Multiobjective Pareto and GAs nonlinear optimization approach for flyback transformer
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Abstract
Design and optimization of highfrequency inductive components is a complex task because of the huge number of variables to manipulate, the strong interdependence and the interaction between variables, the nonlinear variation of some design variables as well as the problem nonlinearity. This paper proposes a multiobjective design methodology of a 200W flyback transformer in continuous conduction mode using genetic algorithms and Pareto optimality concept. The objective is to minimize loss, volume and cost of the transformer. Design variables such as the duty cycle, the winding configuration and the core shape, which have great effects on the former objectives but were neglected in previous works, are considered in this paper. The optimization is performed in discrete research space at different switching frequencies. In total, 24 magnetic materials, 6 core shapes and 2 winding configurations are considered in the database. Accurate volume and cost models are also developed to deal with the optimization in the discrete research space. The biobjective (loss–volume) and triobjective (loss–volume–cost) optimization results are presented, and the variations of the design variables are analyzed for the case of 60 kHz. An example of a design (30 kHz) is experimentally verified. The registered efficiency is 88% at full load.
Keywords
Genetic algorithms NSGAII Pareto front Flyback converter Transformer Core loss Winding lossList of symbols
 V_{in}, V_{out,}P_{in}, P_{out}
Input and output voltage and power
 P_{t}, P_{c}, P_{w}
Transformer, core and winding losses
 D, l_{g}, J
Duty cycle, air gap, current density
 V_{p}, V_{s}, I_{p}, I_{s}
Primary and secondary voltage and current
 N_{lp}, N_{ls}, N_{tp}, N_{ts}
Primary and secondary number of layers and number of turns per layer
 C_{c}, W_{c}
Core cost and winding configuration
 d_{p}, d_{s}
Primary and secondary diameters
 l_{c}, l_{m}
Mean core length and mean turn length
 I_{pmax}, I_{smax}
Primary and secondary maximum currents
 I_{pavg}, I_{savg}
Primary and secondary average currents
 I_{pr}, I_{sr}
Primary and secondary currents ripple
 R_{dcp}, R_{dcs}
Primary and secondary DC resistance
 F_{r}, Ac
Actodc resistance ratio, core section
 x
Diametertoskin depth ratio
 L_{p}, L_{s}
Primary and secondary inductance
 B_{ac}, B_{max}
Swing and maximum flux density
 N_{p}, N_{s}
Primary and secondary number of turns
 V_{tr}, V_{c}, V_{w}
Transformer, core and winding volumes
 µ_{0}, µ_{r}
Vacuum and relative permeability
 I_{prms}, I_{srms}
Primary and secondary RMS currents
 f, f_{u}
Frequency and winding filling factor
1 Introduction
Flyback converter is widely used in lowpower applications such as laptop and mobile chargers. It has several advantages compared to other topologies, especially for lowpower applications. On the other side, this topology presents one major inconvenient which is its limited power capability. The converter is only suitable for low power (< 150 W), and its efficiency degrades as the power increases. This is due to the conflict in the inductance design requirements. From one side, the inductance needs to be increased in order to allow more power storage capability during the ontime, and from the other side, it needs to be decreased because it is inversely proportional to the output power [1, 2, 3, 4, 5, 6]. Several parameters and interdependent variables contribute in the design of the flyback transformer. The conflict in the design of the transformer inductance makes the optimization process a complex task which needs advanced design and optimization techniques to improve its performance.
Classical design techniques of inductive components are mainly the area product method and the core geometry coefficient [6, 7]. Both design approaches do not consider the nonlinearity of the design variables or/and the nonlinearity of the complete problem. Moreover, the variables are very dependent on each other which make the problem impossible to be solved accurately using these methods. Some other techniques were presented to optimize the magnetic devices by determining the optimum magnetic flux density; however, they do not solve the nonlinearity issue [8, 9, 10]. The limitations of these methods are discussed more deeply in Sect. 2.1.
The consideration of more than one objective in the optimization transforms the problem to a multiobjective problem which cannot be solved by the previous presented techniques. Some methods like the weighted sum method and the Min/Max method were used for this kind of problems [11]. However, they can only deal with convex and continuous functions which is not the case of the inductive components design problem.
Genetic algorithms and Pareto optimality concept starts to be a common solution in the design of electromagnetic devices in the last decade [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22]. The main feature of the GAs is their capability to handle numerous optimization variables and objective functions without losing accuracy in the convergence to the optimum Pareto front. They are also very effective in solving problems with discontinuous variables, concave and convex, linear and nonlinear, constrained and nonconstrained objective functions. Additionally, GAs allow also reducing the design time and eventually the cost. All the previous advantages make of the GAs a good candidate for the multiobjective optimization of the inductive components.
From another side, the convergence to the optimum set of solutions doesn’t only depend on the performance of the optimization algorithms but more importantly on the problem formulation. This later includes the selection of the design variables, the design equations, the design constraints and the sequences between them. As an example, in some types of nonlinear problems, one variable or more can be at the same time an objective function and a design variable or it can also be a constraint. Few works were achieved to optimize the inductive components for power electronics such as [13] and [14]; however, no one of them addressed these issues. The goal of this paper is to deal with this problem. Particularly in this work, some crucial variables such as the duty cycle, which has a great effect on the transformer loss, the efficiency, the magnetic material and the winding configurations, are considered as main design variables. Finally, to improve the design objectives, reliable and accurate models of the objective functions, the design constraints and the intermediate design functions, such as core loss, winding loss and leakage inductance, are developed.
