Modeling and control of photovoltaic energy conversion connected to the grid

Abstract

This paper presents modeling and control of a photovoltaic generator (PVG) connected to the grid. The parameters of the PVG have been identified in previous work (series and parallel resistance, reverse saturation current and thermal voltage) using Newton-Raphston and the gradient algorithm. The electrical energy from a PVG is transferred to the grid via two static converters (DC/DC and DC/AC). The objective of the proposed control strategy is to maximize energy captured from the PVG. The adapted control law for extracting maximum power from the PVG is based on the incremental conductance algorithm. The developed algorithm has the capability of searching the maximum photovoltaic power under variable irradiation and temperature. To control the DC/AC inverter, an intelligent system based on two structures is constructed: a current source control structure and a voltage source control structure. The system has been validated by numerical simulation using data obtained from the PVG installed in the laboratory research (INSAT, Tunisia).

Keywords

photovoltaic generator (PVG) / maximum power point tracker / grid-connected / static converters

Introduction

Due to environmental contamination and the depletion of fossil energy, a renewable energy application such as photovoltaic (PV) system has been widely used for a few decades. PV power supplied to the utility grid is gaining more and more visibility, while the demand of power over the world is increasing. It registers an exponential increase with Germany and Japan leading the list of the countries having the largest capacity installed [1]. However, what happened to Japan and the consequences of the tsunami seism, especially the nuclear catastrophes on humanity give raison opinion for giving up nuclear energy. However the efficiency and controllability stand as major drawbacks; in addition, the transmission system operators are imposing tough standards [2], when the photovoltaic generator (PVG) is connected to the grid. The PV energy applications can be categorized into the stand-alone system and the grid-connected system. The stand-alone system requires the battery bank to store the PV energy which is suitable for low-power system. On the other hand, the grid-connected system does not require the battery bank and has become the primary PV application for high power applications. The main purpose of the grid-connected system is to transfer maximum solar array energy into grid with a unity power factor. The PV power is influenced by environmental factors, such as temperature and irradiance [3]. Since the characteristic curve of a solar cell exhibits a nonlinear voltage current characteristic, a maximum power point tracker (MPPT) controller is required to match the solar cell power to the environmental changes. The typical configuration of a three-phase grid-connected PV system consisting of a solar array, an input capacitor C, a three phase inverter, a filter inductor L, and grid voltages is shown in Fig. 1. The input capacitor supports the solar array voltage for a voltage source inverter. The three-phase inverter with the filter inductor converts a DC input voltage into an AC sinusoidal voltage by means of appropriate switch signals to make the output current in phase with the utility voltage and obtain a unity power factor. A typical controller configuration of the grid-connected PV system consists of an MPPT controller [4], to generate power reference. This power is used for current reference directly by the power balance relation in the first step, and for voltage reference in the second step. As a consequence, large research efforts are made to the control of these systems in order to improve their behavior [5,6]. Several PV systems are interfacing the grid through a voltage source inverter and a boost converter. Many control strategies and controller types [7–9] have been investigated. Several research results in literature have been demonstrated that in the case of grid frequency or voltage fluctuations, there are problems in controlling of the grid current and power factor. Tunisia is a country in North Africa that has considerable potential in solar energy [10]. It confirms the ambition of Tunisia in becoming an international platform for production of industrial energy including solar energy. In this paper, a photovoltaic energy conversion system (PVECS) for the solar panels is proposed to employ a control strategy for the static converter to guarantee control of active and reactive power. The mathematical modeling is evolved and the simulation result verifies the proposed scheme.

Description of the PV energy conversion system

Figure 1 illustrates the power converter structure used to interface the PV array with the power grid. The first stage is the boost converter, which raises the relatively low solar voltage to a level suitable (550 V) for the DC link directly connected to the inverter. The second stage is the DC/AC inverter that operates in a current source controlled mode which injects unity power factor current to the grid in the first step, and in second step the DC/AC inverter operates in a voltage source controlled mode. The inverter should be able to supply a continuous power from the DC link bus to a three phase utility line (220 V/50 Hz). An output R-L filter is employed to reduce the ripple components due to pulse width modulation (PWM) switching operation. The interconnection topology structure proposed by the block diagram of the system is constituted of a PVG, a DC/DC boost converter, a DC/AC inverter, and the grid.

