Optimization of an On-Grid Inverter for PV Applications Using Genetic Algorithms

A new approach to the optimal design of power inverters for on-grid photovoltaic systems that uses genetic algorithms (GA) is provided in this article. )e nonlinear average model is adopted to model the conversion stage in order to accurately evaluate and quickly estimate the power losses of the power devices. )e hysteresis current control that guarantees a quasi-sinusoidal output current is applied to generate the inverter control signals.)e design of the solar inverter must meet three contradictory objectives that need to be optimized at the same time. In fact, the aim is to maximize the efficiency of the converter while minimizing its size and price under electrical constraints. )e problem variables are the output current ripple and the passive and active components available on the market (IGBTs/MOSFETs, Diodes, Inductors). NSGA-II (Elitist Nondominated Sorting Genetic Algorithm) is appropriate in the case where discrete design variables are used to search for optimal Pareto solutions. It carries out a systematic and efficient search among the developed databases for a set of components which define the optimal structures of the inverter. )e introduced method makes the design task easier since the best solutions depend on the components available on the market and significantly reduces the time to market for manufacturers.


Introduction
According to the BP Statistical Review of World Energy [1], global energy demand has increased by 2.9% in 2018, nearly twice the average annual growth in demand over the past decade (+1.5%/year) and the fastest since 2010. At this unprecedented increase, a strong growth in the integration of renewable energy sources into power plants worldwide has been recorded [2][3][4][5][6][7][8][9].
us, progressive advances of renewable energy power generation systems have led to the search for solutions to improve the performances of these systems, mainly by taking advantage of the enormous developments in the power semiconductor industry. In fact, power electronics are at the core of power generation systems since they provide the technology between sources and loads that converts energy from its continuous form into alternative or inverse [10]. Recently, photovoltaic (PV) electrical power generation has become more and more visible because it is a very attractive way to produce energy for several reasons, especially for its simplicity and modularity. It is almost entirely in solid form from the PV cell to the generated electricity. Whether it is a power plant with less than 1 W or a 100 MW grid-connected one, all that is needed between the solar system and the load are electronic and electrical components. Compared to other energy sources that humanity has used to produce electricity for a long time, PV is the most flexible and modular. Large PV systems require more electric buses, fuses, and wiring, but the most crucial component between the solar panel and the load is the electronic component that converts and processes electricity: it is the inverter [11,12].
One approach widely adopted for improving the entire grid-connected PV system performances is upgrading the converter by acting on its own design or on its control strategy or on both by benefiting from optimization algorithms as efficient and commonly used tools to obtain the best possible converter [13][14][15][16]. e design, control, and performance problems encountered in the field of power electronics generally involve the simultaneous optimization of multiple objectives which are often in contradiction (e.g., the simultaneous minimization of the power losses and the design cost or volume). In order to resolve conflicts, these cases can be formulated as multiobjective optimization problems.
is paper describes a methodology for designing gridconnected PV inverters that optimizes efficiency and/or volume and/or price, taking into account the quality of output current and the maximum switching frequency of active devices and based on commercially available components. is approach simplifies the implementation phase of the optimal solution since it is based on what we have on the market. e next part highlights the modeling process of the studied system, shown in Figure 1, by using the nonlinear averaged model with discrete semiconductor devices and the hysteresis current control. In the third part, the nondominated sorting genetic algorithm NSGA-II as well as the method used to determine the optimal solution is investigated. e fourth part presents the process used to manipulate the genetic algorithm with discrete variables and defines the design parameters, the constraints, and the fitness of optimization problems. en, optimization results and interpretations are provided and finally, conclusions are given.

