# A Study of a DC/AC Conversion Structure for Photovoltaic System Connected to the Grid with Active and Reactive Power Control

Authors: Houda Brahmi; Rachid Dhifaoui
Journal: Complexity
 (2021)

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## Abstract

DC/AC conversion of photovoltaic energy is in great demand for AC applications; the supply of electrical machines and transfer energy to the distribution network is a typical case. This work is realized in this context and presents a new structure for the transfer of photovoltaic energy to the electricity grid. This structure is based on a push-pull converter connected to a three-phase DC/AC inverter. In particular, a great interest is focused on the steady operating conditions of energy transfer. The study also develops the limits and the feasibility of the PV energy transfer to the grid of the proposed structure. Injecting photovoltaic energy into the grid with maximum active power and zero reactive power is also considered for the dynamic regime. Support simulations are carried out to validate the proposed control strategy.


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## Body

## 1. Introduction

Renewable energies are inexhaustible and clean and have the additional advantage of complementing each other. This favours and encourages their integration in the AC grid. Among these sources, photovoltaic solar energy offers a very promising alternative [1–5]. It is experiencing a wide development, favoured by the awareness of governments and other organizations seriously calling to preserve the environment.Relatively for a long time, photovoltaic solar energy was used as a power source only for certain loads, such as satellites and/or rural areas situated far from conventional electricity transmission lines [6–9]. Nowadays, from the environmental concern of modern society, economical point of view, and technology advances, the interest in solar energy and associated conversion systems has emerged as good future solution. Furthermore, photovoltaic solar energy has the greatest potential for use in various ways. Due to its modular nature, it can be used in the country side and in the city, in small and large places [10–12].Basically, there are two types of solar energy applications: isolated systems and grid connected systems [13, 14]. For the first case, we found a lot of applications: isolated and independent houses or equipment, rural power plants, telecommunication networks, water pumping, lighting systems, computers or cell phones, cameras, calculators, etc. Surely, photovoltaic solar energy can offer a more significant contribution in many countries where electrification levels are close to saturation [15–17]. Furthermore, many countries in Europe and Africa have great chance to produce photovoltaic energy in very high quantities. This is why the second case dealing with connexion of solar photovoltaic energy to the electricity grid shows a high growing rate during the last decades. At present, the governments of various European countries, Japan and the United States, among others, are promoting economic incentives for renewable energies to try to cope with climate change [18–20]. In this sense, there are special credits to finance photovoltaic installations for connection to the grid and remuneration per kWh sold to the electricity company.To generate photovoltaic solar electricity, a photovoltaic generator is needed; that is, a set of solar panels are connected in series and/or in parallel to produce the desired direct current. Then, to produce alternating current (at rated voltage 220 V and rated frequency 50 Hz, for example), a DC/AC inverter is required. Thus, photovoltaic inverters are installed between the photovoltaic generator and the point of connection to the grid. Energy transfer from the DC side to the AC side is submitted to various constraints such as secure operating conditions, economical efficiency, and environmental benefits [21, 22].As part of the specific grid-related operation requirements, the PV inverter must also operate within the output voltage and frequency ranges satisfying the tolerated harmonic distortion of the grid voltage wave. From a signal quality point of view, main technical issues to consider in an inverter are in fact power factor and harmonic distortion. The inverter must also have admissible galvanic insolation level between the network and the photovoltaic installation [23–26]. On the other hand, the global efficiency of a grid connected photovoltaic system depends largely on the efficiency of the inverter that in turn depends on circuit topology and control scheme. Generally speaking, this control must ensure various active power operating points at minimum possible reactive power [27, 28].The study realized in this paper is organized as follows. Section2 deals with the studied energy transfer structure. It gives a general view and a simplified description of main components of this structure. Section 3 details the used mathematical model of solar cells forming the photovoltaic generator. This model is relatively an enhanced one because it includes internal series and parallel resistors and is described by a unique nonlinear equation to solve by iterative numerical process. The first part of Section 4 defines the mathematical model connecting photovoltaic generator to a simple electrical circuit describing the grid bus bar. Then, it presents the steady-state regime and determines useful constraint to ensure the existence of realistic energy transfer solution. The system to solve is also presented and commented. The second part of Section 4 describes the control structure to solve in dynamic regime. The dynamic model of the grid is detailed and the procedure to impose active and reactive powers reference values is outlined. It also introduces how to incorporate corrective terms in voltage reference value. Finally, Section 5 presents and comments on obtained dynamic response signals. The degree of acceptance of the results and their accuracy are also discussed.