Mathematical modeling system components

Description and identification of PV parameters

The solar panel is installed on the roof of the laboratory research unit RME at INSAT, Tunisia as displayed in Fig. 2. It consists of ten elementary solar modules connected in series to provide an open circuit voltage of approximately 200 V. It has a rated power of 50 W and a short circuit current of 3.2 A at standard test conditions (STC). The voltage current characteristic equation of a solar cell is composed of the light generated current source, series resistance, as described in Fig. 2. The selected model responds to

(1)I_p={\mathit{M}}_p{\mathit{I}}_{sc}-{\mathit{M}}_p{\mathit{I}}_s(e^{{\mathit{V}}_d\mathit{/}{\mathit{V}}_T}-\mathrm{1})-{\displaystyle \frac{{\mathit{V}}_d}{{\mathit{R}}_{sh}}},

(2){\mathit{I}}_d={\mathit{I}}_s(e^{{\mathit{V}}_d/{\mathit{V}}_T}-\mathrm{1}),

with

(3){\mathit{V}}_T={\mathit{M}}_s{\displaystyle \frac{\mathit{A}\mathit{K}\mathit{T}}{q}},

(4){\mathit{V}}_d={\mathit{V}}_p+{\mathit{R}}_s{\mathit{I}}_p,

A: Ideality factor (n=1),

I_ph: Current generated by the illumination, it is also called photocurrent,

I_d: Current diode, also called dark current,

I_sh: Current shunt,

I_sc: Short-circuit current,

I_s: Reverse saturation current,

I_p: The current delivered by the solar cell,

K: Boltzmann constant (K=1.38×10^-23 J/K),

M_p: Number of parallel connected modules,

M_s: Number of series connected modules,

q: Electron charge,

R_s: series resistance,

R_sh: shunt resistance,

T: Temperature of the cell/K.

This implicit non-linear equation can be solved with numerical iterative methods, such as the Newton-Raphson algorithm [3], although these systems require powerful mathematical tools and a close approximation of initial parameter values to attain convergence. The parameters are extracted by means of analytical methods. Equation (1) and its derivative are evaluated at the short and open circuit. The extraction of the parameters is based on the obtained expression, by resolving the non-linear I-V equation with Newton-Raphson method, and the selection and the identification algorithm that is focused on the gradient method for the analytic extraction of the parameters with the single exponential model, as demonstrated in Fig. 3.

The parameters {\mathit{R}}_s, {\mathit{R}}_{sh}, {\mathit{V}}_T and {\mathit{I}}_s were extracted previously [3] based on the derivative expressions of Eq. (1) by the gradient iterative method. It is about finding the optimum of the function defined by the Euclidean distance between measured values and those of non-linear model. Gradients to minimize the function are given by the relationships exposed in Eq. (5). The precise determination of the gradient of the voltage-current curve at the said points requires experimentation under certain temperature and irradiance conditions to obtain enough pairs of current-voltage values. In order to adapt the solar module behavior to different conditions of temperature and irradiance, the procedure described beforehand is applied. It is known that the diode saturation current is proportional to temperature, so by the gradient equation proposed, this parameter can be adapted to other temperature conditions. The parameters that are considered variables under temperature and/or irradiance conditions are R_s, R_sh, V_T, I_s, I_sc and V_oc.