1.1 Research contributions
 1.
To develop a multiobjective optimization approach to design a flyback transformer in continuous conduction mode using GAs and Pareto optimality concept.
 2.
To solve the nonlinearity of the design problem resulting from the interdependence between the design variables, the objective functions and the design constraints.
 3.
To include the effect of some crucial variables in the optimization process such as the duty cycle and the efficiency which are neglected in the previous works.
 4.
To analyze the variation of the design and optimization variables with respect to the Pareto front for the biobjective (loss–volume) optimization and triobjective (loss–volume–cost) optimization problems in order to get a clear picture on the variation of these variables with respect to the objective functions.
 5.
To verify the optimization results of the transformer designed for 30 kHz by experiments and comparison with existing approaches.
 6.
To give an insightful picture on the stateoftheart on the design and optimization of inductive components.
2 Review of the highfrequency inductive components design and optimization methods
2.1 Challenges in the design of HF transformer
The major challenges faced today by designers of highfrequency transformers are the selection of: the appropriate magnetic material, the suitable core shape, the required core size and the winding configuration. All the former variables are generally chosen based on the designer experience to satisfy the design constraints and requirements which can reduce the design efficiency. The selection of the switching frequency and the duty cycle is defined arbitrarily. However, it was shown in the literature that the duty cycle presents a great effect on the core loss and the winding loss [22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32]. The frequency has also an important role in the transformer loss and volume. Thus, the consideration of these variables as main design and optimization variables is crucial. The determination of the number of turns, the calculation of the air gap and some other intermediate functions is derived from the relationships between the inductance reluctance model, the energy storage equation and Faraday’s and Ampere’s laws. These variables have nonlinear relationships between each other. Furthermore, the design of highfrequency transformers needs to take into consideration some constraints such as the leakage inductance and the temperature rise. In general, there is a huge interdependence and interaction between the design variables, the intermediate design functions, the design constraints and the objective functions which makes the problem very complex to be solved.
In a given design problem, one combination or more of the previous discussed design variables can be the optimum set of solutions for the objective functions such as the efficiency, the cost and the volume. In the following, the existing methods are analyzed and their limitations to solve the former issues are figured out. The existing methods can be classified into two categories: singleobjective methods and multiobjective methods.
2.2 Singleobjective methods
The area product method was early developed to select the suitable magnetic core for a given application [6, 7, 33]. It is a geometry dimensional factor of magnetic cores which has an equivalent term of electrical variables of the power converter. The equivalent term depends on the maximum magnetic flux density. The geometry term is the product of the core window area and the core cross section. In the literature, there are several forms of the area product. Some correction factors are adopted by industrials in order to meet the design requirements [34, 35, 36]. For example, author in [34, 35] has proposed that the volume of the flyback transformer is twice the one for forward. This correction factor is used because the flyback transformer needs to store energy during the ontime. In [2], a monogram of the area product as a function of the stored energy is given in [36], and the area product was expressed as a function of the magnetizing inductance to take into consideration the effect of the DC current and the stored energy.
The first drawback of the area product (AP) method is that it includes the filling factor which is an unknown and usually estimated to 0.3 or 0.4. Secondly, the product between the core section and the window area doesn’t give a real picture and an accurate characterization of the core shape. In other terms, the area product does not take into account the core shape effect on the design because two different core shapes can have the same AP. Furthermore, the area product method allows only getting an idea about the required core size; however, all the remaining variables, such as the material and the winding configuration, are chosen by the designer using his own experience. In addition to that, this method does not minimize the transformer loss because it doesn’t include the eddy currents effect at high frequency.
The core geometry coefficient method is also a geometry dimensional factor. The expression of the core geometry coefficient is derived from the formula of the optimum magnetic flux density that yields to minimum transformer/inductance loss. It includes the core length in addition to the core window and the core cross section which can improve the accuracy in the selection of the suitable magnetic core. It also includes the Steinmetz parameters, characterizing the core loss of the magnetic materials. This feature offers the possibility to consider the effect of the magnetic material loss in the design process, and therefore the selection of the best material becomes achievable [37].
Although the core geometry coefficient offers better accuracy in the selection of the suitable magnetic core than the area product, it still presents major limitations which are summarized as follows: (1) the method is a singleobjective design approach used to improve the efficiency and not the cost or the volume, (2) the method is only suitable at fixed frequency and fixed duty cycle, and (3) the method needs an iterative approach to include the effect of the core shape because magnetic core with different shapes might have the same core geometry.
Hurley developed an improved expression of the equivalent term of the area product to include highfrequency effects such as the skin and the proximity effects as well as the temperature effect. This was achieved by calculating the optimum magnetic flux density instead of using the maximum magnetic flux density [9, 10].
