Parameter estimation in equivalent circuit analysis of dielectric cure monitoring signals using genetic algorithms

Figures & data

Figure 1. Equivalent circuit of epoxy thermoset.

Figure 2. Real and imaginary impedance spectra typical of a particular given state in epoxy resin cure. The equivalent circuit used is shown in figure 1 and the circuit element values are given in table 1.

Table 1. Circuit element values for the construction of simulated spectra.

R ion (Ohm)	R dip (Ohm)	C dip (F)	C ind (F)	R el (Ohm)	n
10^6	10^4	2×10^−12	10^−11	10^6	0.5

Table 2. Sensitivity to circuit parameters expressed as the average percentage difference from the original case for a ±10% change in the parameter.

		R ion (Ohm)	R dip (Ohm)	C dip (F)	C ind (F)	R el (Ohm)	n
Z′	+10%	8.42	0.58	1.34	6.14	0.25	2.21
	−10%	7.10	0.01	1.03	5.36	2.86	2.61
Z′′	+10%	8.00	0.57	1.35	4.95	0.25	1.12
	−10%	7.56	0.01	1.02	4.96	2.86	2.52

Figure 3. Genetic algorithm flowchart.

Table 3. Objective functions tested without any regularization term.

S 1	Weighted least squares
S 2	Weighted absolute differences
S 3	Logarithmic scaling – least squares
S 4	Logarithmic scaling – absolute differences

Table 4. Search space of the fitting parameters.

	R ion (Ohm)	R dip (Ohm)	C dip (F)	C ind (F)	R el (Ohm)	n
Upper limit	1.25 × 10^6	4.95 × 10^4	1.90 × 10^−12	1.25 × 10^−12	1.95 × 10^6	0.625
Low limit	7.50 × 10^5	5.00 × 10^2	1.00 × 10^−13	7.50 × 10^−13	5.00 × 10^4	0.375
Space size	5.00 × 10^5	4.90 × 10^4	1.80 × 10^−12	5.00 × 10^−13	1.90 × 10^6	0.250

Table 5. Fitting results for the tuning of the genetic algorithm parameters.

Exchange probability value ( p e )	Maximum fitness value	Standard deviation
0.4	95.41	4.242
0.5	95.12	4.052
0.6	94.25	3.351
Number of generations (G max)	Fitness value divided by the maximum fitness	Standard deviation
60	0.996	2.45 × 10^−3
80	0.998	1.22 × 10^−3
100	1.000	0.51 × 10^−3
120	1.000	0.05 × 10^−3
140	1.000	0.02 × 10^−3
Number of individuals (N)	Maximum fitness value	Standard deviation
81	95.32	4.700
101	94.50	4.785
121	94.48	4.126

Table 6. Final values for the genetic algorithm implementation.

Maximum number of generations   ( G max )	Number of solution vectors (individuals) ( N )	Elitism ( N e)	Exchange probability ( p e)	zInitial Mutation Probability ( p m)
100	101	9	0.5	0.025

Table 7. Results from test runs on noisy data for the determination of the objective function.

	R ion  (Ohm)	R dip (Ohm)	C dip (F)	C ind (F)	R el (Ohm)	n
S 1	1 000 391	23 793	1.35 × 10^−12	1.02 × 10^−11	1 010 811	0.503
S 2	1 003 824	14 534	1.68 × 10^−12	1.02 × 10^−11	1 015 095	0.505
S 3	1 004 254	16 695	1.59 × 10^−12	1.03 × 10^−11	1 017 422	0.506
S 4	1 005 266	15 424	1.67 × 10^−12	1.02 × 10^−11	1 019 466	0.507
% Percentage difference compared to the true values
S 1	0.04	137.93	32.55	2.28	1.08	0.69
S 2	0.38	45.34	15.94	2.23	1.51	0.90
S 3	0.43	66.95	20.69	2.78	1.74	1.13
S 4	0.53	54.24	16.00	2.10	1.95	1.41
Standard deviation (normalised by the corresponding means)
S 1	1.25 × 10^−3	1.67 × 10^−1	6.80 × 10^−2	8.02 × 10^−3	9.11 × 10^−3	7.95 × 10^−3
S 2	1.03 × 10^−3	1.10 × 10^−1	5.83 × 10^−2	8.17 × 10^−3	5.93 × 10^−3	5.94 × 10^−3
S 3	1.82 × 10^−3	1.24 × 10^−1	6.48 × 10^−2	7.39 × 10^−3	9.22 × 10^−3	5.93 × 10^−3
S 4	1.28 × 10^−3	1.38 × 10^−1	5.78 × 10^−2	7.96 × 10^−3	9.21 × 10^−3	5.92 × 10^−3

Figure 4. Impedance spectrum with 4 outliers (indicated by crosses at log[ f (Hz)] = 0, 1.5, 1.75 and 2) and 10% random Gaussian noise. The continuous lines are noiseless spectra (same as Fig. 2).

Figure 5. Dependence of circuit parameters on the regularization parameter. The fitted spectrum is shown in figure 4. The correct values are denoted by the grey line.

Figure 6. Geometry of GIA dielectric sensor.

Figure 7. Real impedance spectrum evolution for the isothermal cure of RTM6 at 150°C.

Figure 8. Imaginary impedance spectrum evolution for the isothermal cure of RTM6 at 150°C.

Figure 9. Methodology for modelling consecutive spectra.

Table 8. Adaptive search space of the fitting parameters. Subscript p denotes the parameter values obtained from fitting the chronologically preceding spectrum.

	R ion, new (Ohm)	R dip, new (Ohm)	C dip, new (F)	C ind, new (F)	R el, new (Ohm)	n new
Mean	1.00 × R ion,  p	5.05 × R dip,  p	5.05 × C dip,  p	1.00 × C ind,  p	1.25 × R el,  p	1.00 × np
Range	0.99 × R ion,  p	4.95 × R dip,  p	4.95 × C dip,  p	0.25 × C ind,  p	0.75 × R el,  p	0.25 × np

Figure 10. Equivalent circuit parameters for the isothermal cure of RTM6 at 150°C. Black points show the fitting without regularization. Grey points show the fitting when the regularization term is added to the objective function.

Table

c	= Adjustable parameter for adaptive mutation
C dip	= Dipolar capacitance
C ind	= Induced capacitance
f	= Frequency
F	= Fitness function
F critical	= Fitness value above which the genetic algorithm terminates
G	= Number of generations
G max	= Number of maximum generation
n	= Constant Phase Element exponent
N	= Number of solution vectors
N e	= Number of solution vectors passing directly to the next generation
N r	= Number of solution vectors passing through crossover and mutation
par	= Parameter values
parexp	= Expected parameter values
p e	= Exchange probability
p m	= Mutation probability
P no	= Number of parameters
q	= Number of frequencies in a sweep
r dip	= Dipolar resistance
R el	= Constant Phase Element parameter
r ion	= Ionic resistance
S	= Objective function
t	= Time
w	= Weight factor
Z	= Complex impedance
Z′	= Real impedance
Z′′	= Imaginary impedance
λ	= Tikhonov regularization parameter
ω	= Angular frequency
τdip	= Dipolar relaxation time
Z CPE	= Complex impedance of the Constant Phase Element (CPE)