(5)\{\begin{array}{l}{{\displaystyle G}}_{I_{os}}={\displaystyle \frac{\partial I_p}{\partial I_{os}}}={\displaystyle \frac{1-e^{V_d/V_T}}{D}},\hfill \\ {{\displaystyle G}}_{V_T}={\displaystyle \frac{\partial I_p}{\partial V_T}}={\displaystyle \frac{I_{os}{V}_d{e}^{V_d/V_T}}{V_T^{2}D}},\hfill \\ {{\displaystyle G}}_{R_s}={\displaystyle \frac{\partial I_p}{\partial R_s}}=\left(-{\displaystyle \frac{I_{os}{I}_p}{V_T}}{e}^{V_d/V_T}-{\displaystyle \frac{I_p}{R_{sh}}}\right)/D,\hfill \\ {{\displaystyle G}}_{R_{sh}}={\displaystyle \frac{\partial I_p}{\partial R_{sh}}}={\displaystyle \frac{V_d}{R_{sh}^{2}D}},\hfill \end{array}

with

(6)D=\mathrm{1}+{\displaystyle \frac{R_s}{R_{sh}}}+{\displaystyle \frac{R_s{I}_{os}}{V_T}}{e}^{V_d/V_T},V_d=V_p+R_s{I}_p.

The panel is delivered with experimental characteristics coming down, under different irradiation and two differents temperature.

DC/DC converter

Commonly the shunt converter is current controlled, but it is also possible to control the voltage directly. A cascade structure is developed which composes of a grid power by a PV system via the two static converters in Fig. 1. The solar panel is represented by a current source that varies depending on the irradiance and the temperature. The first converter is a boost converter with load of the electromagnetic energy. This operation is governed by Eq. (7) of the system

(7)\{\begin{array}{l}{\displaystyle \frac{d{V}_{PV}}{dt}}={\displaystyle \frac{I_{PV}-I_L}{C_{PV}}},\hfill \\ {\displaystyle \frac{d{I}_L}{dt}}={\displaystyle \frac{V_{PV}-C\left(t\right)V_{DC}}{L}},\hfill \\ {\displaystyle \frac{d{V}_{DC}}{dt}}={\displaystyle \frac{C\left(t\right)I_L-I_{DC}}{C_{DC}}}.\hfill \end{array}

The output voltage V_DC is expressed by Eq. (8), where \alpha is a duty cycle ratio fixed by the MPPT algorithm.

(8)V_{DC}={\displaystyle \frac{\mathrm{1}}{\mathrm{1}-\alpha }}{V}_{PV}.

DC/AC inverter

The DC/AC is an inverter operating in direct voltage regulator. Controlling the voltage V_{DC}, in presence of voltage variations, the grid current is forced by the controller to have a sinusoidal waveform which is in phase with respect to the corresponding grid voltage [3]. The inverter is used to arrange the interface between the generator PV and the network. This inverter connected to the grid conditions not only the power of PV exit of the generator, but it also contributes to the ordering of the system to ensure a real injection of the power produced by the photovoltaic generator (PVG) in the network. Because of this, the inverter must carefully be designed to allow the PV outlet side of the generator and the parameters of the network to interface. The PWM three-phase inverter developed in this paper adopts two strategies of control, a current source control structure and a voltage source control structure to ensure a response in fast transitory mode and satisfactory characteristics in permanent mode, and the catch in consideration of information coming from controller MPPT.

Grid

The grid is modeled by a thevenin equivalent circuit. This model is nonlinear and time variant. The case of a three-phase source voltage delivered by the inverter feeding an active load is considered. It consists of a sinusoidal voltage source of amplitude {\overline{E}}_{\mathrm{r}} placed behind impedance {\overline{Z}}_{\mathrm{r}} as indicated by the line diagram in Fig. 4.

The operation of this circuit in sinusoidal permanent regime is governed by Eq. (9) and the vector diagram in Fig. 5.

(9)\overline{V}={\overline{E}}_r+\overline{Z_{\mathrm{r}}}\overline{I}.

The active and reactive power delivered to the grid is expressed by Eq. (10), where, Y_{\mathrm{r}} and {\theta }_{\mathrm{r}} are the modulus and argument of the admittance associated to \overline{Z_{\mathrm{r}}} and \delta is the angle between vectors \overline{V} and \overline{E_{\mathrm{r}}}.

