# Analysis of a Three-Phase Grid-Connected PV Power System Using a Modified Dual-Stage Inverter

Authors: Denizar Cruz Martins
Journal: ISRN Renewable Energy
 (2013)

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## Abstract

This paper presents a grid-connected PV system in a centralized configuration constructed through a three-phase dual-stage inverter. For the DC-DC stage the three-phase series resonant converter is chosen thanks to the advantages that it exhibits. However, it is inadequate for the accomplishment of MPPT, due to its efficiency strongly depending on the implemented deadtime and switching frequency. Then, this paper proposes a conceptual modification, that is, a modified dual-stage inverter in which the inverter stage is responsible for both the MPPT and the grid-current control. In addition, the DC-DC converter operates with constant duty cycle and frequency. Such configuration requires a new concept, introduced as Behavior Matching. It serves as a fundamental feature for the DC-DC converter to reproduce the PV arrayI–V characteristic when they are connected, without control action. The maximum power operating point is found by maximizing the direct axis current, obtained by Park's transformation from the inverter, through the perturbation and observation algorithm. Any specific measurement to realize the MPPT is needed. The galvanic isolation is achieved by using a high-frequency transformer. The structure is appropriate for high power applications, above 10 kW.


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## Body

## 1. Introduction

The photovoltaic solar energy represents an emergent technology in function of the continuous fall in the production costs and in the technological progress of the PV modules. This alternative energy can significantly contribute with the reduction in the emission of greenhouse gases in the atmosphere, which attack the environment deeply.Around 75% of the PV systems installed in the world are grid connected [1]. In the grid-connected PV system, DC-AC converters (inverters) need to realize the grid interconnection, inverting the dc current that comes from the PV array into a sinusoidal waveform synchronized with the utility grid [2, 3]. Besides, the DC-AC converter is used to stabilize the dc-bus voltage to a specific value, because the output voltage of the PV array varies with temperature, irradiance, and the effect of MPPT (maximum power point tracking) [4–23]. The DC-AC conversion systems, depending on its topology, can be classified as presented as follows [24, 25].(i)
Single-stage inverter: in one processing stage, MPPT and grid-current control are handled (Figure1).
(ii)
Dual-stage inverter: a DC-DC converter performs the MPPT and a DC-AC one is responsible for the grid-current controlling (Figure2).
(iii)
Multistage inverter: various DC-DC converters are used for the MPPT and only one DC-AC converter takes care of the grid-current control (Figure3).Figure 1
Single-stage inverter without a DC-DC converter.Figure 2
Photovoltaic solar energy flowing through two conditioning converters.Figure 3
Low power DC-DC converters connected to strings and one large DC-AC in the grid interface.The inverters previously shown give an idea about the control and the DC-DC converters’ application. It is worth to discuss in more details how the PV modules are connected with inverters and these are connected with the grid. There are four configurations commercially accepted [26–30].(i)
Central-plant inverter: usually a large inverter is used to convert DC output power of the PV array to AC power. In this system, the PV modules are serially string and several strings are connected in parallel to a single dc-bus. A single or a dual-stage inverter can be employed. Figure4 illustrates this configuration.(ii)
Multiple-string DC-DC converter: each string has a DC-DC converter, which can be galvanically isolated. There is a common DC link, which feeds a transformerless DC-AC converter. As shown in Figure5, only the multistage inverter can implement this configuration.(iii)
Multiple-string inverter: several PV modules are connected in series on the DC side to form a string. The output from each string is converted to AC through a smaller individual inverter. Many such inverters are connected in parallel on the AC side, as shown in Figure6. A single or a dual-stage inverter can be employed in this kind of configuration.(iv)
Module-integrated inverter: each module has a small inverter, and each one is connected in parallel forming an ac-bus, which is connected to the AC grid. Once more, a single or a dual-stage inverter can be used. Figure7 shows this configuration.Figure 4
Central-plant inverter.Figure 5
Multiple-string DC-DC converter.Figure 6
Multiple-string inverter.Figure 7