A similar approach based on the determination of the optimum magnetic flux density was presented by Petkov but without consideration of the highfrequency effects [8].
In [38], the weighted efficiency method was applied to optimize the flyback inverter, including the transformer component, for photovoltaic applications. The optimization of the weighted function was solved by the differential evolution method.
2.3 Multiobjective methods
In the former design approaches, the efficiency is the only objective to deal with as it is the major concern of the designer. In the last decade, designers are much more interested to optimize, not only the converter efficiency, but also the cost and the volume in order to meet the standards and clients requirements. This is called a multiobjective optimization problem. Genetic algorithms are powerful optimization methods to solve this kind of problems. Pareto front is also one insightful and effective tool to represent the optimum solution in the objectives space or variables space. Using Pareto optimality concept, the decision maker can move along the Pareto front to select one of the optimal solutions that meets the required needs. It is also possible to know the evolution of the optimization variables and any intermediate function with respect to Pareto front. Hence, the effect of every variable can be well understood and analyzed. In fact, using GAs and Pareto concept of optimality, we can get a clear picture of the full problem in the objective space and the intrinsic variation of the design variables can be seen at any optimal operating point [11, 19, 20, 21]. In [12], GAs was used to achieve the best tradeoff between volume and loss of PWM inverter output filters. In [13], the loss and weight optimization of medium frequency transformers was performed using genetic algorithms (NSGAII). The main limitation of this work is that the capability of the GAs to solve nonlinear problems is not fully utilized. As an example, in [13], the value of the required inductance was initialized in the design of LLC and DAB converter. GAs was applied to determine the optimum set of the variables given by the inductance reluctance model independently of the converter operation constraints. However, to be more accurate, the value of the required inductance depends on several other variables, depending on the converter topology, such as the duty cycle, the frequency and the efficiency. In a similar way in [14], the magnetizing inductance was considered as a fixed parameter for LLC converter which is not enough accurate design approach.
Generally speaking, in the previous discussed works, the dependency of required inductance on the converter requirements (efficiency, duty cycle, frequency, etc.) was neglected and this can reduce the design accuracy. Another important issue which was not clearly explained and formulated in previous work is the relationship between the different optimization variables, the objective functions and the design constraints which all contribute in the nonlinearity of the design problem. Additionally, the models of the objective functions present lack of accuracy such as the negligence of the temperature effect on the core loss.
In [15], two kinds of evolutionary algorithms (GA and PSO) have been tested to optimize the volume and the mass of EI and UI core inductor, respectively. A continuation of the work presented in [15] is exhibited in [16]. The single optimization (volume) and the biobjective optimization (loss vs. volume and loss vs. cost) of the EI inductor are performed. The triobjective optimization is also presented by optimizing the cost and the volume constrained to the objective function loss. The singleobjective optimization is presented with continuous and discrete variables. GOSET optimization tool, developed by authors in [17, 18], was used to achieve the optimization process. It was shown that GOSET tool allows achieving better minimization with discrete variables. The structure of GOSET is similar to NSGAII but with more optional function that improves its performance. In [19], a multiobjective optimization approach was presented to minimize the losses, the weight and the volume of power inductors for threephase high power density inverter.
2.4 200W flyback converter design complexity
Designing a 200W flyback converter is very complicated due to the reasons discussed previously. In the literature, there are few attempts done to realize a flyback converter at this power level. The only industrial 200W flyback converter was designed by Texas Instruments. The adopted solution consists in using two interleaved transformers with an inductance of 500 µH each, and the switching frequency is 100 kHz. The value of the inductance reflects the significant volume of the used transformers [39]. Another company developed a 140W flyback converter, but it can supply a maximum power of 200 W [40]. The inductance of the transformer is 300 µH. Another study in [41] showed the design of a 180W flyback converter using a regenerative snubber circuit concept. The magnetizing inductance is 102 µH, and the switching frequency is 80 kHz.
In all the previous designs, the use of a big inductance is necessary to reach the 200W output. In the following, GAs and Pareto front optimality concept is used to optimize the 200W flyback transformer.
3 Design approach
3.1 Problem description
Converter characteristics
Output voltage  120 V 
Input voltage  48 V 
Output power  200 W 
Maximum temperature rise  60 °C 
Frequency  10–20–30–40–50–60 kHz 
Several variables and constraints contribute in the design of flyback transformer. As per our knowledge, still there is no study solving the optimization of this magnetic problem with consideration of all variables due to the issues discussed in Sect. 2. Using the advantages of GAs, the optimization of the flyback transformer with consideration of all the design variables becomes possible. All the data of the magnetic cores and magnetic materials are summarized in Tables 5, 6 and 7.
3.2 Core loss prediction
3.3 Winding loss
3.4 Volume modeling
Similarly, the secondary winding is calculated.
3.5 Transformer cost modeling
Parameter k represents the geometry cost effect, and a, b and c are parameters depending on the magnetic material.
3.6 Problem formulation
The objective of this work is to minimize the loss, the volume and the cost of the flyback transformer. Their models are given in the previous subsections. In order to achieve an accurate design, a good understanding of the design equations as well as the dependency and interaction between the optimization variables, the design constraints and objective functions is mandatory.
Design space for the discrete research spaces
Variable  Unit  Type  Interval 