(10)\{\begin{array}{l}P=Y_{\mathrm{r}}\mathrm{c}\mathrm{o}\mathrm{s}\left({\theta }_{\mathrm{r}}\right)V^2+Y_{\mathrm{r}}{E}_{\mathrm{r}}V\mathrm{c}\mathrm{o}\mathrm{s}\left(\delta -{\theta }_{\mathrm{r}}\right),\hfill \\ Q=-Y_r\mathrm{s}\mathrm{i}\mathrm{n}\left({\theta }_r\right)V^2+Y_r{E}_{r}Vsin\left(\delta -{\theta }_r\right).\hfill \end{array}

The norm \overline{V} of voltage source can be calculated and the current vector \overline{I} can be deducted as

(11)\{\begin{array}{l}V=f\left(P,Q\right),\hfill \\ \overline{I}={\displaystyle \frac{P-\mathrm{j}Q}{V}},\hfill \end{array}

(12)V=f(P,Q)=\sqrt{M+\sqrt{M^{\mathrm{2}}-N}},

with

(13)\lambda =\arg [P-jQ-(R_r-j{X}_r)I^{\mathrm{2}}],

(14)\{\begin{array}{l}M=R_{r}P-X_{r}Q+E_r^2/2,\hfill \\ N=Z_r^2\left(P^2+Q^2\right),\hfill \end{array}

And

(15)\delta =\arg \left({\displaystyle \frac{P+jQ}{{\overline{I}}^{*}}}\right).

Presents the phases of \overline{I} and \overline{V} compared to Z_r.

Proposed strategy control of the maximum PV power extraction

Supervisory control algorithm

The objective of the supervisory control system is to control the active and the reactive power injected by the PVG in the grid [10,11]. First, the reference DC voltage and the active power P_ref used in the control can be specified. In this study, the voltage V_ref is constant and equal to the nominal voltage which is used to determine the nominal AC grid voltage. The power P_ref is specified in order to extract the maximum power from the available photovoltaic energy for actual irradiance. Here, the adapting control strategy used for extracting maximum power from the PVG is based on the INC algorithm described below.

The proposed control design

To effective control the PV power generation systems under variable climatic conditions, direct and quadratic currents demand control apply the research results from the desired regime to inverter controls. To ensure this result, the objectives of the proposed control system are summarized in the structure in Fig. 1. In the proposed structure, three control loops are suggested:

1) Modelling and identification of PVG parameters, which acts essentially on determination of series and shunt resistances, reverse saturation current, thermal voltage under different irradiance and temperature [12-14]. Those parameters vary with climatic change, so causes variation of the transferred power to the grid.

2) A controller of extraction maximum PV power. So a control algorithm of MPPT in Refs. [15,16-18] has to be applied. In this work INC algorithm was used. This controller of the duty cyclic ratio of the DC/DC converter controls the voltage of the DC link bus of the inverter to the reference value in Ref. [19].

3) A controller of energy transferred to the grid; that permits to inject in the grid the totality of the power available to the frequency 50 Hz and under a factor of unit power as shown in Figs. 19 and 26 [20,21]. From the values of the reference power to transfer P_{\mathrm{r}\mathrm{e}\mathrm{f}}, this controller calculates at a first time the amplitude and the phase of a reference current or of the reference voltage to be assured by the inverter. The controller determines at a second time the state of the keys of the inverter.

To evaluate the reliability performances of the developed control scheme for different ranges of climatic conditions, a period of two days was picked up randomly from each season namely: winter (December), and summer (July). The simulation results show clearly that the proposed control mode fulfilled a higher MPPT efficiency and achieved the highest power efficiency. The improvement of this work is to validate the identified PVG with mathematic tools using experiment data loaded in the research laboratory RME. So the tracking of the maximum point was observed during the grid connection operation. In fact, the proposed control mode with identified PVG supplies a new value of reference voltage, tracked through the DC/DC stage so that the overall system can keep operating under optimal conditions. That is to say, the power injected into the main utility is equal to the maximum point.