## 2. Presentation of the Proposed Structure

The studied structure is shown in Figure1 and it is composed of the following:(i)
A photovoltaic generator whose voltage and current are denoted, respectively, in the usual way byVp and Ip. Thus, its output power is(1)Ppv=VpIp.(ii)
A DC/DC push pull voltage converter with a fixed ratiom. It is assumed that it delivers a sufficiently smooth and constant voltage Vdc proportional to input voltage Vp such that(2)Vdc=mVp.(iii)
A DC/AC three-phase inverter considered with a large inductance to sufficiently smooth the current supplied by the high voltage DC side of the DC/DC inverter(iv)
A three-phase voltage source representing the grid (distribution network)Figure 1
The studied energy conversion structure.Some working hypotheses are here considered. Surely, the goal is first to facilitate the study. On the other side, we keep in mind that the final target is focused on control feasibility and not to study detailed models of structure components. Thus, we assume that the two power inverters are lossless and alternative current signals are very close to a sinusoidal wave form so that they are treated within the framework of the first harmonic. This hypothesis can be validated by insertion of a filter circuit. Finally, we consider that the input impedance of the grid is very low to ovoid inadmissible voltage drop. In this context, the rms voltageV of the grid is linked to photovoltaic input voltage Vp by the following expression where ρ holds for the adjustable duty cycle that varies between 0 and 1:(3)V=ρVdc2=mρVp2=γVp,γ=mρ2.


## 3. PV Array Modelling

Figure2 presents the refined model of a PV cell [29–32]. The equivalent electrical circuit is formed by a current source depending on the solar radiation, a diode characterized by a saturation current I0 and an idealistic factor q, and two resistors modelling the joule losses in the cell Rs and Rsh, respectively. Current Iph emitted by the cell under the effect of solar radiation (the current of photons) is strongly related to the intensity of the solar radiation but it depends very slightly on the temperature variation. The current ID of the diode is strongly related to the temperature and to the energy of the gap of the junction. It also depends on the voltage across the diode. The shunt current Ish corresponds to the current flowing through the shunt resistance Rsh usually considered very high.Figure 2
Electrical model of a solar cell.Within the meaning of this electrical model given in Figure2 and the considered notations listed in Abbreviations, we can write the following relations:(4)Iph=ID+Ip+Ish=Id+Ip+Vp+RsIpRsh,ID=IsexpVp+RsIpVT−1,VT=KIKBqTj.The electrical behaviour under load of the solar cell will be described by the following relation:(5)fIp,Ip=Ip−Iph+IsexpVp+RsIpVT−1+Vp+RsIpRsh=0.This system being nonlinear, we solved it by the iterative Newton–Raphson method which is the most appropriate tool for this case of systems.


