Design of an Optimum Single Phase Inverter for a Grid Tie PV System

Power converter optimization by genetic algorithm (GA) is used to provide simpler and more reliable converter design for high efficiency, small size and low cost. This paper presents a Computer-Aided Design Optimization Tool based on GA to determine the optimal structure of single-phase voltage source inverter devoted to grid-connected photovoltaic applications. An accurate non-linear averaged model was used to model the power converter. The hysteresis technique was used to control the output sine wave current of the inverter while the Elitist Non-dominated Sorting Genetic Algorithm NSGA-II was used to search the Pareto optimal front and the best design in terms of efficiency, volume and cost under electrical constraints. The converter model and the NSGA-II algorithm are developed in the MATLAB/Simulink environment. The problem formulation was detailed. It was shown that the optimization of a power converter, working in a given application without the need of tedious and expensive experimental tests classically used to build this converter, is possible by mean of simulation. This will decrease time to market phase for manufacturers.


Introduction
Due to rising prices and environmental impacts of fossil fuels, renewable energy resources have been considered as one of the most efficient solutions to meet global energy demand [1]- [3]. Photovoltaic (PV) solar technology is the fastest growing energy source for renewable energy production at an annual average rate of 8.3% [4]. Grid integrated PV energy sources have been widely used to produce electricity, but their performances need to be improved. To access the grid, the power electronic converter is an extremely important element for photovoltaic systems. To maximize performance, an optimized design of the conversion stage becomes the key to achieve the above goal.
A general block scheme of a PV system connected to the grid is shown in Fig. 1. In such a structure, the inverter is a basic component of the grid connected PV systems. Its main task is to convert the direct current to a quasi-sinusoidal form to be injected into electrical grid.
The design of power electronic converter or its components by optimization using metaheuristic methods, such as genetic ones, is an effective and widely used approach to obtain an optimal system with better performance [5]. In [6], the authors proposed an optimal design of a DC/DC cascaded converter dedicated to gridconnected photovoltaic systems to maximize efficiency or minimize volume. The obtained results showed that the best solution is a topology with two boosts converters in cascade. References [7]- [9] set out a design methodology based on the multi-objective genetic algorithm of DC/DC converters for grid-connected PV systems to minimize losses, size and cost. Multi-objective optimization was used to obtain the optimal design of a distributed maximum power point tracking synchronous boost converter in [10]. In detail, the efficiency and reliability of the power converter are maximized while its price is minimized. In [11], a study of modelling approaches of several components for power electronic converter was performed. In [12] and [13], the authors adopted a multi-objective GA-based methodology to design and select the appropriate DC/DC stage for a module-integrated inverter. In [14], a systematic methodology using multi-objective evolutionary algorithm and multi-objective covariance matrix adaptation evolution strategy are used in the extraction of power diode parameters. 224 Design of an Optimum Single Phase Inverter for a Grid Tie PV System The optimization of a power system itself is the final goal of applying evolutionary algorithm, but this task is also a means of modelling, forecasting, control and simulation. Although previous works have taken into account the characteristics of optimization, they still have some limitations. The used models are simplified and the behavior of the power electronic converter is mistreated. The control strategy is shortened, even limited to a constant duty cycle for DC/DC converters [6]- [9]. This paper presents a modeling and optimization approach to reveal the optimal design of the DC/AC converter in a grid-connected photovoltaic system using the popular and effective multi-objective algorithm NSGA-II. Optimization focuses on fulfilling three objectives: maximizing inverter efficiency, minimizing its size and reducing its cost. To model the power converter, an advanced non-ideal averaged model was used.
Various approaches have been directed toward power loss estimation in semiconductor components. The most widely used approach is the semiconductor refined models. This approach gives accurate results. Unfortunately, the simulation cost of these models is unaffordable especially in converter optimization context. Simplify models of power semiconductor devices are therefore essential for the analysis and the design in different applications. Averaged circuit modeling of switching power converters has been a major topic of research since 1976. The average model of the semiconductor devices is a simplified representation of a switch cell that is common in different converter topologies [15]- [16]. Non-linear 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 [17]- [20]. Thus, the authors have proposed an advanced PWM-Switch Model including semiconductor device nonlinearities.
Hence this model has been used in this study to estimate power losses in different semiconductors devices. The main advantage of the average model is that the simulation time is greatly reduced and accuracy is good enough. This is due to the fact that the commutation of the switch and the diode is averaged.
The use of the optimization algorithms requires cheap (in term of time) simulation models, particularly if multi-objective algorithms are used.
Since the power converter control is a crucial phase to think over the overall function of a grid-connected inverter, the hysteresis current control was used to generate switch gate drives [22]- [24]. This control strategy offer a high current tracking accuracy, robustness and simplicity but it's usually used with models based on electrical scheme of power converters which don't take into consideration the nonlinearities of power semiconductor devices. In this work, we use the hysteresis current control with the non-linear averaged model to combine the benefits of the two adopted methods. This paper is structured as follows. Section 2 describes the basic idea of the study, including the hysteresis current control strategy and the electrical power converter modelling. Section 3 highlights the principle of the NSGA-II multi-objective genetic algorithm. Section 4 focuses on the formulation of optimization problem under study including the definition of design variables, constraints and objectives. Section 5 describes the results of continuous mono, bi and tri-objective optimization under electrical constraints to achieve higher efficiency, smaller size and/or lower price.