Impact of the PVG connected to the grid

The connection of a PVG to the electrical grid cannot be carried out without conforming to the standards in force and the best practices of the state of the art.

One of the technical difficulties resides in the lack of standards which are unanimously recognized in this field. Moreover, experiment in regard to PV coupled with the network has had the existing programs in Japan (70000 generators), Germany (cost of purchase of the electricity of 0, 5 Euro/(kW·h)), and Italy (10000 solar roofs)<FootNote>

Guide de rédaction du cahier des charges techniques des générateurs photovoltaïques connectés au réseau (N°ADEME/PVC/V1). http://www.mctparis.com/fr/images_db/Guide_Ademe_PVC.pdf

</FootNote>. The constraints to be subjected to are of three natures:

1)　Security toward users, the public and workers;

2)　Subject to the electrical supply network;

3)　Integration in construction.

In a more total context of control energy, it is a significant element which makes it possible to ensure greater autonomic energy to a building.

In front of the fast development of these technologies, in May 2006, the International Energy Agency published a study on the energy profitability of PV electricity in 41 towns of 26 countries, Europeans for the majority, the average lifespan of the PV modules is 30 years, the calculation of the return time on invested energy and that of the profitability energy factor takes into account all the expenditure of energy for manufacture, installation, disassembling and recycling of the PV systems Fig. 6 [22].

The access to the energy of the populations representing a strong engine of development, the sector of electricity have not escaped there.

Generation of the references of the control structure

References voltage and duty cycle ratio of DC/DC converter

This study concerns the cascade structure composed of a network powered by a PV system via two static converters. The solar panel is represented by a current source varies depending on the sunlight. The first converter is a booster chopper with storage of electromagnetic energy. The second is an inverter operating in direct voltage regulator. The frequency of this inverter is synchronized with the network. The average output voltage V_{DC} is expressed in Eq. (8) where α is a duty cycle ratio.

The reference voltage of the continuous bus of the inverter and the reference voltage of the DC/DC converter define the reference duty-cycle ratio α with which the DC/DC converter will be commanded:

(16){\alpha }_{ref}=\mathrm{1}-{\displaystyle \frac{V_{pv}}{V_{DC}}}.

Reference voltage injected in the grid

The transfer of powers deals with two approaches, the first with reference voltage injected in the grid, and the second with a reference current injected in the grid. The injected voltage in the network must be of the amplitude and phase allowing the transferred power P_{\mathrm{r}\mathrm{e}\mathrm{f}} with unit power factor. The vector voltage \overline{V} is generated by the inverter. It provides to the receiver an active power reference P_{\mathrm{r}\mathrm{e}\mathrm{f}} and a reactive power reference Q_{\mathrm{r}\mathrm{e}\mathrm{f}}. The instantaneous values of reference voltage is denoted by \overline{V_{ref}}(t). To assure the required balance power, this reference instantaneous voltage must verify the relationship of Eq. (17) with {\beta }_r(t) as the phase vector of the instantaneous emf \overline{e_{\mathrm{r}}}(t):

(17){\overline{V}}_{ref}(t)=V_{ref}{e}^{j({\delta }_{ref}+{\beta }_r(t))}.

The transfer of powers deals with a control voltage of the inverter. Figure 6 exhibits the structure of the voltage control approach, while Fig. 7 represents the simulation scheme of this approach.

Reference current injected in the grid

The second approach to transfer power to the grid is achieved by injecting current provided by the transferred power to the grid P_{ref} . The instantaneous values of reference current is denoted by {\overline{i}}_{ref}(t). To assure the equilibrium of power required, this reference instantaneous current must verify the relationship of Eq. (18) with {\beta }_r(t) as the phase vector of the instantaneous emf {\overline{e}}_r(t). Figure 8 demonstrates the structure of current control approach.

(18){\overline{i}}_{ref}(t)=I_{ref}{e}^{j({\lambda }_{ref}+{\beta }_r(t))}.

