Application of the Firefly Algorithm for Optimizing a Single-switch Class E ZVS Voltage-Source Inverter's Operating Point.

I. INTRODUCTION

High current circuits of modern electric equipment contain power electronic units, which require precise control, especially at high operational frequencies. This is also the case of induction heating appliances, to which belongs the single-switch class E voltage-source quasiresonant inverter shown in Figure 1.

The converter presented in the paper is a class E inverter. It is described in literature, especially in connection with its applications in induction cookers operating usually in the range of 20 kHz / 50 kHz [1-8].

A much higher frequency is often necessary in induction heaters and furnaces to achieve the expected results of heating (e.g. surface heating), or as a result of technological requirements. However, the converter's switching frequency is limited from above, owing to its commutation capabilities. Therefore, the issue of minimizing the switching losses in the converter is of crucial importance, especially in high-frequency units . The references [10-12] discuss the inverter in Figure 1 operating at frequencies of several hundred kilohertz.

An analysis of the inverter circuits and optimizing the operation point using the classical Hook-Jeeves method is described in . Under optimum conditions, measurements were made for several types of charge (magnetic or nonmagnetic rods of various diameters). The results of measurements were used to investigate the effect of the charge and the system capacitor on the inverter maximum voltage, current and power, and finally the system losses . Through the optimization of the inverter's operating point, the switching losses are minimized. The paper  analyzed energy the efficiency of the system and discussed the sources of losses. The effect of the system capacitance on the total electrical efficiency was also investigated.

Determining the optimal operating conditions of the inverter requires the solution of two nonlinear equations, which generate a complex surface search with many local minima. In , the optimum operating conditions of the inverter was assigned using the Hook-Jeeves optimization method. Unfortunately, the effectiveness of this optimization method is strongly dependent on the choice of the starting point. An improperly selected starting point means that the algorithm gets stuck in a local minimum.

For this reason, it was decided to use one of the modern methods of optimization, one of the artificial intelligence family methods, which is increasingly being used in practical applications, such as the firefly algorithm , cuckoo search , ant colony optimization , genetic algorithm , particle swarm optimization [17-19], crow search algorithm , Simulated Annealing [18-19], Gravitational Search Algorithm , Harmony Search Algorithm , Neural Network  and others.

All these methods were created by the observation of nature. Nature often inspires us. We try to imitate what evolution has created.

The firefly's algorithm belongs to the family of intelligent swarm methods that can be used to solve non-linear equations, becoming independent of the choice of the starting point.

[mathematical expression not reproducible] (1)

[mathematical expression not reproducible] (2)

[mathematical expression not reproducible] (3)

where

[mathematical expression not reproducible] (4)

Thus, the main contribution of this paper is presenting the real application of a modified firefly algorithm to solving nonlinear equations with very satisfying results. The algorithm's modification makes it possible to speed up the optimization and gain certainty of finding the global optimum.

In this paper, the algorithm has been applied to determine the optimal operating conditions of the inverter in Figure 1, and the results were confirmed by simulation and verified experimentally. However, the field of applications of the presented algorithm is very wide, and it can be used for solving a number of theoretical and practical problems.

The paper is organized as follows. Sections I and II present an introduction and basic operation aspects of the inverter that is the object of the optimization.

In Section III, the firefly algorithm is presented. Section IV shows the results of the optimization of the inverter in two operating points, using the modified firefly algorithm. Section V contains simulation and experimental verification of the results obtained in Section IV. The most important conclusions are presented in Section VI.

II. OPERATION OF THE INVERTER

Each switching cycle [T.sub.S] in steady-state operation of the inverter can be divided into two time intervals (Figure 2). During the time interval [T.sub.1] (mode I), the transistor is on, and the current [i.sub.S] rises exponentially. The voltage across capacitor C remains constant, being equal to [U.sub.d].

During the time interval of [T.sub.2] (mode II), the transistor is off. Current [i.sub.S] does not flow and capacitor C discharges in the series-resonant circuit [L.sub.0]C[R.sub.0]. This mode lasts until [u.sub.C] reaches the supply voltage [U.sub.d] again at zero current [i.sub.0].

[mathematical expression not reproducible] (6)

[mathematical expression not reproducible] (7)

The sum of times [T.sub.1] and [T.sub.2] gives the duration of the switching cycle [T.sub.s].

At optimum operating point, the transistor is turned on at zero voltage across and zero current; therefore, the turn-on losses in the transistor are zero. Moreover, it is turned off at zero voltage; hence, the turn-off losses are low. This means, that optimum operation guarantees minimizing of power losses and achieving maximum efficiency. Other operating ways (non-optimum operation, sub-optimum operation) are presented in , .

The mathematical analysis of the inverter (Figure 1) is based on the following assumptions:

* switch T and all the other circuit components are ideal,

* supply voltage [U.sub.d] is constant in one switching cycle,

* inverter operates in steady-state with soft switching.

