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Optimization of the Complex-RFM optimization algorithm

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Abstract

This paper presents and compares different modifications made to the Complex-RF optimization algorithm with the aim of improving its performance for computationally expensive models with few variables. The modifications reduce the required number of objective function evaluations by creating and using surrogate models (SMs) of the objective function iteratively during the optimization process. The chosen SM type is a second order response surface. The performance of the modified algorithm is demonstrated for both analytical and engineering problems and compared with the performance of a number of existing algorithms. A meta-optimization of the algorithm is also performed to optimize its performance for arbitrary problems. To emphasize the fact that the modified algorithm uses metamodels it is denoted Complex-RFM.

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Authors and Affiliations

  1. Division of Machine Design, Linköping University, Linköping, Sweden

    Johan A. Persson & Johan Ölvander

Authors
  1. Johan A. Persson
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  2. Johan Ölvander
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Corresponding author

Correspondence to Johan A. Persson.

Appendices

Annex 1: pseudo code for the Original Model/SM activity of Complex-RFM

figure a

Annex 2: pseudo code for the SM check activity of Complex-RFM

figure b

Annex 3: coefficients of the Hartmann6 function

See Table 8.

Table 8 Parameters for Hartmann6
Full size table

Annex 4: a description of the electric motorcycle model

The electric motorcycle model consists of four major parts—a battery, an electric motor, a gearbox and the motorcycle itself. The parameters of the model can be seen in Table 9.

Table 9 Parameters for the electric motorcycle model
Full size table

4.1 The battery model

The voltage of the battery is modeled as a linearly decreasing function of the energy it has delivered, see Eq. 11.

$$ V(t)=U_{max}-K_{bat}\int\limits_0^t Adt. $$
(11)

However, the battery can only deliver a certain amount of energy. When the maximum number of Ah is reached, the battery is considered empty and the voltage is set to zero.

4.2 The electric motor

The equations used to model the electric motor are shown in Eq. 12.

$$ L {\frac{di}{dt}}=C_{in} U-k_E \omega-R \cdot i, $$
(12)
$$ T_{out}=k_mi. $$

A picture of the motor model is shown in Fig.5. The control signal to the motor (C in in Eq. 12) enters the model at the top and is used to reduce the voltage to the motor. The control signal is between −1 and 1 where 1 means full throttle.

Fig. 5
figure 5

A screenshot of the electric motor submodel from MATLAB Simulink

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Fig. 6
figure 6

A screenshot of the gearbox submodel from MATLAB Simulink

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4.3 The gearbox

The gearbox has two gears and it is modeled using three parameters, u 1, u 2 and v shift . The gear ratio is changed from u 1 to u 2 at the rotational speed v shift . In the gearbox, the output torque is amplified by a factor u 1 (or u 2) and the torque is then reduced by a factor of 0.95 which models all mechanical losses of the transmission. The model of the gearbox is shown in Fig. 6.

4.4 The motorcycle model

The motorcycle itself is modeled using Newton’s second law, where the input torque is divided by the wheel radius to give the force driving the motorcycle forward. From this force a rolling resistance and the air drag are subtracted, see Eq. 13.

$$ \frac{T_{in}(t)}{r}-C_{d}A\rho{\frac{v(t)^2}{2}}-\mu\cdot mg=m\cdot a(t). $$
(13)

The acceleration a(t) is then integrated to obtain the velocity, which in turn is integrated in order to calculate the distance travelled by the motorcycle, see Fig. 7.

Fig. 7
figure 7

A screenshot of the motorcycle submodel from MATLAB Simulink

Full size image

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Persson, J.A., Ölvander, J. Optimization of the Complex-RFM optimization algorithm. Optim Eng 16, 27–48 (2015). https://doi.org/10.1007/s11081-014-9247-9

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  • DOI: https://doi.org/10.1007/s11081-014-9247-9

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