Nonlinear Average Model with Discrete Semiconductor
Devices. Modeling and dynamic simulation are essential tools to analyze, design, control, and optimize a power electronics structure. Averaging techniques are well used to model power converters since it ensures both accuracy and rapidity, two criteria that must always be met while modeling power systems especially when dealing with multiobjective optimization problems. e average model of power electronic devices is a simplified presentation of a switching cell that is common in several converter topologies [17,18]. Nonlinear effects of the power semiconductor devices are not included in most of averaged models because they use ideal switches instead of semiconductor device models. Considering the unsatisfactory situation in the averaged modeling, an effort was undertaken in the averaged models presented in [19][20][21]. us, the authors have proposed an advanced PWM-switch model including semiconductor device nonlinearities. Figure 2(a) presents the inverter leg to be modeled with two complementary transistors (IGBTs or MOSFETs) controlled by external gate signals and two freewheeling diodes. In Figure 2(b), the equivalent representation of power converter leg using averaged model is illustrated. In fact, upper switches are replaced by a controlled voltage source V 1 while lower ones are substituted by a controlled current source I 1 given by V 1 � 〈U as 〉, (1) e main advantage of the proposed nonideal average model is to supply accurate losses values with acceptable simulation costs. e power losses of semiconductors (P transistor and P diode ) comprising conduction and switching losses and taking into account the various conduction and switching times are derived from switching characteristics of power devices presented in Figure 3 and can be given by [20,21] With X � 0 if the transistor is a MOSFET 1 if the transistor is an IGBT T s � 1/f s is the switching frequency. To perform average model simulations with discrete semiconductor devices, it is crucial to determine the static and dynamic characteristics of each component. For that, the determination of the model parameters is done using the technical data sheets of the semiconductor devices and based on study developed in [22].

Hysteresis Current Control.
Hysteresis control is one of the simplest nonlinear mechanisms to ensure spontaneous current regulation in power systems [23,24]. It will directly define the state of the switches in the H-bridge topology to track the output current I out to its reference I ref with a fixed value for the current ripple given by ΔI. e band delimited by (I ref ± ΔI) is called the hysteresis band and any violation of these two limits results in a change of power switch state from conduction to blocking or vice versa. Figure 4 reveals the principle of this control strategy where the on-time T on and off-time T off depends on the sign of grid voltage V s and the output voltage V out . e use of hysteresis control with model based on electrical circuit does not pose any problems since the switches are controlled by the generic gate signals (high for conduction state and low for blocking one). However, this is not the case for the average models where switches are monitored by duty cycle.
is developed method is explained in more detail in [25,26].

Maximum Switching Frequency.
One of the most critical parameters in a power converter is the switching frequency which can vary from a few kHz to a few MHz (20 kHz-2 MHz). A high value reduces the size of associated components such as inductors, transformers, resistors, and capacitors. On the other hand, the switching frequency directly affects power dissipation in semiconductor devices (diodes, MOSFETs, IGBTs, etc.), inductive and capacitive elements, and electromagnetic interference. Moreover, when discussing the switching frequency of a semiconductor, it is obvious to be concerned with its internal temperature since these two parameters are interdependent. In fact, an uncontrollable increase in the switching frequency can lead to an unexpected rise in temperature and subsequently to the instantaneous failure of the component and, at best, a decrease in its life cycle. For this reason, an estimation of the maximum switching frequency f s,max as function of power system specifications should be done.

Elitist Nondominated Sorting Genetic Algorithm
Genetic algorithms are random optimization techniques derived from approximate modeling of the natural evolution of races. ey are based on the principle of species evolution mentioned by Darwin which shows how, since their appearance, species have been able to reproduce in an innovative and flexible way to better adapt to the environment, by allowing only individuals well adapted to their environment to evolve [27]. e basic principle, although simple, is nevertheless powerful.
One of the most suitable genetic algorithms for solving multiobjective optimization problems is NSGA-II introduced by Deb et al. in [28,29]. Figure 5 is a schematic explanation of this algorithm process. NSGA-II first randomly creates an initial population of N individuals, each of whom is characterized by a set of design parameters ( Figure 5(a)). Based on objective functions, the NSGA-II ranks the N individuals using the concept of "nondomination". In fact, all individuals are classified into groups of different levels of nondominance in the fitness space ( Figure 5(b)). In addition to the fitness assignment, it is also necessary to maintain the diversity of the population in the nondominated front. For this, a density-estimation metric, known as crowding distance, is calculated for each individual ( Figure 5(c)). After individuals ranking and their crowding distances allocation, the binary tournament selection is affected according to their objective function and their crowding distance. NSGA-II first selects parents from the 1 st rank in the current generation population. en, individuals with a greater crowding distance are selected as parents while those with a smaller distance are rejected ( Figure 5(d)). e selected parents produce the corresponding offspring through the crossover and mutation operators. e newly produced offspring population is then merged with the current generation population ( Figure 5(e)). Finally, an elitist process is carried out on the combined population in order to identify the best solutioxns, based on two criteria: the individuals' ranks and their crowding distances. e best solutions of the current generation are the individuals of the next generation. en, all the above processes (the binary tournament selection, crossover operator, mutation operator, etc.) are iterated until the defined maximum number of generations is reached. Finally, NSGA-II produces a set of N optimal solutions towards a global Pareto front ( Figure 5(f )). Once the Pareto front is available, the decision maker should calculate the distance between each Pareto solution and the ideal solution as follows: where d i (i � 1 . . . N; N: population size) is the distance between the ideal point and the i th individual, Solution espace Randomly selection of N chromosomes   And then the closest individual to ideal vector is defined as the best solution.