The current and voltage waveforms are determined by the equations (1) - (3) . Substituting the requirements for optimum operation (5)

[mathematical expression not reproducible] (5)

into (1) - (3), we obtain a system of equations (6) and (7). The quality factor Q (4) of the series circuit L0CR0 is a parameter in the equations (6) and (7). The decision variables are normalized transistor conduction time (duty cycle) D (4) and normalized switching angular frequency [[omega].sub.sn] (4). Both variables are included in the interval (0, 1).

The objective of the optimization is calculating these variables (D and [[omega].sub.sn]) using (8), which determines the optimum operating point of the inverter, with minimum switching losses.

y(D, [[omega].sub.sn]) = |[f.sub.1](D, [[omega].sub.sn])| + [f.sub.2](D,[[omega].sub.sn]) (8)

The inverter in Figure 1 was investigated thoroughly with various loads and at different operating frequencies, and its circuit parameters were calculated based on the measurements results. Two cases are considered in this paper:

a) [L.sub.0] = 0.293 [micro]H, C = 0.235 [micro]F, [R.sub.0] = 87.4 m[OMEGA], yielding Q = 12.78 and [f.sub.0] = 606.07 kHz, (case 1),

b) [L.sub.0] = 0.432 [micro]H, C = 0.235 [micro]F, [R.sub.0] = 352.88 m[OMEGA], yielding Q = 3.84 and [f.sub.0] = 495.26 kHz, (case 2).

For both considered cases, equation (8) is minimised numerically in Matlab with the aid of fminsearch function using the Nelder-Mead algorithm.

Finding a solution of equation (8) is heavily dependent on the starting point of the search. Having knowledge of the shape of the surface that is searched (Figure 3), we can point to a starting point that will find the correct solution.

It should be noted, however, that the shape of the surface depends on the quality factor Q and requires imaging to select a potentially good starting point.

Using the procedure described above (starting point D = 0.1, cosn = 0.99) the following solutions were found:

a) D = 0.125840636, [[omega].sub.sn] = 0.97998184, y = 79.88e-6, [[omega].sub.s] = 3731792 rad/s, [f.sub.s] = 593,933 kHz, [T.sub.1] = 0,211877 [micro]s, [T.sub.2] = 1.471814 [micro]s

b) D = 0.322942367, [[omega].sub.sn] = 0.83198313, y = 122.03e-6, [[omega].sub.s] = 2588987 rad/s, [f.sub.s] = 412,050 kHz, [T.sub.1] = 0.783745 [micro]s, [Tsub.2] = 1.643144 [micro]s

where [f.sub.S] = [[omega].sub.s]/(2[PI]) is switching frequency.

III. FIREFLY ALGORITHM

The firefly algorithm was described by Xin-She Yang in 2008  and later, in 2010 . The algorithm is based on the assumption that each firefly represents a potential solution of the problem and illuminates proportionally to its quality. The better the solution, the brighter the firefly light that attracts a larger number of fireflies. Thanks to this, the area with better fireflies is intensively searched by a larger number of fireflies.

The firefly algorithm is based on the following concept:

* all fireflies are themselves attractive,

* the attractiveness of a firefly is proportional to its brightness, which means that less bright fireflies will fly in the direction of fireflies glowing brighter,

* the attractiveness of the firefly decreases with increasing distance,

* the brightness of firefly light affects the landscape that is the objective function to be optimized.

There are two important concepts in the firefly algorithm: the brightness of the firefly light and its attractiveness. It can be concluded that the firefly illuminates proportionally to the quality of the proposed solution. The attractiveness of firefly A to firefly B is dependent on the:

* brightness of both fireflies (the firefly which illuminates brighter is more attractive),

* light absorption rate [gamma],

* distance between fireflies r.

[beta] = [[beta].sub.0] x exp(-[gamma] x [r.sup.m]) (9)

where:

[[beta].sub.0] - attraction coefficient base value,

[gamma]- light absorption coefficient,

r - distance of fireflies A and B,

m - distance exponent.

The equation (9) shows that the attraction of firefly decreases exponentially with the m power of the distance r, which is defined by the formula (10)

[mathematical expression not reproducible] (10)

where k is the number of design variables, and the vectors [x.sub.A] and [x.sub.B] are sets of values of these variables for fireflies A and B, respectively. The firefly's A movement towards a more attractive firefly B is determined by equation (11).

[mathematical expression not reproducible] (11)

where:

[x.sub.A] - the position of the firefly A,

[x.sub.Anew] - potential new position of the firefly A,

[x.sub.B] - a better position of the firefly in the direction to which firefly A is shifted, rand(a, b) - a random value within the range (a, b),

[alpha] - mutation coefficient,

max, min - the maximum and minimum values of design variables.