## 4. Transfer of Photovoltaic Energy to the Grid

### 4.1. Study of Transfer Conditions in Steady State

In a general and simplified way, we deal with the circuit of Figure3 representing a voltage source vt supplying a AC load composed by a second voltage source et placed behind an internal impedance. Such a circuit could correspond to a portion of an electrical energy distribution network [33, 34]. The most important thing here is to consider that the short-circuit power of this load is very important compared to the limit power of the source, that is to say, the conversion chain formed by the photovoltaic generator and the two static converters. This assumption makes it possible to consider the voltage source constant in amplitude and frequency whatever the operating point.Figure 3
Single-phase equivalent diagram of the studied load.During the steady state, Park’s vectors merge with Fresnel’s ones. Thus, for such a regime, we establish vector diagram of Figure4 and consequently deduce the relations below. It is to be noted that these equations are generated in a synchronously rotating reference frame associated with voltage vector E¯. In this situation, direct and quadrature voltage and current components are noted by capital characters.(6)Vd=rId−xIq+E,Vq=rIq+xId.Figure 4
Vector diagram of the load in steady state.Moreover, taking into account relation (1) and the power conservation imposed by the assumption of ideal static converters, system (7) is established where P holds for the power delivered by the photovoltaic generator:(7)V2=γ2Vp2=rId−xIq+E2+rIq+xId2,Pg=VpIp+rI2=EId+rId2+Iq2.For aVp,Ip point taken on the P=fV curve of the photovoltaic generator and for a specified value of the control variable γ, the two equations above are sufficient to determine the two variables Id and Iq and to deduce from them the values of active power and reactive power at the source bus bar E¯ and at the inverter output terminals. At the source bus bar E¯, the active power P and the reactive power Q are calculated by(8)P=EId,(9)Q=−EIq.The choice of the control valueγ plays a crucial role in the existence of a solution; this is on one hand, and on the other hand, it plays a crucial role in the degree of satisfaction with the solution if it exists. In fact, we could find a solution which would correspond, for example, to an inadmissible balance in reactive power and in this case the selected value is rejected. When the photovoltaic generator works in current generator mode, that is to say, when its voltage is low, the voltage at the output of the inverter would also be low and the load would be called upon to supply reactive power to the conversion chain. This is of course an undesirable regime on several levels: current overload, drop in voltage, and active power.The strategies largely preferred in the field of integrating photovoltaic energy into the network consist in imposing zero reactive power on the bus bar of this network. It is precisely this type of control that we are interested in, in this study. A control structure is developed to impose a reference active power valuePref, which is as high as possible, and a zero reference reactive power Qref at the point where the source E representing the electrical network is located. According to (8), to obtain null reactive power at the input of the electrical network, the quadrature component Iq of the current must also be zero. So, by imposing this condition in (7)–(9), we obtain(10)γ2Vp2=z2Id2+2rEId−2xIqE+E2,(11)Pg=VpIp=EId+rId2.By multiplying (10) by r/z2 and subtracting (11), we get(12)Id=z2VpIp−rγ2Vp2−E2x2−r2E.Then, by combining (11) and (12) and thus eliminating the current component Id, one obtains the following unique equation:(13)x2−r22Pg=x2−r2z2Pg−rγ2Vp2−E2+rE2z2Pg−rγ2Vp2−E22.For a set of parametersr,x,E chosen suitably, this equation defines in the plane Vp,Pg a curve branch which could cut the characteristic of the photovoltaic generator at a certain operating point as illustrated in Figure 5.Figure 5
Zero reactive power control solutions.If we place at the point of the no-load regime, that is to say,Vp=Voc and P=Q=0, the preceding equation implies the trivial solution:(14)γVp=γVoc=E.It follows that the minimum theoretical value ofγ guaranteeing the existence of an operating point of the photovoltaic generator for the climatic working conditions is(15)γmin=EVoc.Any voltage ratio inferior to this critical value makes it impossible to transfer active power to the sourceE while respecting the zero reactive power. However, it is obvious that the best choice of γ is the one allowing the transfer of the maximum power. Let us say, for example, that we are at the optimum operating point of the photovoltaic generator where voltage, current, and power are, respectively, Vm, Im, and Pm. From (10), we can calculate the value of the direct current Id and deduce the optimum power. Substituting the result found in (10) allows us to deduce the value of the control variable γ to be used. If the short-circuit impedance of the network is low enough, which is usually the case, this value would almost correspond to the optimal value:(16)γm=EVm.As the global voltage ratio includes the turns ratiom of the push pull transformer, it is evident that the feasibility condition (15) turns into a constraint on m. This means that a particular attention must then be given to the sizing of the minimum value of the DC/DC inverter. If such a precaution is not taken into account, the duty cycle of the inverter checking the operating point would be greater than 1 and the control structure would also fail because this ratio is of course saturated at 1. From this we understand that it is important to carry out a preliminary scan taking into account the limiting climatic variations in order to decide on the minimum value of the DC/DC converter ratio m. In a formal way, we pose the constraint below where γc is a value verifying the transfer of the optimal power within the largest variation in climatic conditions:(17)m>2γc.The consideration of grid parametersr,x,E in the analysis of the possibilities of integrating photovoltaic energy is an aspect of great importance. To be able to transfer the power available in the photovoltaic source to the grid, it is essential to know in advance the limit level of this power and to check the effective possibility of transferring it. By this we mean that a preliminary calculation of the transmissible power is very necessary to ensure the transfer properly. In the case, for example, where the capacity for transferring power to the network is less than the power to be transferred, the control structure would regain certain failure. In terms of the power limit, and therefore in terms of the existence of a transfer solution, the following condition must be respected:(18)P<E22z−r−rI2.Finally, the operating point corresponds to the simultaneous resolution of the three equations of the system (19) below by the Newton–Raphson method in terms of the three variables Vp, Ip, and Id, for a specified value of γ. The details of this method are given in Figure 6. From the obtained value of Id, one easily deduces the value of the power P transmitted to the source E of the network.(19)Ip−Iph+IsexpVp+RsIpVT−1+GshVp+RsIp=0,z2Id2+2rEId+E2−γ2Vp2=0,EId+rId2−VpIp=0.Figure 6
Chart diagram of Newton–Raphson method.