Discrete Optimization Problem Formulation
Opposed to continuous optimization where all problem parameters are continuous quantities, the discrete optimization deals with choosing an optimal solution using either pure discrete design variables or a discrete and continuous combination. e appropriate process for grid-tied PV inverter optimization problem with discrete design parameters is illustrated in the flowchart in Figure 6. As presented, the NSGA-II identifies the design parameters from databases that are then used in the simulation model to calculate the constraints of the problem and determine the values of the objective functions that are then returned to the genetic algorithm for evaluation.
is will be repeated for each individual in the population and for each generation until the maximum number of generations is reached.
us, the optimization problem can be formulated by the following general system: T and D are, respectively, the transistor and the diode. N L , N T , and N D are the sizes of the database of inductances, transistors, and diodes, respectively.

Design Parameters.
ese are the numerical quantities that define each individual and are modified during the optimization process to achieve fitness functions while meeting the constraints. e design parameters of our optimization problems with discrete variables are well described in Table 1. Component databases are defined as an input to the optimization algorithm. ey are implemented from text files to facilitate updating, maintenance, and setup if necessary. e databases of power transistors (IGBT, MOSFET), freewheel diodes, and output inductors required for the design and available on the market, their technical characteristics, and their associated unit costs are given in Tables 2-4.

Problem Constraints.
Depending on the technical characteristics of the system to be studied, limits on its operation are always imposed and must be taken into account when formulating an optimization problem. In the case of grid-tied PV inverter optimization, constraints on the quality of the output power and the switching frequency of the converter must be considered. us, two constraints are defined: the total harmonic distortion of output current which should not exceed 5% in normal operation and the maximum switching frequency f s,max which should not overpass a limit value f s,lim chosen equal to 50 kHz.

Fitness Functions.
ree contradictory objective functions are taken into consideration which are the power losses of the grid-tied inverter, its volume, and its cost. In fact, the losses of a power converter are very significant since they characterize its conversion efficiency. e volume specifies the space to be allocated by the converter in relation to the other equipment of the PV installation so it must be taken into account because it is not profitable enough to design a very large converter even if its efficiency is the best or its price is the lowest. e cost describes the economic aspect of the converter that needs to be considered without degrading the other performances.
Power losses are caused mainly by the semiconductor devices and the output inductor. Semiconductor power losses are given by (2) and (3) and the load inductor losses are defined by the following: P cu are the inductor copper losses and can be expressed by R DC is the inductor DC resistance obtained from the database of inductors, and I L,rms is the rms value of the output current.
V L is the volume of output inductor and is determined from the database of inductors while P core are the timeaverage power losses per unit volume which are assumed to be given by the well-known Steinmetz equation [30][31][32]: We notice that all the inductors values in Table 4 are supposed to be independent of frequency.
B is the peak induction of sinusoidal excitation with frequency F and K, α, and β are Steinmetz parameters.
us, the total power losses of the inverter can be defined as P Inverter � P L + 4 P transistor + P diode . (9) e volume of the DC/AC converter for purely electrical optimization problem is mainly due to the inductor volume V L and the DC-link capacitor volume V Cap : e total cost of the DC/AC converter is the sum of the prices of its components determined from the databases and is given by