Magnetic material  U  Integer  [1–24] 
Duty cycle  –  Integer  [1–72] 
Core reference  U  Integer  [1–n] 
W _{c}  U  Integer  [1–2] 
N _{p}  U  Integer  [1–100] 
N _{s}  U  Integer  [1–100] 
d _{p}  U  Integer  [1–60] 
d _{s}  U  Integer  [1–60] 
l _{g}  M  Real  [1e−4, 4e−3] 
η _{i}  –  Real  [0.95–0.99] 
3.6.1 Design equations of the transformer
3.6.2 Design constraints
 The core saturation, the gain and the air gap are as follows:$$ B_{\rm{max} } \le 0.7 B_{\text{s}} $$(14)$$ {\text{Gain}} = \frac{{N_{\text{s}} D}}{{N_{\text{p}} \left( {1  D} \right)}} $$(15)$$ l_{\text{g}} = \frac{{\mu_{0} I_{\text{pmax}}^{2} L_{\text{p}} }}{{A_{\text{s}} B_{\rm{max} }^{2} }}  \frac{{l_{\text{c}} }}{{\mu_{\text{r}} }} $$(16)
 Conductor diameters: They are determined by [31].$$ d_{\text{p}} = \sqrt {\frac{{ I_{\text{pmax}} }}{7.2}} ;\quad d_{\text{s}} = \sqrt {\frac{{I_{\text{smax}} }}{7.2}} $$(17)
 The winding area A_{w} should be capable of allocating the two windings. f_{u} is equal to 0.4.$$ A_{\text{w}} \ge \frac{{\pi \left( {N_{\text{p}} d_{\text{p}}^{2} + N_{\text{s}} d_{\text{s}}^{2} } \right)}}{{f_{\text{u}} }} $$(18)
 Temperature rise T_{r}:$$ T_{\text{r}} = \frac{{53 \left( {P_{\text{c}} + P_{\text{w}} } \right)}}{{V_{\text{c}}^{0.53} }} \le 60 \,^\circ {\text{C}} $$(19)
 Currents ripple factor: it should be lower than r_{max}.$$ I_{\text{pr}} \le r_{\rm{max} } ;\quad I_{\text{sr}} \le r_{\rm{max} } I_{\text{savg}} $$(20)

