Abstract
Supercritical carbon dioxide (S-CO2) Brayton cycle has a wide application prospect. It is significant to optimize the aerodynamic performance of the core part turbine. In this paper, we develop an expected hypervolume improvement (EHVI)-based active learning (EHVI-AL) framework and focus on multi-objective optimization for output power and isentropic efficiency. Firstly, the crowding distance is considered in the candidate samples generation of EHVI evaluation. The power and efficiency of the Pareto front solutions obtained by the EHVI-AL framework are higher than those of other optimization algorithms, and the Pareto front can provide more diverse choices of optimized solutions. Then, considering the cycle economy and output power, the target range of efficiency and power is obtained by unconstrained multi-objective optimization with fewer steps, and the optimization is carried out within this range based on truncated expected hypervolume improvement (tEHVI), which reduces the computational cost by 50%. Finally, considering the turbine mass flow variation may cause surge or choke of the compressor as well as uneven pressure ratio distribution in multi-stage compressor, the range of the turbine mass flow is set and the multi-objective optimization with the mass flow constraint is realized by the constrained expectation hypervolume improvement (cEHVI). The EHVI-AL framework adopted in this paper provides an efficient way to optimize the aerodynamic performance of a S-CO2 turbine, and the modification of EHVI can effectively take into account the effects of Brayton cycle and deal with the power, efficiency, and flow constraints in real situations.
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Abbreviations
- RGP:
-
Real gas properties
- SST:
-
Shear stress transfer
- \(f\) :
-
Objective function
- \(c\) :
-
Constraints
- \(L\) :
-
Lower boundary of design parameters
- \(U\) :
-
Upper boundary of design parameters
- \(\mu ,{{\varvec{\upmu}}}\),:
-
Mean or mean matrix
- \(\sigma ,{{\varvec{\upsigma}}}\) :
-
Standard deviation or standard deviation matrix
- \({\mathbf{k}}\) :
-
Covariance matrix
- \({\mathbf{x}}^{*}\) :
-
Test input vector
- \({\mathbf{X}}\) :
-
Train set input matrix
- \({\mathbf{Y}}\) :
-
Response to input matrix
- \(\lambda_{d}\) :
-
Lebesgue measure
- \({\mathbf{y}}\) :
-
Newly added vector
- \({\mathbf{r}}\) :
-
Reference point
- \({\mathbf{P}}\) :
-
Pareto set
- \({\text{HVI}}\) :
-
Hypervolume improvement
- \({\text{PDF}}\) :
-
Probability density function
- \({\text{tPDF}}\) :
-
Truncated probability density function
- \(l_{c}\) :
-
Crowding distance
- \(l\) :
-
Lower boundary of slice area
- \(u\) :
-
Upper boundary of slice area
- \(x\) :
-
Profile geometry parameter
- \(s\) :
-
Standard deviation of the random search
- \(R_{{{\text{in}}}}\) :
-
Inlet radius of rotor
- \(R_{{{\text{out}}}}\) :
-
Outlet radius of rotor
- \(H\) :
-
Height of blade
- \(n\) :
-
Number of blades
- \(d\) :
-
Distance between two adjacent points
- \(P\) :
-
Output power
- \(T_{z}\) :
-
Torque on rotor blade
- \(\eta_{is}\) :
-
Isentropic efficiency of turbine rotor
- r :
-
Rotation speed
- \(\dot{m}\) :
-
Mass flow
- \(\eta_{s}^{*}\) :
-
Isentropic efficiency of compressor stage
- \(\pi^{*}\) :
-
Stage inlet and outlet pressure ratio
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Acknowledgements
This work was support by 111 Project of Chinese Ministry of Education (Grant No. B16038).
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PW contributed to methodology, formal analysis, investigation, resources, and writing—original draft. YW contributed to conceptualization, software, and data curation. TL: contributed to validation, writing—review and editing, and visualization. DZ contributed to supervision, project administration, and funding acquisition.
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Wang, P., Wang, Y., Liu, T. et al. Active learning-based multi-objective optimization for aerodynamic performance of a supercritical carbon dioxide turbine. Struct Multidisc Optim 65, 267 (2022). https://doi.org/10.1007/s00158-022-03391-x
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DOI: https://doi.org/10.1007/s00158-022-03391-x