Module-integrated inverter.Table1 summarizes the implementation options of a grid-connected PV power system.Table 1
Implementation options of a grid-connected PV power system.
Configurations
Topologies
Single-stage inverter
Dual-stage inverter
Multistage inverter
Central-plant inverter
✓
✓
✗
Multiple-string DC-DC converter
✗
✗
✓
Multiple-string inverter
✓
✓
✗
Module integrated inverter
✓
✓
✗The high efficiency is one of the most important characteristics of a PV inverter. Thus, whenever possible, these inverters are nonisolated electronic circuits, since a transformer imposes an efficiency drop. This efficiency drop is 2% larger for a low than that for a high-frequency transformer [1]. Hence, when grid isolation is mandatory, the incorporation of a high-frequency transformer is a trend. This implies the need for a DC-DC converter in the structure of the PV power system. The isolated ZVS Full-Bridge DC-DC converter [31–38] is usually used at power levels above 750 W [24], to performer both the MPPT and the galvanic isolation. Commonly, its efficiency ranges from 92% to 93% under a 45% to 100% load condition [39]. This performance is not recommended for high power industrial applications. With nonisolated versions, the efficiency can be increased from 96% to 98% [39].As an alternative to Full-Bridge based converters, the three-phase conversion has some advantages, such as the following [40].(i)
Reduced switching stresses of the power semiconductor devices.(ii)
Reduced size and ratings of associated reactive components.(iii)
Better transformer copper and core utilization.As important as high efficiency, it is the inverter cost. The study in [24] indicates the centralization of inverter (central-plant inverter) for reducing cost, according to plant shown in Figure 4.From the context previously presented, and considering to improve the volume and weight of the whole system, a dual-stage inverter configured in a central-plant, according to Figures2 and 4, is proposed in this paper, however, with two basic differences.(
1
) Uses a three-phase DC-DC converter in place of a single-phase. In [40], it is demonstrated that the three-phase conversion has some advantages such as the following.(i)
High-frequency transformer reduction in comparison to the transformer used in a Full-Bridge DC-DC converter, operating with the same switching frequency.(ii)
Increase of three times in the input and output current frequencies, reducing the size of the filters components.(iii)
Better distribution of the losses.The three-phase isolated DC-DC series resonant converter (SRC3) [41, 42] was applied to act as DC-DC stage. Despite the galvanic isolation, the measured efficiency of the DC-DC stage was limited to 96%–97.5% under a 45%–100% load condition.(
2
) The MPPT will not be carried out by a DC-DC stage; it will be performed by the inverter, which is also responsible for the grid-current control. It is important to emphasize that any measurements of voltage or current in the photovoltaic array are made. It is an indirect MPPT, possibly due to the behavior matching between the DC-DC converter output I–V characteristic and the PV array I–V characteristic, when they are connected. Then, behavior matching serves as a fundamental feature for DC-DC converters to reproduce the PV array I–V characteristic without control action. The studies in [43–45] indicate this possibility.The traditional dual-stage inverter implements some control strategy in the DC-DC stage [46–50]. In this paper, a conceptual modification is proposed that is called Modified Dual-Stage Inverter [51, 52]. In this new approach, there is no control in the DC-DC stage, the SRC3 operates with constant frequency and duty-cycle, representing a very high efficiency operation. All the control, MPPT, and grid-current are implemented in the DC-AC stage (inverter) that consists of a three-phase bidirectional power flow PWM voltage source inverter (VSI3). This is the principal power electronics circuit of a Three-Phase Grid-Connected PV Power System. Figure 8 shows the basic idea of a modified dual-stage inverter.Figure 8
Modified dual-stage inverter.The DC-AC stage performs the MPPT through the P and O method [22, 49, 53–58] to maximize the direct axis current, Id, required for the grid current control. The current Id reflects the active power delivered by the photovoltaic array and is expressed through the inverter modeling, using the Park transformation [59, 60]. Then, the inverter output power is maximized without additional sensors. In a single-stage inverter, this principle can also be used. Figure 9 presents the proposal topology for the dual-stage inverter in a three-phase configuration.Figure 9
Three-phase dual-stage inverter in centralized configuration.