Inverter Current Control Strategy
The work presented in this paper consists of the study and the optimization of a single phase inverter used in a grid-connected photovoltaic system. The chain is composed by a photovoltaic generator, a power conversion stage (inverter) and the grid. Fig. 2 presents a single phase grid-tied voltage source inverter which is directly connected to a photovoltaic generator via a bias capacitor C dc . The DC/AC converter consists of four transistors (IGBT or MOSFET), four freewheel diodes and a control system. The design of the control system requires a detailed modelling of the converter. The non-linear average model is used to model the actual operation of the converter since it provides a very approximate estimation of the converter's dynamic behavior. In our case, the four switches operate so that the driving signal S 12 is complementary to S 11 at fundamental frequency (f grid =50 Hz) and S 22 is complementary to S 21 at switching frequency (f s ). The hysteresis control allows an instantaneous adjustment of output current to be maintained below the hysteresis band limited by the upper limit of (I ref + ΔI) and the lower limit of (I ref -ΔI) where I ref is the reference current and ΔI is the output current ripple. When using a model with ideal semiconductor devices, the implementation of hysteresis control is simple. However, in case when the non-linear averaged model is used, it appears necessary to develop a method to estimate the duty cycle in order to control the active switches. One way to attend such a target is to estimate the value of the switch on-time (T on ) and off-time (T off ) and then determine the duty cycle. Fig. 3 depicts the principle of hysteresis current regulation. As shown in this figure, the T on and T off values depend on the sign of the grid voltage (v s ) and of the output voltage (v out ). Therefore, an estimation of these two values during fundamental period can be expressed as follows [25] (1) (2) Design of an Optimum Single Phase Inverter for a Grid Tie PV System Where (E on ) P , (E off ) P , (E on ) N and (E off ) N are respectively the steady states turn-on and turn-off modulated voltages which depend on the inverter output voltage sign and are given by Then, the switching period (T s ) and the duty ratio (ρ) can be defined respectively as follows (5)   (6) The switching frequency is determined as the inverse of (5) and it is given by (7). Fig. 4 shows the effects of the current ripple and duty cycle on the switching frequency variation. It can be seen that the switching frequency admits a maximum value (f s,max ) that must be controlled to protect semiconductor devices and reduce switching losses. This maximum frequency is sensitive to any variation in current ripple rate and output inductance value. By resolving the derivative of (7) equal to zero, with respect to time t, and by replacing the solution in (7), the maximum switching frequency can be easy determined.