### 4.2. Control Structure in Dynamic Regime

It is important to show the feasibility of the problem in variable regime through a control strategy. This is where the objective of this section lies. To approach this problem in the simplest and clearest way possible, only the dynamics of the control loop of the inverter and that of the load are here taken into account. It is assumed that the dynamics of the DC/DC converter upstream of the inverter is relatively slow because of its output large filter capacitor.The proposed control structure is indicated by Figure7. This structure comprises a set of modules associated with the necessary mathematical models. Module 1 relates to the photovoltaic generator. Its inputs are the variables illustrating the climatic conditions: ambient temperature Ta and solar radiation Es. Its outputs are current Ip and voltage Vp which depend on the power level of the load and input variables Ta and Es. Module 2 represents the DC/DC push pull converter, its output voltage is linked to photovoltaic generator voltage by equation (2). Module 3 generates reference voltage v¯ref vector with a module ρref and phase δref with respect to the grid synchronous angle. These two quantities come from module 7 where the control part resides. Module 4 realises space pulse width modulation (SPWM) signals corresponding to switching times of power transistor of the inverter schematised by module 5. Output signals of this module are two quadratic voltage components in Concordia’s representation. Module 6 solves the differential equation (20) below written in the stationary Concordia’s reference frame. When they appear, Concordia’s voltage and current components are noted with small characters. Relation (21) is to consider that the voltage at grid bus bar is purely sinusoidal.(20)di¯dt=v¯−ri¯−e¯ℓ,(21)e¯=Eexpjωt.Figure 7
The developed control structure.In Park’s rotating reference frame synchronized with the network voltage sourceE¯, the dynamics of the components of the current transmitted is expressed by(22)dIddt=Vd−E−rId+xIqℓ,dIqdt=Vq−rIq−xIdℓ.These current components must converge in steady state towards their reference values imposed by the required active powerPref and reactive power Qref:(23)Idref=PrefE,Iqref=QrefE=0.The output voltage components of the inverter are also known in steady state:(24)Vdref=E+rIdref,Vqref=xIdref.For the transient state, one uses these relations to introduce corrective terms compensating the effect of the dynamic variation of the current components. The added corrective terms are as follows, whererc is a constant whose value will be provided in the part reserved for the simulation:(25)ΔVdref=rcIdref−Id,ΔVqref=−rcIq.We then consider a reference voltage such as(26)Vdref=E+rIdref+rcIdref−Id,Vqref=xIdref−rcIq.This fully defines the inputs of the inverter control module which are the voltage duty cycleρref and its phase δref in the grid synchronous frame:(27)ρref=2Vdref2+Vqref2mVp,δref=tg−1VqrefVdref.