## 2. Photovoltaic Array Modelling

It was commented previously that the proposed MPPT is based on the behavior of the photovoltaic array by means of temperature and irradiation variations. Thus, the mathematical model of the PV cells is implemented in the form of a current source controlled by voltage, sensible to two input parameters, that is, temperature (°C) and solar irradiation power (W/m2).An equivalent simplified electric circuit of a photovoltaic cell is presented in Figure10.Figure 10
Equivalent electric circuit of a simplified single-diode model (SSDM) of a PV solar cell.Although it is a simplified version, this equivalent circuit is enough to represent different types of photovoltaic cells when the temperature effects are considered [61]. From [62], it is verified that the cells of polycrystalline material are contemplated. This material is distinguished because information gotten of datasheet of a polycrystalline module is used in the simulation studies; however, the relevant aspects for the control, shown from the modeling, are not only applied to this type of material.In a more complete version, the equivalent circuit of Figure10 has two electrical resistors, Rs and Rp [63–65]. According to [66, 67], both resistors can be neglected. However, it is demonstrated that the series resistor, Rs, has a great impact on the inclination of the I–V characteristic curve, becoming it more accurate between the maximum power operating point and the open circuit voltage. This information can also be found in [68].Expression (1) can be obtained from Figure 10:
(1)I=Iph-Ir·[eq·(V+I·Rs)/npn·k·T-1],
where V and  I: voltage and current across the cell; Iph: photocurrent; Ir: cell reverse saturation current; q: charge of an electron; Rs: intrinsic series resistance of the cell; npn: ideality factor of the p-n junction; k: Boltzmann’s constant; T: temperature.The photocurrent depends on the solar irradiation and the temperature, given by (2):
(2)Iph=[Isc+α·(T-Tr)]·Psun1000,
where Isc: short-circuit current; α: temperature coefficient of the short-circuit current; Tr: reference temperature, for standard condition; Psun: irradiance level. The standard power is 1000 W/m2.The reverse saturation current varies according to the temperature, as shown in (3):
(3)Ir=Irr·(TTr)3·e[((q·EG)/(npn·k))·(1/Tr-1/T)],
where Irr: cell reference reverse saturation current; EG: band-gap energy of the semiconductor used in the cell.These equations can be found in [66, 69]. The solution of (1) takes the characteristic curve for only one photovoltaic cell. However, the model is such that, if connected in a PV array form, it can be treated as only one cell with multiple associations in series and parallel [68]. Thus, the photovoltaic array, corresponding to two parallel-connected strings, is simulated. Each string contains ten modules, which approximately produce the operation voltage of 263 V. Therefore, it is found that a 4 kWp array formed by KC200GT modules from Kyocera. Figures 11 and 12 reflect the behavior obtained with the PV array modeling which is connected to DC-DC converter.Figure 11
Current-voltage characteristics of photovoltaic array at various irradiance levels.Figure 12
Current-voltage characteristics of photovoltaic array at various temperatures.


## 3. Dual-Stage Inverter

It is desired the system to be suitable for high power applications, above 10 kW. Thus, the two stages of the inverter are three-phase configurations.


### 3.1. DC-DC Stage

When the single-phase DC-DC converter is replaced by a three-phase one, several advantages appear [40]:(i)
faster response times;(ii)
low stresses on active devices;(iii)
filters components reduction;(iv)
high-frequency transformer reduction.Amongst the three-phase DC-DC converters, most attractive they are those that present soft commutation. This characteristic is important due to be high switching frequency in which they can operate, keeping the high efficiency and the heat sinks size in reasonable levels. In addition, with the rise of the switching frequency, a significant reduction in the transformer size and weight is obtained.Various three-phase DC-DC converters with soft commutation are available [41, 42, 70–72]. Each of them was evaluated about the efficiency in different input power levels, components number, EMI emission, performance under unbalanced conditions, and power range. The topology that appears to relate the best characteristics for the proposal application in this paper is presented in Figure 13, obtained from [42]. The transformer was replaced by their leakage inductances, Ld.Figure 13
DC-DC three-phase series resonant converter (SRC3).The switches are gated by 6 phase-shifted signals. Neglecting the deadtime between two switches in each inverter leg, all switches are turned on exactly half a period.When the switching frequency,fs, is equal to resonance frequency, fr, the converter operates in ZCS. If fs>fr, the converter operates in ZVS. In this condition, the efficiency is much reduced for low-power transfer [42, 73]. Then
(4)fs=fr=12·π·Ld·Cr,(5)Iin=Ipv=6π2·Rloss(Vin-Vdc′),
where Iin: average input current; Ipv: PV array current; Vdc′: average output voltage of the three-phase bridge rectifier, referred to the primary side; Vin: average DC-DC converter input voltage; Rloss: take all losses into account, such as the conduction and switching losses of the switches and diodes, the dielectric losses of the capacitors, the copper and iron losses of the three-phase transformer, and the conduction losses of wires and connections; Ld: leakage inductance of the transformer; Cr: resonant capacitor.The transformer turn’s ratio can be defined by the following:(6)n=N2N1,
where N1: number of turns on the transformer primary winding; N2: number of turns on the transformer secondary winding.From (5) and (6), (7) and (8) can be found:
(7)Rloss=6π2·Iin(Vin-Vdcn),(8)Vin=π26·Rloss·Iin+Vdcn.A peculiarity about this converter is that, despite the many advantages that it presents, it is inadequate for the accomplishment of MPPT, due to that its efficiency extremely depends on the implemented deadtime and switching frequency [42]. These variables should not vary. That deficiency can be neglected thanks to the Behavior Matching [44, 45, 51, 52, 74].