The most important element of the firefly algorithm is to shift the fireflies. In the classical algorithm, potential shifts are generated in directions of all better fireflies, according to equation (11). Among those potential shifts generated, the one defining the greatest improvement in the quality of the proposed solution is selected by the shifting firefly.

Figure 4 shows a diagram of the firefly algorithm.

In this study, a modification of the algorithm is applied, by moving the firefly in all directions set by the better fireflies, if such a shift causes improvement of the represented firefly solution. In the classical algorithm, the firefly is moved only towards one firefly causing the greatest improvement of the objective function. This modification accelerates the search for a solution. Each firefly in the population undergoes such a procedure.

All the optimizations have been performed with the following configuration:

* range of variation of decision variables: D, [[omega].sub.sn] = (0 - 1),

* number of fireflies - 20,

* number of iterations - 100,

* attraction coefficient base value [[beta].sub.0] = 2,

* light absorption coefficient [gamma]= 1,

* exponent distance m = 2,

* mutation coefficient [alpha] = 0.01,

* mutation coefficient damping ratio [[alpha].sub.damp] = 0.98.

In the formula (11), there is a rate [alpha] present responsible for part of the random firefly shift. This rate decreases with each iteration of the algorithm by multiplying the rate of the previous iteration and parameter [[alpha].sub.damp].

IV. RESULTS OBTAINED

To properly control the inverter (Fig. 1), the switching frequency [f.sub.s] and transistor conduction time [T.sub.1] have to be known. They have been computed, using a firefly algorithm, based on decision variables [[omega].sub.sn] and D (4), obtained as a result of minimizing variable y in equation (8) for both considered cases of the inverter's circuit parameters (12).

[mathematical expression not reproducible] (12)

where:

[D, [[omega].sub.sn]]* - optimal values of decision variables, with constraints: 0 < D < 1, 0 <[[omega].sub.sn] < 1.

The optimization algorithm shifts the fireflies towards the descending objective function. The location of the firefly x represents the potential solution to the problem posed and the value of the objective function y represents its quality. In this article, the position of the firefly x is represented as a two-element vector x = [D, [[omega].sub.sn]]. The position of the firefly changes according to the dependence (11).

The following results were obtained:

a) D = 0.125863124, [[omega].sub.sn] = 0.98001186, y = 0.945e-6, [[omega].sub.sn] = 3731906 rad/s, [f.sub.s] = 593,951 kHz, [T.sub.1] = 0,211908 [micro]s, [T.sub.2] = 1.471731 [micro]s,

b) D = 0.322946189, [[omega].sub.sn] = 0.83195220, y = 0.823e-6, [[omega].sub.s] = 2588890 rad/s, [f.sub.s] = 412,035 kHz, [T.sub.1] = 0.783784 [micro]s, [T.sub.2] = 1.643196 [micro]s.

Figure 5 shows the process of optimizing using firefly algorithm. Already, around the thirtieth iteration of the algorithm, a solution was found similar to that found by fminsearch function. Repeatedly, the firefly algorithm was run, and each time it found similar solutions. It should be emphasized that the fminsearch function found a solution only at a correctly selected starting point, which depends on the parameter Q.

The calculation time of the algorithm is worth noting. Fireflies needed 2.86 seconds to determine the solution. The calculations were performed on a computer processor i76700HQ, 8GB DDR3 RAM, HD SSD, Windows 10 64-bit, Matlab 2016a.

In the firefly algorithm, apart from the number of iterations and number of fireflies, there are other parameters that can affect the rate of finding a solution, i.e: [gamma], [beta]0, [alpha], adamp and m. During tests with different values of these parameters, there was no significant influence of the parameters [alpha], [beta]0 and m on the speed of the search for solutions. Parameters [alpha] and especially adamp, are very influential on the speed of searching. They allow you to quickly find the precise solution.

Figure 6 shows the effect of the adamp factor on the speed of finding a solution. For each [[alpha].sub.damp] value from 0.01 to 1, optimization was performed 50 times. The figure shows the best solution found. The solution for [[alpha].sub.damp] = 0.01 is found the fastest. Unfortunately, from [[alpha].sub.damp] = 0.78 upwards, decreasing this value increases the number of cases of not finding the optimal solution, i.e. the value of the objective function y does not drop below [10.sup.-15].

The following solutions were found:

a) D = 0.125862934, [[omega].sub.sn] = 0.98001227, y = 2.29e-16, [[omega].sub.s] = 3731907 rad/s, [f.sub.s] = 593,951 kHz, [T.sub.1] = 0,211908 [micro]s, [T.sub.2] = 1.471731 [micro]s,

b) D = 0.322946467, [[omega].sub.sn] = 0.83195196, y = 1.11e-16, [[omega].sub.s] = 2588890 rad/s, [f.sub.s] = 412,035 kHz, [T.sub.1] = 0.783785 [micro]s, [T.sub.2] = 1.643196 [micro]s.

