© The Institution of Engineering and Technology
Since uncertainties in variables are unavoidable, an optimal solution must consider the robustness of the design. The gradient index approach provides a convenient way to evaluate the robustness but is inconclusive when several possible solutions exist. To overcome this limitation, a novel methodology based on the use of first- and second-order gradient indices is proposed introducing the notion of gradient sensitivity. The sensitivity affords a measure of the change in the objective function with respect to the uncertainty of the variables. A Kriging method assisted by algorithms exploiting the concept of rewards is employed to facilitate function predictions for the robust optimisation process. The performance of the proposed algorithm is assessed through a series of numerical experiments. A modification to the correlation model through the introduction of a Kriging predictor and mean square error criterion allows efficient solution of large scale and multi-parameter problems. The three-parameter version of TEAM Workshop Problem 22 has been used for illustration.
References
-
-
1)
-
L. Lebensztajn ,
C.A.R. Marretto ,
M.C. Costa ,
J.-L. Coulomb
.
Kriging: a useful tool for electromagnetic devices optimization.
IEEE Trans. Magn.
,
2 ,
1196 -
1199
-
2)
-
4. Ren, Z.Y., Pham, M.T., Song, M.H., Kim, D.H., Koh, C.S.: ‘A robust global optimisation algorithm of electromagnetic devices utilizing gradient index and multi-objective optimisation method’, IEEE Trans. Magn., 2011, 47, (5), pp. 1254–1257 (doi: 10.1109/TMAG.2010.2080664).
-
3)
-
15. Santner, T.J., Williams, B.J., Notz, W.I.: ‘The design and analysis of computer experiments’ (Springer, 2003).
-
4)
-
2. Kim, K.S., Lee, S.J., Cho, S.G., et al: ‘Multi-response Taguchi robust design of back electromotive force and cogging torque considering the manufacturing tolerance for electric machine’. 13thInt. Conf. Optimization of Electrical and Electronic Equipment (OPTIM), Brasov, Romania, 24–26 May 2012.
-
5)
-
3. González, I., Sánchez, I.: ‘Optimal centering and tolerance design for correlated variables’, Int. J. Adv. Manuf. Technol., 2013, 66, pp. 1499–1510 (doi: 10.1007/s00170-012-4434-3).
-
6)
-
9. Xiao, S., Rotaru, M., Sykulski, J.K.: ‘Exploration versus exploitation using Kriging surrogate modelling in electromagnetic design’, COMPEL, 2012, 31, (5), pp. 1541–51 (doi: 10.1108/03321641211248291).
-
7)
-
13. Xiao, S., Rotaru, M., Sykulski, J.K.: ‘Adaptive weighted expected improvement with rewards approach in Kriging assisted electromagnetic design’, IEEE Trans. Magn., 2013, 49, (5), pp. 2057–2060 (doi: 10.1109/TMAG.2013.2240662).
-
8)
-
6. Ren, Z.Y., Pham, M.T., Koh, C.S.: ‘Robust global optimisation of electromagnetic devices with uncertain design parameters: comparison of the worst-case optimisation methods and multi-objective optimisation approach using gradient index’, IEEE Trans. Magn., 2013, 49, (2), pp. 851–859 (doi: 10.1109/TMAG.2012.2212713).
-
9)
-
5. Ren, Z.Y., Zhang, D.H., Koh, C.S.: ‘Reliability evaluation of the electromagnetic device based on the second order sensitivity analysis’. Sixth Int. Conf. Electromagnetic Field Problems and Applications (ICEF), Dalian, China, 2012.
-
10)
-
19. Ho, S.L., Yang, S.Y., Ni, G.Z., Wong, H.C.: ‘An improved Tabu search for the global optimizations of electromagnetic devices’, IEEE Trans. Magn., 2001, 37, (5), pp. 3570–3574 (doi: 10.1109/20.952664).
-
11)
-
10. Xiao, S., Rotaru, M., Sykulski, J.K.: ‘Strategies for balancing exploration and exploitation in electromagnetic optimisation’, COMPEL, 2013, 32, (4), pp. 1176–1188 (doi: 10.1108/03321641311317004).
-
12)
-
16. Ho, S.L., Yang, S.Y., Ni, P.H., Wong, K.F.: ‘An adaptive interpolating MLS based response surface model applied to design optimisations of electromagnetic devices’, IEEE Trans. Magn., 2007, 43, (4), pp. 1593–1596 (doi: 10.1109/TMAG.2006.892107).
-
13)
-
22. Yang, S.Y., Bai, Y.N., Zhang, G.H., Vecchi, L.Wu.: ‘An improved population-based incremental learning method for inverse problems’. Automation Congress, 2008, pp. 1–4.
-
14)
-
11. Sykulski, A., Adams, N., Jennings, N.: ‘On-line adaptation of exploration in the one-armed bandit with covariate problem’. Proc. Ninth Int. Conf. Machine Learning and Applications, 2010.
-
15)
-
17. Holland, J.H.: ‘Adaption in natural and artificial system’ (MIT Press, Cambridge, 1975).
-
16)
-
18. Hu, N.: ‘Tabu search method with random moves for globally optimal design’, Int. J. Numer. Methods Eng., 1992, 35, pp. 1055–1071 (doi: 10.1002/nme.1620350508).
-
17)
-
23. Xiao, S., Rotaru, M., Sykulski, J.K.: ‘Correlation matrices in Kriging assisted optimisation of electromagnetic devices’, IET Sci. Meas. Technol., 2014, .
-
18)
-
14. Alotto, P.G., Baumgartner, U., Freschi, F., et al: .
-
19)
-
7. Sykulski, J.K.: ‘New trends in optimisation in electromagnetics’. IET Seventh Int. Conf. Computation in Electromagnetics (CEM 2008), 2008, pp. 44–49.
-
20)
-
1. Weng, W.C., Yang, F., Demir, V., Elsherbeni, A.: ‘Optimization using Taguchi method for electromagnetic applications’. Proc. European Conf. Antennas and Propagation ‘EuCAP 2006’, (ESA SP-626), Nice, France, 6–10 November 2006.
-
21)
-
S. Kirkpatrick ,
J.C.D. Gelatt ,
M. Vecchi
.
Optimization by simulated annealing.
Science
,
4598 ,
671 -
680
-
22)
-
21. Hajji, O., Brisset, S., Brochet, P.: ‘A new Tabu search method for optimization with continuous parameters’, IEEE Trans. Magn., 2004, 40, (2), pp. 1184–1187 (doi: 10.1109/TMAG.2004.824909).
-
23)
-
G. Hawe ,
J. Sykulski
.
Consideratons of accuracy and uncertainty with kriging surrogate models in single objective electromagnetic design optimisation.
IET Sci. Meas. Technol.
,
1 ,
37 -
47
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