### 3.2. DC-AC Stage

Two new trends for this stage are discussed in [75, 76], using LCL filters on a grid interface and replacing the conventional three-level PWM inverter by a multilevel inverter. The objectives are the same: lower total harmonic distortion (THD) and more compact designs. However, for the application presented in this paper, the three-phase current-controlled voltage-source inverter with L filter is adequate.The inverters used for grid interfacing are broadly classified as voltage-source inverters (VSI) and current-source inverters (CSI). The control schemes can be classified as current-controlled inverters (CCI) and voltage-controlled inverters (VCI).PV solar arrays are fairly good approximation to a current source. Then, most PV inverters are voltage source. In addition, with vector modulation [77–82], the THD measured is insignificant when harmonics below the switching frequency are considered.The current-controlled scheme objective is to control active and reactive components of the current fed into the grid. The current controllers are better suited for the control of power export from PV inverters to the utility grid since they are less sensitive to errors in synchronizing sinusoidal voltage waveforms [83]. The structure of the three-phase current-controlled voltage-source inverter is shown in Figure 14.Figure 14
Three-phase current-controlled VSI.Performing a Park transformation, its dynamical model in the dq-frame can be described by the following:(9)dIddt=VdL+ω·Iq-RL·Id-VdcL·Dd,dIqdt=-ω·Id-RL·Iq-VdcL·Dq,dVdcdt=-IdcCf2+IdCf2·Dd+IqCf2·Dq,
where Id, Iq, Vdc: state variable; Dd, Dq: control variables; L=L1=L2=L3; R=R1=R2=R3: intrinsic resistance of the inductance L1,L2,L3, respectively; Vd,  ω: utility grid parameters (Vd=3·Vg, and Vg is the grid rms phase voltage, and ω=2πf, where f is the grid frequency); Idc: average output current of the three-phase bridge rectifier, that is, the average input current of the three-phase VSI.SincePg=Vd×Id represents the power that the inverter injects in the grid, the MPPT can be performed perturbing Vdc and observing Id, according to the following:
(10)Vdc(k+1)=Vdc(k)+ΔV·sign[Vdc(k)-Vdc(k-1)]·sign[Id(k)-Id(k-1)].
Setting Vdc, the DC-AC stage defines the DC-DC converter’s input characteristic behavior, who determines the PV array operation point. When Id is maximized, the PV array operates on MPOP. As noted, Id and Vdc, used by MPPT, are state variables employed in the grid-current control, that is, additional hardware is not need. Section 4 explains this behavior in detail, which represents the behavior matching of the proposed dual-stage inverter.