## 4.1. Study of Transfer Conditions in Steady State

In a general and simplified way, we deal with the circuit of Figure3 representing a voltage source vt supplying a AC load composed by a second voltage source et placed behind an internal impedance. Such a circuit could correspond to a portion of an electrical energy distribution network [33, 34]. The most important thing here is to consider that the short-circuit power of this load is very important compared to the limit power of the source, that is to say, the conversion chain formed by the photovoltaic generator and the two static converters. This assumption makes it possible to consider the voltage source constant in amplitude and frequency whatever the operating point.Figure 3
Single-phase equivalent diagram of the studied load.During the steady state, Park’s vectors merge with Fresnel’s ones. Thus, for such a regime, we establish vector diagram of Figure4 and consequently deduce the relations below. It is to be noted that these equations are generated in a synchronously rotating reference frame associated with voltage vector E¯. In this situation, direct and quadrature voltage and current components are noted by capital characters.(6)Vd=rId−xIq+E,Vq=rIq+xId.Figure 4
Vector diagram of the load in steady state.Moreover, taking into account relation (1) and the power conservation imposed by the assumption of ideal static converters, system (7) is established where P holds for the power delivered by the photovoltaic generator:(7)V2=γ2Vp2=rId−xIq+E2+rIq+xId2,Pg=VpIp+rI2=EId+rId2+Iq2.For aVp,Ip point taken on the P=fV curve of the photovoltaic generator and for a specified value of the control variable γ, the two equations above are sufficient to determine the two variables Id and Iq and to deduce from them the values of active power and reactive power at the source bus bar E¯ and at the inverter output terminals. At the source bus bar E¯, the active power P and the reactive power Q are calculated by(8)P=EId,(9)Q=−EIq.The choice of the control valueγ plays a crucial role in the existence of a solution; this is on one hand, and on the other hand, it plays a crucial role in the degree of satisfaction with the solution if it exists. In fact, we could find a solution which would correspond, for example, to an inadmissible balance in reactive power and in this case the selected value is rejected. When the photovoltaic generator works in current generator mode, that is to say, when its voltage is low, the voltage at the output of the inverter would also be low and the load would be called upon to supply reactive power to the conversion chain. This is of course an undesirable regime on several levels: current overload, drop in voltage, and active power.The strategies largely preferred in the field of integrating photovoltaic energy into the network consist in imposing zero reactive power on the bus bar of this network. It is precisely this type of control that we are interested in, in this study. A control structure is developed to impose a reference active power valuePref, which is as high as possible, and a zero reference reactive power Qref at the point where the source E representing the electrical network is located. According to (8), to obtain null reactive power at the input of the electrical network, the quadrature component Iq of the current must also be zero. So, by imposing this condition in (7)–(9), we obtain(10)γ2Vp2=z2Id2+2rEId−2xIqE+E2,(11)Pg=VpIp=EId+rId2.By multiplying (10) by r/z2 and subtracting (11), we get(12)Id=z2VpIp−rγ2Vp2−E2x2−r2E.Then, by combining (11) and (12) and thus eliminating the current component Id, one obtains the following unique equation:(13)x2−r22Pg=x2−r2z2Pg−rγ2Vp2−E2+rE2z2Pg−rγ2Vp2−E22.For a set of parametersr,x,E chosen suitably, this equation defines in the plane Vp,Pg a curve branch which could cut the characteristic of the photovoltaic generator at a certain operating point as illustrated in Figure 5.Figure 5
Zero reactive power control solutions.If we place at the point of the no-load regime, that is to say,Vp=Voc and P=Q=0, the preceding equation implies the trivial solution:(14)γVp=γVoc=E.It follows that the minimum theoretical value ofγ guaranteeing the existence of an operating point of the photovoltaic generator for the climatic working conditions is(15)γmin=EVoc.Any voltage ratio inferior to this critical value makes it impossible to transfer active power to the sourceE while respecting the zero reactive power. However, it is obvious that the best choice of γ is the one allowing the transfer of the maximum power. Let us say, for example, that we are at the optimum operating point of the photovoltaic generator where voltage, current, and power are, respectively, Vm, Im, and Pm. From (10), we can calculate the value of the direct current Id and deduce the optimum power. Substituting the result found in (10) allows us to deduce the value of the control variable γ to be used. If the short-circuit impedance of the network is low enough, which is usually the case, this value would almost correspond to the optimal value:(16)γm=EVm.As the global voltage ratio includes the turns ratiom of the push pull transformer, it is evident that the feasibility condition (15) turns into a constraint on m. This means that a particular attention must then be given to the sizing of the minimum value of the DC/DC inverter. If such a precaution is not taken into account, the duty cycle of the inverter checking the operating point would be greater than 1 and the control structure would also fail because this ratio is of course saturated at 1. From this we understand that it is important to carry out a preliminary scan taking into account the limiting climatic variations in order to decide on the minimum value of the DC/DC converter ratio m. In a formal way, we pose the constraint below where γc is a value verifying the transfer of the optimal power within the largest variation in climatic conditions:(17)m>2γc.The consideration of grid parametersr,x,E in the analysis of the possibilities of integrating photovoltaic energy is an aspect of great importance. To be able to transfer the power available in the photovoltaic source to the grid, it is essential to know in advance the limit level of this power and to check the effective possibility of transferring it. By this we mean that a preliminary calculation of the transmissible power is very necessary to ensure the transfer properly. In the case, for example, where the capacity for transferring power to the network is less than the power to be transferred, the control structure would regain certain failure. In terms of the power limit, and therefore in terms of the existence of a transfer solution, the following condition must be respected:(18)P<E22z−r−rI2.Finally, the operating point corresponds to the simultaneous resolution of the three equations of the system (19) below by the Newton–Raphson method in terms of the three variables Vp, Ip, and Id, for a specified value of γ. The details of this method are given in Figure 6. From the obtained value of Id, one easily deduces the value of the power P transmitted to the source E of the network.(19)Ip−Iph+IsexpVp+RsIpVT−1+GshVp+RsIp=0,z2Id2+2rEId+E2−γ2Vp2=0,EId+rId2−VpIp=0.Figure 6
Chart diagram of Newton–Raphson method.