From the point of view of practical implementation, the improvement of the accuracy of system parameters is negligible. Values have been improved around 1.8.[10.sup.-2]%.

However, from a theoretical point of view, a more accurate solution was found and in a shorter time by adjusting one of the parameters of the algorithm.

Figure 7 shows the optimizing process for a changed value of [[alpha].sub.damp] = 0.5.

Figure 8 shows the normalized switching angular frequency [[omega].sub.sn] and duty cycle D as a function of quality factor Q for optimum operation. The results have been obtained using the firefly algorithm. The graphs are limited by the minimum value of quality factor [Q.sub.min], above which the optimum operation is possible. [Q.sub.min] is equal to 2.606, which, for the assumed values of [L.sub.0] and C corresponds to maximum resistance [R.sub.0max] = 0.4285 [OMEGA] for case 1, and [R.sub.0max] = 0.5203 [OMEGA] for case 2. For small values of Q, small variations of [R.sub.0] and [L.sub.0] cause large changes of parameters [[omega].sub.sn] and D.

V. VERIFICATION OF THE OBTAINED RESULTS

The main goal of the authors was to develop a theoretical model of the inverter presented in Sections I and II. The firefly algorithm was used to solve the required set of nonlinear equations. This section contains results of simulations, which has been verified by experimental measurements on the basis of the self-developed real model of the inverter (figure 9).

Figure 10 depicts the theoretical waveforms for optimum operation, computed based on the equations (1) - (3) and optimum parameters marked in Figure 8, for the above presented cases (case 1 and 2) and for different quality factors.

The waveforms confirm the correctness of the calculation results. For example, the voltage across the transistor reaches zero before it is turned on again, which is required for optimum operation. Moreover, a significant load current distortion in the circuit of lower quality factor (higher damping effect in the resonant circuit) is visible.

Figure 11 shows current and voltage waveforms registered in a real inverter for induction heating. A 10-wire inductor was used and the load was as follows: a 10 mm diameter non-magnetic rod and a 4 mm diameter magnetic rod. An original control system was used, able to choose and maintain the optimal operating point for varying load parameters.

The current and voltage waveforms registered in Figure 11 are comparable to those obtained by simulation (Figure 10). The transistor is turned on at zero current and practically zero voltage, and it turns off at relatively low voltage. Therefore, the switching losses are relatively low.

However, an influence of non-modeled parasitic impedances in the circuit is clearly visible. The voltage peak across the transistor after its turn-off, which results from the impact of parasitic inductances in the circuit, is especially remarkable.

VI. CONCLUSIONS

The discussed inverter can operate optimally in class E with a much higher frequency than that presented in most articles. The optimum operation of the system ensures minimum power losses in the power-electronic switch and the highest efficiency of the inverter.

An advantage of the inverter is using only one power-electronic switch, and its disadvantage is the high voltage stress of the switching. However, the application of a new generation switch, like e.g. a SiC transistor, makes this disadvantage less significant.

Determination of the optimal operating conditions of this inverter requires the solution of two nonlinear equations. The effectiveness of classical optimization methods, e.g. the Nelder-Mead or Hooke-Jeeves methods are strongly dependent on the choice of the starting point. Classical optimization methods may get stuck in a local extremum. The firefly algorithm allows the search of a larger area in the search of the global extremum.

In this article, a modified firefly algorithm was used to solve a practical problem, which is the main contribution of the paper. Modifying the algorithm consisted in moving fireflies in all directions for improving the solution instead of choosing only one of the best shifts in each iteration of the algorithm. This allowed to speed up finding solutions.

Moreover, the effect of parameters on the firefly algorithms was checked against the speed of the search for solutions. For an example of this inverter circuit, parameters [gamma], [[beta].sub.0] and m change did not have a great effect. Conversely, a change in the parameter [alpha] and particularly parameter [[alpha].sub.damp] significantly affected the rate of finding solutions and their accuracy. The firefly algorithm proved to be a very effective and accurate optimization method, especially in the case of obtaining of characteristics D = f(Q ) and [[omega].sub.sn] = f(Q).

The additional resistance in the branch with the power electronic switch has not been taken into account. This resistance has a significant impact on the optimum operation parameters, especially for small values of quality factor Q. This issue will be the object of future analysis and research.

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Ryszard KLEMPKA, Zbigniew WARADZYN, Aleksander SKALA

Faculty of Electrical Engineering, Automatics, Computer Science and Biomedical Engineering, AGH--University of Science and Technology, Krakow, Poland klempka@agh.edu.pl

This paper was supported by the statutory research of AGH University of Science and Technology (no. 11.11.120.353).

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