## 3.1. DC-DC Stage

When the single-phase DC-DC converter is replaced by a three-phase one, several advantages appear [40]:(i)
faster response times;(ii)
low stresses on active devices;(iii)
filters components reduction;(iv)
high-frequency transformer reduction.Amongst the three-phase DC-DC converters, most attractive they are those that present soft commutation. This characteristic is important due to be high switching frequency in which they can operate, keeping the high efficiency and the heat sinks size in reasonable levels. In addition, with the rise of the switching frequency, a significant reduction in the transformer size and weight is obtained.Various three-phase DC-DC converters with soft commutation are available [41, 42, 70–72]. Each of them was evaluated about the efficiency in different input power levels, components number, EMI emission, performance under unbalanced conditions, and power range. The topology that appears to relate the best characteristics for the proposal application in this paper is presented in Figure 13, obtained from [42]. The transformer was replaced by their leakage inductances, Ld.Figure 13
DC-DC three-phase series resonant converter (SRC3).The switches are gated by 6 phase-shifted signals. Neglecting the deadtime between two switches in each inverter leg, all switches are turned on exactly half a period.When the switching frequency,fs, is equal to resonance frequency, fr, the converter operates in ZCS. If fs>fr, the converter operates in ZVS. In this condition, the efficiency is much reduced for low-power transfer [42, 73]. Then
(4)fs=fr=12·π·Ld·Cr,(5)Iin=Ipv=6π2·Rloss(Vin-Vdc′),
where Iin: average input current; Ipv: PV array current; Vdc′: average output voltage of the three-phase bridge rectifier, referred to the primary side; Vin: average DC-DC converter input voltage; Rloss: take all losses into account, such as the conduction and switching losses of the switches and diodes, the dielectric losses of the capacitors, the copper and iron losses of the three-phase transformer, and the conduction losses of wires and connections; Ld: leakage inductance of the transformer; Cr: resonant capacitor.The transformer turn’s ratio can be defined by the following:(6)n=N2N1,
where N1: number of turns on the transformer primary winding; N2: number of turns on the transformer secondary winding.From (5) and (6), (7) and (8) can be found:
(7)Rloss=6π2·Iin(Vin-Vdcn),(8)Vin=π26·Rloss·Iin+Vdcn.A peculiarity about this converter is that, despite the many advantages that it presents, it is inadequate for the accomplishment of MPPT, due to that its efficiency extremely depends on the implemented deadtime and switching frequency [42]. These variables should not vary. That deficiency can be neglected thanks to the Behavior Matching [44, 45, 51, 52, 74].


## 3.2. DC-AC Stage

Two new trends for this stage are discussed in [75, 76], using LCL filters on a grid interface and replacing the conventional three-level PWM inverter by a multilevel inverter. The objectives are the same: lower total harmonic distortion (THD) and more compact designs. However, for the application presented in this paper, the three-phase current-controlled voltage-source inverter with L filter is adequate.The inverters used for grid interfacing are broadly classified as voltage-source inverters (VSI) and current-source inverters (CSI). The control schemes can be classified as current-controlled inverters (CCI) and voltage-controlled inverters (VCI).PV solar arrays are fairly good approximation to a current source. Then, most PV inverters are voltage source. In addition, with vector modulation [77–82], the THD measured is insignificant when harmonics below the switching frequency are considered.The current-controlled scheme objective is to control active and reactive components of the current fed into the grid. The current controllers are better suited for the control of power export from PV inverters to the utility grid since they are less sensitive to errors in synchronizing sinusoidal voltage waveforms [83]. The structure of the three-phase current-controlled voltage-source inverter is shown in Figure 14.Figure 14
Three-phase current-controlled VSI.Performing a Park transformation, its dynamical model in the dq-frame can be described by the following:(9)dIddt=VdL+ω·Iq-RL·Id-VdcL·Dd,dIqdt=-ω·Id-RL·Iq-VdcL·Dq,dVdcdt=-IdcCf2+IdCf2·Dd+IqCf2·Dq,
where Id, Iq, Vdc: state variable; Dd, Dq: control variables; L=L1=L2=L3; R=R1=R2=R3: intrinsic resistance of the inductance L1,L2,L3, respectively; Vd,  ω: utility grid parameters (Vd=3·Vg, and Vg is the grid rms phase voltage, and ω=2πf, where f is the grid frequency); Idc: average output current of the three-phase bridge rectifier, that is, the average input current of the three-phase VSI.SincePg=Vd×Id represents the power that the inverter injects in the grid, the MPPT can be performed perturbing Vdc and observing Id, according to the following:
(10)Vdc(k+1)=Vdc(k)+ΔV·sign[Vdc(k)-Vdc(k-1)]·sign[Id(k)-Id(k-1)].
Setting Vdc, the DC-AC stage defines the DC-DC converter’s input characteristic behavior, who determines the PV array operation point. When Id is maximized, the PV array operates on MPOP. As noted, Id and Vdc, used by MPPT, are state variables employed in the grid-current control, that is, additional hardware is not need. Section 4 explains this behavior in detail, which represents the behavior matching of the proposed dual-stage inverter.