DC/AC Converter Modelling
Modelling a power converter is the first step necessary in order to analyze its dynamic behavior in various applications. The averaging method is the widely used technique since both accuracy and rapidity are required especially for long time simulation and for complicated circuits. In spite of classical averaged model where the converter is assumed to be a linear system using ideal switches [26], the non-linear averaged model uses semiconductor device models where both static and dynamic characteristics of the switch are taken into account. In [17] and [18], the authors proposed an advanced pulse width modulation (PWM) switch model to account for nonlinearities in semiconductor devices. Fig. 5(a) shows the studied inverter leg with two active switches (IGBTs or MOSFETs) directly controlled by external control signals and two passive switches (DIODEs). In Fig. 5(b), the adopted leg circuit based on the used averaged model is presented.
In this developed model, the leg switches are replaced by a controlled voltage source V 1 in series with a controlled current source I 1 given by (7)   (8) With and are the time averaged values of the instantaneous terminal waveforms of U as (t) and i e2 (t) respectively over one cycle T s (switching period of the controlled switches).
(9) Figure 5. (a) The PWM-switch; (b) The corresponding averaged model, [18] 1 as Design of an Optimum Single Phase Inverter for a Grid Tie PV System Figure 6. Adopted switching characteristics of the switch and diode components in the PWM-switch, [19] Fig. 6 shows the adopted switching waveforms of the active switch (U as (t), i e1 (t)) and the passive switch (U bs (t), i e2 (t)) during T s . e g1 and e g2 are the control signals of T 1 and T 2 respectively.
Based on this analytical representation of the switching characteristics and the study developed in [18], the power losses of semiconductors (P switch and P diode ) including both conduction and switching losses and considering the various conduction and switching times can be given by (10) and (11).

Multi-Objective Optimization
Over the last few years, stochastic optimization techniques using evolutionary algorithms have received attention in power electronic optimization. Contrary to the conventional methods, genetic algorithms are considered to be an effective way of finding solutions close to the global optimum without being trapped in local minima and they are less dependent upon the initial starting point of the search.
The genetic algorithm is a well-known metaheuristic research method derived from the natural evolution process. It successively executes three genetic operators (selection, crossover and mutation) to give birth to an offspring population. Quite often, optimization problems such as those associated with the development of power electronic converters require a multi-objective approach since at least two conflicting objectives, under certain constraints, must be satisfied simultaneously.
According to [27] and [28], the multi-objective optimization problem can be defined as the problematic of finding a vector of decision parameters that meets the constraints and optimizes a vector of criteria whose elements are objective functions. Thus, the result is not a single solution, but rather a set of optimal compromise solutions. This front is obtained by the Pareto optimality theory. Proposed by Deb et al. [29]- [30], the NSGA-II is used to find a family of solutions that best satisfies the established requirements (objectives). It is equipped with a sorting procedure based on Pareto's optimal approach, which is an elitist method that preserves the diversity of populations and keeps the best solutions found in previous generations. It is supplied also with a comparison operator based on a crowding distance calculation to manage diversity in Pareto front [31]. The pseudocode for NSGA-II is shown in the flowchart in Fig. 7.
After drawing the Pareto front, it is necessary to determine the point adopted as the optimal design. The decision maker can pick an individual, among the individuals contained in the Pareto front, depending on the application and regarding the importance of the different criteria, a more difficult task for more complex applications. That is why a method to determine the best solution is required. The ideal is to have a solution that includes the optimum for each objective considered independently reached at the same specifications, variables and optimization constraints that is called the ideal objective vector. In general, this vector corresponds to a non-existent solution. One way to overcome this problem is to identify the closest solution to this ideal point. To do this, the distance between the ideal solution and each optimal Pareto solution is calculated as follows (12) Where is the distance between the ideal point and the i th individual, are the problem objectives for the i th Pareto optimal individual, and is the ideal objective vector. Power converters are often optimized for minimal losses, size and cost. The appropriate optimization process is illustrated in the flowchart in Fig. 8. As presented, the NSGA-II identifies the design parameters that are 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. This will be repeated for each individual in the population until the maximum number of generations is reached.
Design of an Optimum Single Phase Inverter for a Grid Tie PV System

Design Parameters
Design parameters are the numerical quantities that can be modified to achieve the objectives while respecting the constraints. Before starting any optimization study, it is crucial to define the problem parameters and their limits. The design variables of the system under investigation are the output current ripple (ΔI) and the output inductance (L). Note that the bias capacitor is not considered as an optimization parameter. It is designed to insure an input voltage ripple of approximately 5V.