Optimization Results
e evolution of single objective optimizations over iterations of power losses, volume, and cost of the grid-connected PV inverter is illustrated in Figure 7. e three fitness functions decrease according to generations and converge towards their minimal values; the inverter power losses (Figure 7(a)) converge towards 32.72 W and the volume (Figure 7(b)) converges towards 645.7 cm 3 while the cost (Figure 7(b)) converges towards 77.94 €. Table 5 presents the results of the single objective optimizations. For each single-objective optimization problem, the converter optimal design depends on the objective considered, although we have kept the same design variables and optimization constraints in all cases. e first structure is the most efficient because it has the lowest losses, the second is the smallest design which corresponds to the lowest volume, and the third is the most economical, which correctly reflects the concept of minimization.
Nevertheless, single objective optimization treats only one performance criterion and neglects the others. Improving one objective often means degrading others. is is why the concept of compromise is mentioned in optimization in order to obtain an optimal solution as close as possible to the ideal solution which is defined by the optimal solutions of each objective function considered independently (i.e., the monoobjective optimization problems) and which is given in our case for the biobjective optimization problem by the vector: e result of biobjective cost vs. losses optimization is shown in Figure 8(a), which is a set of optimal solutions rather than a single optimal one and a trade-off between the inverter losses and its cost. As can be seen, the Pareto front is delimited by the two points "A" and "B." e first solution is the most efficient of all Pareto optimal solutions but it is the most expensive, whereas the second has the highest power losses but it is most economical. e choice of one solution or another depends on the preferences of the designer and the application for which the converter will be used. However, if the best of the best solutions is desired, the closest solution to the ideal one must be taken, which is indicated in Figure 8(a) and its characteristics are given in Table 6.
To go further and to have a more powerful inverter, we consider the triobjective optimization problem which aims to minimize power losses, volume, and cost in order to obtain a structure closer to the ideal one defined by   Figure 8(b) depicts the Pareto front for the triobjective optimization problem which is delimited by the two points "C" and "D." e first represents the most efficient inverter design based on the components available on the market, while the second is the most economical with a lower volume. e switching frequency at point "D" is higher than at point "C," resulting in higher power losses, but the volume and cost decrease because they are inversely proportional to the frequency. e best inverter structure for this optimization problem is detailed in Table 6.
To validate the approach using discrete variables, a comparison for the same triobjective optimization problem with the one using continuous parameters is presented in Figure 9. As shown, the result of the continuous         Figure 9: Comparing the Pareto front for discrete triobjective optimization problem to the Pareto space for a continuous one. optimization problem is a Pareto space that certainly contains unfeasible solutions while the discrete variable problem is a Pareto front where each solution matches a set of components present on the market. In addition, we notice that the discrete variable problem delivers solutions that dominate those of the continuous variable problem so it gives better results. is is mainly due to the use of a welldefined pair (transistor; diode) in the continuous case that limits the optimization task and the attachment to the model used to estimate the costs and volumes of passive components that remains more or less close to reality.

Conclusion
e power converter is a key stage in any simple or complex, small or large scale, stand-alone or grid-connected power generation system. erefore an accurate design of this conversion stage influences the performance of the entire system. is task is too difficult since the power converter is the seat of several physical phenomena (electrical, thermal, electromagnetic, etc.) which leads to the development of the predesign approach of the power converters.
In fact, optimization algorithms are applied to determine the optimal design of the converter before proceeding with its implementation. e purpose of this paper is to exploit genetic algorithms to overcome the difficulties encountered in the design of power converters, to take advantage of the accuracy and speed of nonlinear average models in conjunction with the utility and simplicity of hysteresis current control, and finally, to use the active and passive components available on the market to find the optimal structures of the solar inverter in a grid-connected PV application depending on considered fitness functions. e approach developed in this article will considerably facilitate the design task of power converters for a given application.

Conflicts of Interest
e authors declare that they have no conflicts of interest.