## 4.2. Control Structure in Dynamic Regime

It is important to show the feasibility of the problem in variable regime through a control strategy. This is where the objective of this section lies. To approach this problem in the simplest and clearest way possible, only the dynamics of the control loop of the inverter and that of the load are here taken into account. It is assumed that the dynamics of the DC/DC converter upstream of the inverter is relatively slow because of its output large filter capacitor.The proposed control structure is indicated by Figure7. This structure comprises a set of modules associated with the necessary mathematical models. Module 1 relates to the photovoltaic generator. Its inputs are the variables illustrating the climatic conditions: ambient temperature Ta and solar radiation Es. Its outputs are current Ip and voltage Vp which depend on the power level of the load and input variables Ta and Es. Module 2 represents the DC/DC push pull converter, its output voltage is linked to photovoltaic generator voltage by equation (2). Module 3 generates reference voltage v¯ref vector with a module ρref and phase δref with respect to the grid synchronous angle. These two quantities come from module 7 where the control part resides. Module 4 realises space pulse width modulation (SPWM) signals corresponding to switching times of power transistor of the inverter schematised by module 5. Output signals of this module are two quadratic voltage components in Concordia’s representation. Module 6 solves the differential equation (20) below written in the stationary Concordia’s reference frame. When they appear, Concordia’s voltage and current components are noted with small characters. Relation (21) is to consider that the voltage at grid bus bar is purely sinusoidal.(20)di¯dt=v¯−ri¯−e¯ℓ,(21)e¯=Eexpjωt.Figure 7
The developed control structure.In Park’s rotating reference frame synchronized with the network voltage sourceE¯, the dynamics of the components of the current transmitted is expressed by(22)dIddt=Vd−E−rId+xIqℓ,dIqdt=Vq−rIq−xIdℓ.These current components must converge in steady state towards their reference values imposed by the required active powerPref and reactive power Qref:(23)Idref=PrefE,Iqref=QrefE=0.The output voltage components of the inverter are also known in steady state:(24)Vdref=E+rIdref,Vqref=xIdref.For the transient state, one uses these relations to introduce corrective terms compensating the effect of the dynamic variation of the current components. The added corrective terms are as follows, whererc is a constant whose value will be provided in the part reserved for the simulation:(25)ΔVdref=rcIdref−Id,ΔVqref=−rcIq.We then consider a reference voltage such as(26)Vdref=E+rIdref+rcIdref−Id,Vqref=xIdref−rcIq.This fully defines the inputs of the inverter control module which are the voltage duty cycleρref and its phase δref in the grid synchronous frame:(27)ρref=2Vdref2+Vqref2mVp,δref=tg−1VqrefVdref.


## 5. Simulations and Results

To check the performance of the proposed command in dynamic regime, a MATLAB/Simulink model of the PV system was initially developed. The parameters to be used are inspired from a practical project now under development in our research laboratory ERCO (Energy, Robotics, Control and Optimisation) situated at INSAT, Tunis.The photovoltaic generator has been described in a previous work [31]. It is formed by 10 solar panels referenced TITAN STP-12-50, Multi-Cristallin technology. Table 1 reports the necessary parameters at standard climatic conditions.Table 1
Parameters of the PV generator used in simulation.
Solar panel TITAN-12-50cell type: EFG Multi-CristallinNumber of series cells:ns36Number of parallel PV arrays: Np10Number of series PV arrays: Ns1Open circuit voltage in STC (per one PV array)21 VShort-circuit current en STC (per one PV array)3.2 AOptimal voltage in STC (per one PV array)17.2 VOptimal current in STC (per one PV array)2.9 AOptimal power in STC45 W per moduleTemperature25°CRadiation1000 W/m2The DC/DC inverter is a part of a real camping power supply. The reference of this device is KEBO IPS 1200. It is a rated 12 V/240 volts and a rated 500 W power device. This inverter is designed to be connected to a car battery of 12 volts. To meet this voltage constraint, solar panels are connected in parallel. In addition, the DC/DC inverter has being modified in our laboratory so that some input chemical capacitors support 25 dc volts. At a very low load, close to open circuit, we have verified that the voltage ratio ism=22.2.The global structure given by Figure7 is simulated under the following conditions:(i)