Problem Constraints
Constraints are restrictions imposed by the particular characteristics or nature of the problem under study. These limits must be fulfilled to obtain acceptable solutions. For the current problem, two electrical constraints are taken into account which are the total harmonic distortion (THD) and the maximum switching frequency of semiconductor devices (f s,max ). The THD for the injected grid current should not exceed 5% in normal operation in order to meet the grid harmonic requirements and to avoid negative effects on other equipment connected to it. The f s,max should be lower than a selected value (fixed here to 50 KHz) to operate the power converter in a given switching frequency range. This is due to the dependence of the switching losses on this frequency and the limitation of the maximum switching frequency of power semiconductors since any over-increase in the switching frequency leads to an increase in the temperature of the semiconductor and, consequently, the failure of the device.

Inverter Losses Models
Power losses calculation is necessary in the design of power converters, since it characterize their energy efficiency, which should be maximized. The converter losses are mainly caused by conductive and switching losses in semiconductor devices (P switch and P diode ) as well as core and copper losses in the load inductor. The losses in DC capacitor P Cap are neglected because its value is too low compared to the semiconductor losses P SC and the inductor losses P L . Semiconductor power losses are derived from (10) and (11). The iron losses of the load inductor are supposed to be given by the well-known Steinmetz equation [32]- [34] (13) Where is the peak induction of sinusoidal excitation with frequency F, P core is the time-average power losses per unit volume, and K, α, β are Steinmetz parameters, And the inductor winding losses are obtained by (14) Then, the total losses are given by (15) With V L and P L are respectively the inductor volume and losses.

Inverter Volume Models
The volume of the DC/AC converter when the studied optimization problem is purely electrical is mainly due to the inductor volume V L and the DC capacitor volume V Cap (16) Where V L is defined as (17)   (18) With A W is the winding area, A C is the core section, I L,max is the peak value of the inductor current, J rms is the rms value of the current density, K L and K W are the geometrical parameters of the magnetic circuit.
And V Cap is given by (19) For example, A Cap = 12.632 cm 3 and B Cap = 90794 cm 3 /F for an EVOX RIFA electrolyte capacitor [35].

Inverter Cost Models
When designing a power converter, its price must be taken into consideration to provide a more economical solution.
In our case, the inverter cost is defined as (20) The cost of DC Bus capacitor is expressed as With , and is the capacitor stored energy ( is the capacitor rated voltage) [35]. The cost of the load inductor depends on its volume and can be defined by (22) With A Ind = 0.07008 €, B Ind = 0.3904 €/cm 3 and C Ind = 2.16x10 -4 €/cm 6 [35]. In addition, semiconductor devices costs should be taken into account. Thus, the devices that meet the studied system requirement are selected (IGBT SGP15N60 (15A, 600V) and PIN diode 15ETH06PbF (15A, 600V)) in our application. Fig. 9 shows the convergence of single-objective optimization of losses, volume and cost by number of iterations, while Table 1 summarizes the results obtained from this work. The converter losses decrease from 71.23 W (A1) to 63.45 W (B1) (Fig. 9 (a)). Based on (7), the switching frequency is inversely proportional to the product L∆I. In fact, this product is lower in point A1 (L=10.68 mH, ∆I≈0.56 A) than in point B1 (L=13.49 mH, ∆I ≈ 0.6 A), which results in the diminution of the switching frequency from A1 to B1 and, consequently, the reduction of power losses.

Mono-objective Optimization
The inverter volume drops by 33.3% from 671.3 cm 3 (A2) to 447.9 cm 3 (B2) when the number of iterations increases ( Fig. 9 (b)). This can be explained essentially by the diminution of the inductance value since the inductor is the bulkiest component of the converter. Another reason of this decrease is the increase of the switching frequency which yields to the power losses rise and respectively the size reduction.
The cost function decreases from 255.3 € (A3) to 137.2 € (B3) with a profit of 118.1 € (Fig. 9 (c)). This advantage is obtained by reducing the inductance value and increasing the switching frequency which reduce the size of the inverter and therefore its price.
The main disadvantage of the above study is that mono-objective optimization deals with one criterion while neglecting the others. A large size implies a heavy inverter with a large occupied space, even if it is more efficient. In addition, high losses considerably reduce the efficiency of the converter, although it becomes smaller and cheaper. On the other hand, the cost must be kept at a reasonable level even if the converter's capacity is reduced. Indeed, improving one goal often means degrading others. This is why the concept of compromise is often mentioned in optimization where there is not only one objective function to optimize but multiple. Before studying the tri-objective optimization, it is necessary to solve the bi-objective problem in order to determine the optimal designs closest to the ideal vectors for Volume vs. Losses and Cost vs. Losses optimization, which are defined respectively by