## 4. Behavior Matching and the MPPT Proposal Technique

The indirect MPPT is a type of tracking that uses the connection between measured variables and the position of the maximum power operating point (MPOP). Some variables as temperature and PV generator’s open circuit voltage are used. It is not a true MPPT technique.The direct MPPT requires, in principle, a measurement of generator voltage and current as well as a multiplication of these variables [84–95].In both cases, specific measurements for MPPT are made. In the proposed grid-connected dual-stage inverter, the direct axis current,Id, is observed, which serves for the inverter stage to set Vdc. These actions define the DC-DC converter’s input characteristic behavior, which determines the PV array operation point [96]. When Id is maximized, the PV array operates on MPOP. As noted, the variables used in the MPPT are Id and Vdc, initially employed in the grid current control, that is, any specific measurement for MPPT’s purpose is made.The DC-DC converter operates with constant duty cycle and frequency. It is designed to work with an efficiency of 97% coupled to the PV array shown in Section  3.1, working under Standard Test Conditions (STC) (irradiance of  1 kW/m2,  spectrum of  1.5 air mass, and cell temperature of 25°C), resulting in Rloss = 0.32 Ω and Vdc′ = 255 V. Thus, (5) results in
(11)Iin=1.9(Vin-255).Equation (11) is drawn in Figure 15. This illustration also demonstrates that when MPPT is put in action by the inverter stage, Vdc vary, it shifts the DC-DC converter input I–V characteristic.Figure 15
DC-DC converter inputI–V characteristic.The Behavior Matching depends on the DC-DC converter input characteristic at an operation point. Several single and three-phase converters exhibit appropriate input characteristic to present Behavior Matching [45, 50, 94, 97–101]. In the case of SRC3, the input characteristic curve is a straight line with a small inclination.It is important to register that this inclination favors the MPPT on irradiation variations. If this straight line fall together with the line formed by union of the MPOPs, would not be necessary the intervention of the inverter stage is necessary just in the temperature variations, whenVd must be changed in order to move the straight line for a new position, searching the MPOP. Putting upon Figures 11 and 15, the approaching between the DC-DC converter’s input characteristic and the MPOPs can be found, as shown in Figure 16.Figure 16
Crossing between SRC3 and 4 kWp PV arrayI–V characteristic curves.The PV array operation point is defined by the crossing of its respectiveI–V characteristic curve with the DC-DC converter input characteristic curve. Then, Ipv and Idc are also defined. Figure 17 presents simulation results where it can be seen that the PV array behavior, represented by its I–V characteristic, is reproduced on the DC-DC stage output terminals, represented by Idc×Vdc. From the controller point of view, Idc×Vdc is similar to Ipv×Vin. Thus, MPPT task is to extract the maximum power of the DC link, or to inject the maximum power in the grid. The strategies are different, but the result is the same: both tune the DC-DC stage input I–V characteristic to the PV array MPP. Therefore, Figure 17 shows that the DC-DC stage’s output has behavior matching that of the PV array’s output terminals.Figure 17
Behavior matching verifying.The Behavior Matching Technique bring significant advantages always that the input characteristic curve of the DC-DC converter is located near the MPP of the PV array, for all tracking algorithm range. Step-down DC-DC converters, with similar input Buck behavior, are eligible to integrate a modified dual-stage inverter.


### 4.1. MPPT Proposed Technique Using P and O Algorithm

The MPPT proposed technique, using P and O algorithm, has the following procedure.The temperature and the solar irradiation in the PV array modules vary in different ways. The clouds movement can result in brusque alterations in the irradiance level. However, the temperature tends to vary much more slowly.The MPOP can be tracked by a DC-DC converter through its input characteristic. Since temperature remains constant, there is no necessity for the inverter intervention, whose task is to keep the voltageVdc clamped in a reference.However, periodically, the inverter, through a P and O algorithm, regulates the value ofVdc in order to shift the DC-DC converter input characteristic, which can be crossing with the array I–V characteristic far from the MPOP due to a temperature variation. Besides P and O, others algorithms can also be used.The topology of the inverter is presented in Figure14, where Va, Vb, and Vc represent the AC grid phase voltages. It is verified that P and O block is fed by the direct axes current, Id, obtained from Park’s transformation. According to [102], the power that the inverter injects in the grid is given by Pg=Vd×Id, where Vd is constant. Thus, the MPPT can be indirectly performed, perturbing Vd and observing Id. The Vd perturbation at the (k + 1)th sampling is given by (10).