Extraction of the maximum power: MPPT operation

To determine the operating point corresponding to the maximum power for different insulation levels and different temperatures, a function on line giving the feature I_p=f(V_p) is identified. A maximum power point tracking algorithm used in this work was developed to remove the maximum converted power to the grid [13]. This algorithm is based on the incremental conductance (ING) algorithm (Eq. (19)). The power output from the PVG for any operating point is P = VI. At the MPPT, the partial derivative of extracted power with respect to the supplied voltage is equal to zero:

(19){\left({\displaystyle \frac{dP}{dV}}\right)}_{Mopt}=\mathrm{0.}

This condition can further be written in terms of the array current and voltage as

(20){\left({\displaystyle \frac{\partial I}{\partial V}}\right)}_{Mopt}+{\left({\displaystyle \frac{I}{V}}\right)}_{Mopt}=\mathrm{0.}

The first left term of Eq. (20) illustrates the dynamic conductance because it gives the internal sensitivity of the current to the voltage. The second left term of Eq. (20) designates the static conductance because it corresponds to viewed external conductance. Let these two terms be noted:

(21){\left({\displaystyle \frac{\partial I}{\partial V}}\right)}_{Mopt}=G_d

and

(22){\left({\displaystyle \frac{I}{V}}\right)}_{Mopt}=G_s.

Therefore, for MPPT operation of the PVG subsystem the sum of the dynamic conductance and the static conductance must be equal to 0. Thus, the set point G* of this control subsystem is adjusted at 0, the actual value of G=G_s+G_d can be determined by computing both G_s and G_d, while G_s can be calculated as the instantaneous array conductance {(I/V)}_{Mopt}. Also G_d is calculated as the incremental array conductance {(\partial I/\partial V)}_{Mopt}.

The flowchart of the proposed method is shown in Fig. 9. If it is taken that (dI/dV)_PV= tan γ and (I/V) _PV = tan β, where γ is the tangent line angle and β is the load line angle, both at the operating point of the I-V characteristic curve of the PVG array (Fig. 10), Eq. (20) becomes

(23)\tan \gamma +\tan \beta =\mathrm{0},

while solving Eq. (21), Eq. (24) can be obtained:

(24)\gamma =-\beta +k\pi ,k\in \mathbf{Z}.

Therefore, for MPPT operation of the PVG subsystem the sum of the load line angle and the tangent line angle must be equal to π. Thus, the set point {\theta }_{ref} of this control subsystem is adjusted at π, the actual value of θ = γ + β can be calculated by calculating both γ and β, while β can be calculated as the arctangent of the instantaneous array conductance (I/V)_PV. Also γ is calculated as the arctangent of the incremental array conductance (dI/dV)_PV.

Simulation results

First case of simulation

What is represented in this section is the results of the identification algorithm applied that is focused on the gradient method to characterize PV solar module, particularly for the determination of electrical parameters. The panel is delivered with experimental caracteristics coming down, the irradiation E is equal to 800, 700, 600, 500 and 400 W/m² respectively, as given in Table 1, at the ambient temperature 15ºC in December 2009, and for another insolations E is equal to 400, 450, 650, 700, 800, 900, 1000 W/m², as given in Table 2, at the ambient temperature of day 32ºC in July 2010. In order to construct the model, the four parameters, V_T, R_s, R_sh and I_s, involved in Eq. (2) have to be extracted. The light-generated current (photocurrent) is accepted as linearly dependent on solar irradiance. It is important to note that the improvement of this work in which the adjustement results are significantly improved by introducing the value for thermal potential and reverse saturation current with a minimum precision, and that the model is very sensitive to these factors. In order to determine the values of the these parameter, which yield the best adjusment between the various experimental curves and the theoretical model, good analytical gradient expressions have to be chosen for initial adjusted parameters and the iterations steps. Figures 11 and 12 show the experimental and mesured caracteristics of the PV pannel.