Bi-Objective Optimization
The bi-objective optimization has been carried out and the Pareto optimal solutions are shown in Fig. 10. The Fig.  10 (a) represent the Volume vs. Losses curve, it is clear here that the reduction in volume leads to an increase in energy losses and vice-versa. The curve is limited by the two point A4 and B4 where the point A4 corresponds to the minimum losses while the point B4 represents the minimum size. Moving along the Pareto front from A4 to B4, a diminution of both the inductance value and the current ripple can be recorded. This decrease will leads to the increase of switching frequency and thus, the losses augmentation and the size reduction.
The optimal solution for Volume vs. Losses optimization is chosen according to the method presented in Section III. The minimum distance between the ideal and optimum point is equal to 0.6889 and the best design is obtained for an inductance value of 4.48 mH and a current ripple of 0.581 A. This design represents the best compromise between the efficiency of the inverter and its volume with losses in the order of 93 W and a size of 643.9 cm 3 .
For Cost vs. Losses optimization, shown in Fig. 10 (b), the Pareto front is bounded by points A5 and B5. Solution A5 is the best of all optimal Pareto solutions in terms of efficiency but it is the most expensive, while B5 is the least efficient but the most economical solution. By moving along the Cost vs. Losses curve from A5 to B5, the inductance value and current ripple are reduced, resulting in an increase of switching frequency and thus an increase of losses and a decrease of price. The best design for this bi-objective optimization is obtained at a minimum distance of 0.857 for which the inductance value is approximately 3.6 mH and the output current ripple is about 0.57 A. A trade-off between cost and losses targets can be found in this optimal point with a losses of 102.6 W for an inverter priced at nearly 206.7 €.   Table 2 contains the optimal designs obtained from bi-objective Volume vs. Losses and Cost vs. Losses optimization.
A more efficient solution can be achieved by optimizing the converter design not only by meeting two objectives (Volume vs. Losses or Cost vs. Losses) but by satisfying all three objectives at once, this is the subject of the next section. Fig. 11 shows the Pareto front of the continuous triobjective optimization for the DC/AC converter. The choice of one individual or another among this front depends on the designer preference. If a high efficiency of the inverter is desired despite its size and price, the solution A6 is the most suitable one; however, if the volume or cost minimization is preferred independently to efficiency, the solution B6 is the most appropriate. Sliding from A6 to B6 along the Pareto front, there is a simultaneous diminution of the inductance and current ripple values. These decreases will lead to a rise of the switching frequency and subsequently an increase of converter losses. On the other hand, they have a direct effect on the reduction of converter size and its cost. The ideal objective vector for tri-objective optimization is defined as follows:

Tri-Objective Optimization
The minimum distance between this point and the optimal Pareto solution is then evaluated to about 0.8836 and the optimal structure is obtained for an inductance of 3.36 mH and a current ripple of 0.573 A. This design results in losses around 105.4 W, a size of approximately 565 cm 3 and a price of about 196.4 €, as presented in Table 3.

Conclusions
Grid-connected photovoltaic systems are becoming an increasingly active player in the power generation systems of the future, which are connected by a wide range of electronic power converters. In order to improve the specifications of these systems, high requirements have been imposed on the entire photovoltaic installation, in particular for the power conversion stage. The design of power converters often involves a high level of technical and scientific knowledge in several technical fields. Thus, the electrical, thermal, mechanical, volume, control and cost constraints imposed by manufacturers require a design approach under multi-physical constraints. This approach should take into account the different aspects that affect the efficient operation of converters and their integration into a real environment. This approach was well described in this paper. This paper proposes a methodology for the pre-design of single-phase DC/AC power converters in photovoltaic systems in terms of efficiency, volume and cost based on genetic algorithm. The non-linear average model associated with the hysteresis current regulator is used for the inverter modeling and the multi-objective algorithm NSGA-II is used to reveal the trade-off curves. The results obtained are very useful for easily design an optimal power converter structure according to the specifications and constraints of the desired system. Multi objective optimization of a power converter working in a given application without the need of tedious and expensive experimental tests is possible if an accurate model of the converter is used. This decrease time to market phase for manufacturers.