Peak MOSFET voltage: the maximum allowable voltage across the MOSFET time must be lower than 400 V.

Leakage inductance: it is kept lower than 2% of the magnetizing inductance. Model presented in [42] is used.
3.6.3 Optimization variables
The optimization variables are highlighted in Table 2. They are of two types: integer and real.
4 Multiobjective optimization in discrete research space
4.1 Biobjective optimization
Details of the design solutions
30 kHz  40 kHz  50 kHz  60 kHz  

Losses (W)  4.88  4.37  4.02  3.66 
Cost ($)  5.68  3.9764  3.96  3.32 
Volume (cm^{3})  63.41  44.11  44.12  42.75 
Material/W_{c}  F/2  F/2  F/2  F/2 
Shape/Ref  ETD/7  ETD/6  ETD/6  ETD/6 
d_{p} (mm)  1.82  1.8  1.82  2.05 
d_{s} (mm)  0.91  1.02  1.02  0.91 
N_{p}/N_{s}  23/47  24/27  17/32  9/37 
l_{g} (mm)  1  0.7  0.6  0.5 
5 Comparison with classical area product method and experimental verification
The optimum magnetic material that is selected by the optimization algorithm is F material provided by Magnetics. But, at the time of the experimental implementation, this selected core was not available in the market. For that reason, the optimization process was run a second time and the results show that R material is the new optimum one. No changes were registered for other variables (core reference, number of turns, duty cycle, etc.).
5.1 Comparison with the area product methods
The objective of this section is to compare the outputs of the design case 30 kHz with the outputs of two singleobjective design approaches in order to verify the reliability and the accuracy of our results and to get an approximated idea about the efficiency of the design.
Comparison of the design outputs between the proposed approach and the classical techniques (30 kHz design case)
Proposed approach  Lloyd approaches [36]  Sanjaya approach [34]  

1st approach  2nd approach  
Primary inductance (μH)  233  216.45  216.45  
Maximum swing flux (T)  0.1  0.1  0.1  
Area product (cm^{4})  –  17.78  10.24  2.09 
Selected core  ETD59  EE 65  ETD59  ETD49 
Selected material  F  F  F  
Turns ratio  0.48  0.4  0.4  
Primary/secondary number of turns  23/47  15/38  22/55  38/95 
Primary/secondary diameter (mm)  1.82/1.45  1.28/0.77  1.28/0.77  
Primary/secondary number of layers  1/1  1/1  2/2  
Primary/secondary resistance (mΩ)  81.12/327.52  76.54/396  112.25/574  663/2100 
Winding loss (W)  4.72  5.44  7.94  39 
Core loss (W)  0.17  0.27  0.17  0.08 
Transformer loss (W)  4.88  5.71  8.11  39.08 
It is clear from Table 4 that the proposed approach yields to better theoretical efficiency than other design techniques. However, Sanjaya approach leads to the worst efficiency resulting from the high number of winding layers. The second approach of Lloyd allows getting the same magnetic core as the one selected by the proposed approach; however, the transformer loss is bigger because of the reduced size of the winding conductors. The first approach of Lloyd has comparable efficiency to our design but with bigger magnetic core.
In comparison with the design solutions given in Sect. 2.3, the inductance of the designed transformer is 233 µH at 30 kHz which is much lower than the one designed by Texas and the one proposed by ON Semiconductor.
5.2 Experimental results
An IGBT STGFW30V60F is used, and it is controlled using a driver TC4429. A capacitor of 63 V 2200uF is used in the input, and a 400 V 47 μF for the output filter. The rectifier diode is Mur1560. The sensing resistors in the input and output are 0.1 Ω, 5 W.
The obtained waveforms of the input and output currents, the IGBT voltage and the diode voltage are shown in Fig. 8. The efficiency reaches up to 88% at full load. This efficiency is very acceptable for a 200W flyback converter operating at 30 kHz. The obtained results can be improved by better selection of the switching devices. However, we are limited to use only devices available in the laboratory because it is not the objective of this work to optimize the full converter.
6 Conclusion

The approach solves the limitations of the existing works by considering the effect of the duty cycle and the efficiency as main optimization variables. The approach takes also into account the nonlinear relationships between variables, objective functions and design constraints.

The proposed approach allows better minimization of the loss and volume in comparison with existing classical methods.

The approach is very effective as it improves the accuracy of the results compared to other techniques, and it reduces the design time. The optimization needs 30 min to get the optimum Pareto fronts which can take longer time using classical techniques.

Results show that the optimal magnetic material and optimal core shape depend on the switching frequency.

Pareto fronts of the objective functions (loss, cost and volume) are inversely proportional to the switching frequency.
Notes
Acknowledgements
Open access funding provided by Mid Sweden University.
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