Second case of simulation:

A complete Simulink-Matlab simulation of the power electronics converters (PECs) including the global and command previously studied has been carried out with the following parameters: first for control transferred power with reference voltage injected in the grid, second for control transferred power with reference current injected in the grid.

Voltage grid 220 V/50 Hz,

Filter and grid impedance: R_r = 0.5 Ω, L_r = 7.25×10^-4 H,

Boost converter:　 L_PV = 0.2 H, C_PV = 0.003 F

DC link capacitance: C_DC = 0.02 F.

The grid is represented by a source voltage at fixed frequency. Its impedance is composed of a resistor R_{\mathrm{r}} in series with a reactance X_{\mathrm{r}} at the fundamental frequency. The voltage inverter is powered by a DC voltage V_DC issued by the booster chopper. This converter uses a bang-bang regulator current.

Control power with reference voltage injected in the grid

The identified PV generator which is composed of 60 parallel and 20 series polycrystalline modules is used. Each module is formed by 36 series cells. For fixed radiation equal to 1000 W/m² at 25°C the active and reactive powers converge to their references. The difference comes from natural losses in the transmission line, modelled by the impedance {\overline{Z}}_{\mathrm{r}}, (Figs. 13 and 14).

For a scenario of insulation of Fig. 15, the reference power P_{\mathrm{r}\mathrm{e}\mathrm{f}} is defined by the radiation. It is noted that the control is at zero reactive power Q_{\mathrm{r}\mathrm{e}\mathrm{f}}. The active and reactive powers converge to their references. The difference comes from natural losses in the transmission line, modelled by the impedance {\overline{Z}}_{\mathrm{r}} (Fig. 16). Figure 17 gives the evolution of voltages (reference and inverter) for the proposed scenario. These quantities evolve according to the MPPT operation. The output voltage of the inverter is taken out after and before filtering, as shown in Fig. 18. Effeciency of transferred power to the grid with voltage control is shown in Fig. 19.

Control power with reference current injected in the grid

For the same PV generator used in voltage control and for a fixed radiation equal to 1000 W/m² at 25 °C the active and reactive powers converge to their references. The difference comes from natural losses in the transmission line, modelled by the impedance {\overline{Z}}_{\mathrm{r}} ( Fig. 20). Figures 21 and 22 represent the evolution of currents (reference/ measure) injected in the grid and the zoom in evolution of currents (reference/measure) injected in the grid respectively.

Filter and grid impedance is chosen:

R_r = 0.1 Ω, L_r =2.5×10^-3 H.

For the same scenario of insulation indicated in Fig. 15, the reference power P_{\mathrm{r}\mathrm{e}\mathrm{f}} is defined by the radiation. It is noted that the control is at zero reactive power Q_{\mathrm{r}\mathrm{e}\mathrm{f}}. The active and reactive powers converge to their references (Fig. 23). The difference comes naturally losses in the transmission line, modelled by the impudence {\overline{Z}}_{\mathrm{r}}. The zoom in evolution of current (reference/measure) injected in the grid, the evolution of the current injected in the grid , and the efficiency of the transferred power to the grid with current control are, respectively, shown in Figs. 24, 25, and 26.

Conclusion

A control of a PV energy conversion connected to the three-phase grid has been proposed. An algorithm based on the gradient method to determine model parameters of the PVG from a set of experimental measurement is presented. It is important to note that the series and the shunt resistances depend on the radiation. These changes are taken as a function of the radiation in the control algorithm. From the experimental characteristics, a relationship has been deduced between the optimal values V_p and I_p which are used in simulation computation where the MPPT controller updates the reference voltage at every zero-crossing point of the grid voltage using the average array voltage and the average estimated current to get the solar array power which is transferred to the grid using two approaches. In the first approach, the PVECS with voltage reference is computed with a proposed scenario of radiation. It is noted that the active and reactive powers converge to their references. The proposed system is simple and shows superior performances under the parameter variation environments. In the second approach, the PVECS with current reference is employed and good performance is obtained. However it shows some harmonic distortion.

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