample response vector Y form a sample data set D= { X, Y };
s2, constructing a model: constructing a Gaussian process proxy model between the design variable vector and the sample response value by using the sample data set D; to obtain a predictive distribution of any new sample points outside the sample dataset D within the design envelope;
s3, calculating potential optimal balance of the sample data set DCoefficient set vectorn is the number of potential optimal sample points; the method comprises the following steps:
s3-1, calculating each sample point x in the sample data set D i Near uncertainty vector
Uncertainty vectors around individual sample pointsBy leaving a cross-validation standard deviationMeasuring, wherein->Is the sample data set D with the sample point x removed i After that, the sample point x obtained in step S2 i Prediction standard deviation at location;
s3-2, according to uncertainty vectorAnd sample response vector Y, selecting a potential optimal sample point vectorn is the number of potential optimal sample points, +.>Representing the mth potential optimal sample point; selecting a potentially optimal sample point by solving the following inequality>
Wherein,to balance the coefficients, y (x i ) And->Respectively represent sample points x i The response value of (c) and leave a cross-validation standard deviation, i=1.. t is; />And->Respectively represent potential optimal sample points->A cross validation standard deviation is reserved; y is max Epsilon is a small non-negative number for the optimal response value in the sample dataset D;
s3-3, arranging the potential optimal sample points according to the numerical value of the sample uncertainty from large to small;
s3-4, calculating the slope of adjacent potential optimal sample pointsIs the absolute value of the potential optimal trade-off coefficient, wherein And->Respectively represent potential optimal sample pointsA cross validation standard deviation is reserved;
s3-5, increasing two trade-off coefficient values with the size of 0 and approaching infinityAnd->And the balance coefficient values are arranged in reverse order to obtain a potential optimal balance coefficient set vector +.>
S4, outputting an optimal sample point vector and a response value thereof; selecting point iterative optimization by utilizing a self-adaptive confidence upper bound strategy based on optimal balance to obtain an optimal sample point vector and a response value thereof; obtaining the geometric parameters and aerodynamic index values of the corresponding supercritical carbon dioxide centrifugal compressor according to the obtained optimal sample point vector and the response value thereof; the method comprises the following steps:
s4-1, selecting a weighing coefficient by using a pointer circulation methodSelecting optimal weighting coefficient set vector +/according to pointer theta loop with initial value of 1>Trade-off coefficient +.>
S4-2 based on trade-off coefficient valuesConstruction of an adaptive confidence upper bound acquisition function α (x * The method comprises the steps of carrying out a first treatment on the surface of the D) The method comprises the following steps:wherein μ (x * ) Sum sigma (x) * ) Respectively new sample points x * Is a prediction mean and a prediction standard deviation of (1); the objective function of constructing the acquisition function according to the constraint condition is as follows:
Maximize α(x * ；D)
wherein,respectively the target flow working condition pressure ratio and the design flow working condition efficiency of the initial compressor model, wherein M is the designated flow, tt, is represents the total-to-total isentropic condition, a represents the target working condition, b represents the design working condition, and x represents the design working condition * Representing a new sample point outside of any one sample dataset D,/for>Representing a compressor model at a new sample point x * Target flow operating mode pressure ratio at +.>Representing a compressor model at a new sample point x * Design flow operating efficiency at->Representing a compressor model at a new sample point x * A maximum flow value at;
s4-3, selecting an expansion sample point x by utilizing an objective function of a particle swarm optimization maximization acquisition function t+1 By supercritical CO 2 One-dimensional pneumatic calculation program solving response value y (x t+1 )；
S4-4, response value y (x t+1 ) Performing stop judgment, and expanding the sample point x if the iteration termination condition is met t+1 And its response value y (x t+1 ) Outputting the optimized centrifugal compressor structure and the aerodynamic performance thereof as an optimal result; if the iteration termination condition is not satisfied, executing the next step;
s4-5, sample Point x will be expanded t+1 And its response value y (x t+1 ) Adding the data into the current sample data to expand the data sample set D, wherein the obtained updated sample data is D=DU { x } t+1 ，y(x t+1 ) And executing the steps S2-S3, carrying out reset judgment on the pointer, if the numerical value of the pointer is smaller than the number of the weighing coefficient values, adding 1 to the numerical value of the pointer theta, otherwise, resetting the pointer theta to be 1, and executing the step S4.
2. Supercritical CO based on optimal trade-offs according to claim 1 2 The Bayesian optimization method for the centrifugal compressor is characterized in that the design variables comprise: two-stage supercritical CO 2 The centrifugal compressor comprises a first stage rim radius, an outlet width, a blade root height, a blade top height, a diffuser outlet height and a second stage rim radius.
3. Supercritical CO based on optimal trade-offs according to claim 1 2 The Bayesian optimization method for the centrifugal compressor is characterized in that the proxy model in the step S2 is a Gaussian process model, wherein the Gaussian process is described as follows:
y(x)～GP(m(x)，k(x，x′))，
wherein x and y (x) represent sample points and corresponding response values, and the mean function m (x) is a zero-order polynomial regression function, i.e., m (x) =1; the covariance function k (x, x ') is a radial basis function representing the correlation between any two sample points x and x' in the sample data set D, i.eWherein (1)>z j ，z′ j Is a design variable; delta and l j The kernel hyper-parameters determined for optimizing the log-marginal likelihood function, delta representing the output ratio, l j The input ratio of the j-th dimension of the design variable is represented, exp is an exponential function with a base number e, d is the number of the design variables, j=1, 2.
Combining the sample data set D= { X, Y }, obtaining a new sample point X at any position except the sample data set D * Is a predictive distribution of:
p(y * |D)＝Normal(μ(x * )，σ 2 (x * ))，
wherein μ (x * )＝m(x * )+k(x * ，X)(k(X，X)) -1 (Y-m (X)) is the new sample point X * Mean value of the position prediction, sigma 2 (x * )＝k(x * ，x * )-k(x * ，X)(k(X，X)) -1 k(x * ，X) T For a new sample point x * The variance is predicted.
4. Supercritical CO based on optimal trade-offs according to claim 1 2 The Bayesian optimization method for the centrifugal compressor is characterized in that t groups of assignments are selected through a Latin hypercube sampling experimental design method to obtain t sample points
5. Supercritical CO based on optimal trade-offs according to claim 1 2 A Bayesian optimization method for a centrifugal compressor is characterized in that epsilon is 0 or 0.01 in the step S3-2.
6. According to the weightsSupercritical CO based on optimal trade-offs according to claim 1 2 The Bayesian optimization method of the centrifugal compressor is characterized in that the iteration termination condition in the step S4-4 is the maximum function evaluation times N max And relative error E:
wherein y is max Is the maximum value in the sample response value vector Y, Y global Is a known global maximum.
7. Supercritical CO based on optimal trade-offs according to claim 6 2 The Bayesian optimization method for the centrifugal compressor is characterized in that the maximum function evaluation times N max =20d, d represents the number of design variables.
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