Climatic conditions: temperatureTa=25°C and a solar radiation Es=1000W/m2(ii)
Push-pull converter with fixed ratio:m=22.2(iii)
PWM frequency:5kHz(iv)
Reference reactive powerQref=0(v)
Reference active powerPref=408W: this equals 97% of the maximum power of the photovoltaic generator(vi)
Filtering period:m=200μs(vii)
Grid parameters:r=1Ω,x=5ΩFigures8 and 9 show the structure dynamic response in terms of active power and reactive power control. These variables suitably converge towards the requested references in a sufficiently short response time. Active power convergence is good. However, reactive power oscillates around zero. This is due to the fact that the internal impedance of the source representing the network is low, which generates a phenomenon of permanent current ripple around its fundamental sinusoidal component. We can easily observe that absolute reactive power variation remains within 20 vars that is inferior to 10% of final active power value. In general point of view, this performance gives full satisfaction. In theoretical point of view, oscillations can be justified by the fact that the reference voltage is defined in a pseudostatic form on one hand and the value of the compensation resistor rc is sufficiently large on the other hand.Figure 8
Dynamic evolution of active power.Figure 9
Dynamic evolution of reactive power.Figure10 shows the evolution of the control variable γ which is proportional to the duty cycle of the inverter. We can easily see that this value remains within the range specified by the steady-state study. In fact, it converges to a value around γ≃15 which means that the voltage ratio is about ρ≃95%.Figure 10
Evolution of the variable controlγ.The phase shift of the voltage of the inverter with respect to the synchronous axis of the network is of the order ofδ = 2.5° in steady state as indicated by Figure 11. This value, which seems very low, is well justified. The maximum power that can be transmitted to the network is inversely proportional to its internal impedance; see expression (27). This power is here very large as compared to the power requested.Figure 11
Dynamic evolution of the angleδ.Figure12 gives a zoom on the instantaneous waveform of the current. Although the average value of the current remains generally sinusoidal, we observe the chopping due to the sudden switching of the output voltage of the inverter. Surely, this waveform is not of sufficient degree of acceptance. But we are also sure that the result can be enhanced. Three factors can contribute to reaching this goal. In fact, adding a filter circuit, realising accurate calculation of the transfer impedance, and finding optimal switching frequency could be a real solution.Figure 12
The steady state of the load current.Figure13 gives a zoom on the steady state of the voltage signals. The central curve of this figure indicates the instantaneous average value which naturally characterizes the reference voltage. Figures 14 and 15 correspond, respectively, to the evolution of the current and the voltage of the photovoltaic generator. We can also prove through these curves the convergence of the power delivered towards the requested reference.Figure 13
The steady state of the load voltage.Figure 14
The steady state of the load current.Figure 15
Evolution of the voltage of the photovoltaic generator.We have previously shown that is possible to obtain a steady-state operating point characterized by an active powerPref=408W and null reactive power with a global voltage ratio of γ≃15. Imposing γ=12 in an open loop simulation case leads to inadmissible oscillations and shows impossibility to transfer the requested power. This result is summarised in Figures 16 and 17.Figure 16
Dynamic evolution of the active power forγ=12.Figure 17
Dynamic evolution of the reactive power forγ=12.


## 6. Conclusion

The aim of this analysis was to show the feasibility of transferring photovoltaic energy to the electrical grid with control of active and reactive powers. The proposed structure is based on a push-pull converter connected to three-phase AC-DC inverter. The steady-state study of the structure in question showed the limits of the energy transfer. Indeed, for a pointVp,Ip considered on the characteristic of the photovoltaic generator and for a specified value of the control variable γ, we can deduce the value of the active power and of the reactive power both at the level of the alternative current bus bar and at the output terminals of the inverter.The choice of the value ofγ plays a crucial role on one hand in the existence of a solution and on the other hand in the degree of satisfaction of the solution if it exists. Indeed, one could find a solution which would correspond, for example, to an inadmissible balance in reactive power. When the photovoltaic generator works in current generator mode, the voltage at the output of the inverter would also be low and the load would be called upon to supply reactive power to the conversion chain. This is obviously an undesirable regime on many levels: current overload, drop in voltage and active power, and being very profitable on the network side of the distribution. On the contrary, when the photovoltaic generator works in voltage generator mode, with the contribution of the adjustment on the converter side, we could have an output voltage greater than the voltage E of the network. In such a case, the conversion chain would produce reactive power. The mathematical model of the whole system with a three nonlinear equations describing the behaviour in static mode was formulated. The method of Newton–Raphson was used in order to solve this system numerically. The last point addressed in this work is the study of the dynamic regime during energy transfer. The command developed aims to ensure the transfer of maximum active power and zero reactive power. We have detailed the control strategy adopted; simulations in dynamic regime have shown the effectiveness of the control law.



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*Source: 9967577-2021-07-07.xml*