## 4.1. MPPT Proposed Technique Using P and O Algorithm

The MPPT proposed technique, using P and O algorithm, has the following procedure.The temperature and the solar irradiation in the PV array modules vary in different ways. The clouds movement can result in brusque alterations in the irradiance level. However, the temperature tends to vary much more slowly.The MPOP can be tracked by a DC-DC converter through its input characteristic. Since temperature remains constant, there is no necessity for the inverter intervention, whose task is to keep the voltageVdc clamped in a reference.However, periodically, the inverter, through a P and O algorithm, regulates the value ofVdc in order to shift the DC-DC converter input characteristic, which can be crossing with the array I–V characteristic far from the MPOP due to a temperature variation. Besides P and O, others algorithms can also be used.The topology of the inverter is presented in Figure14, where Va, Vb, and Vc represent the AC grid phase voltages. It is verified that P and O block is fed by the direct axes current, Id, obtained from Park’s transformation. According to [102], the power that the inverter injects in the grid is given by Pg=Vd×Id, where Vd is constant. Thus, the MPPT can be indirectly performed, perturbing Vd and observing Id. The Vd perturbation at the (k + 1)th sampling is given by (10).


## 5. Design Example

The PV array consists of two parallel strings, each with ten KC200GT modules from Kyocera. This PV array defines the nominal input power of the  dual-stage  inverter, whose  value isPin = 4 kW,  with  Vin = 263 V  and Iin = 15.2 A. The estimated efficiency for the DC-DC stage is ηDC = 97%. Then, the output power of the DC-DC stage is Pdc = 3880 W and the same value in the input of the DC-AC stage is obtained.  The switching frequencies of the DC-DC  and  DC-AC  stages are  fDC = 40 kHz and  fAC = 20 kHz,  respectively. The grid rms phase voltage is Vg = 220 V.


### 5.1. PV Array Parallel Capacitor (Cf1)

The PV array parallel capacitor is obtained from (12), where ΔVin is the peak-to-peak ripple on the DC-DC stage input voltage. The adopted value is ΔVin = 1.0% Vin = 2.63 V:
(12)Cf1≥Iin215·fDC·ΔVin=15.2215·40k·2.63≥672.03nF.A polyester capacitor of 680 nF is used.


### 5.2. DC-AC Stage Input Capacitor (Cf2)

According to (13) [73], the input capacitor of the DC-AC stage is obtained. The ripple ΔVCf2 needs to be a magnitude perturbation’s percentage involved in the P and O algorithm. The adopted value is ΔVCf2 = 0.2 V, while the magnitude perturbation is 4 V:
(13)Cf2≥Pdc12·2·Vg·ΔVCf2·fAC=3880122·220·0.2·20k≥259.81μF.The adopted value of Cf2 is 333μF, obtained by a series association of three electrolytic capacitors of 1000 μF.


### 5.3. Grid Line Inductors(L=L1=L2=L3)

The grid line inductors are calculated by (15), assuming phase current ripple equal to ΔIL = 0.42 A. This value corresponds to 5% of the peak line current, obtained by (14) and is sufficient to keep the THD below 5% [73]:
(14)ILpk=2·Pdc3·Vg=2·38803·220=8.31A,(15)L=2·Vg4·fAC·ΔIL=2·2204·20k·0.42⇒L=9.3mH.


### 5.4. Resonant Capacitor (Cr)

The three-phase transformer is constructed with three single-phase transformers, in wye connection. The turns ratio is defined byn=N2/N1=3.2. The leakage inductances have been measured to Ld = 3.6 μH. Applying (4), this results in a resonant capacitor of Cr = 4.4 μF, for ZCS operation. Polypropylene capacitors were adopted.


### 5.5. Average DC-Link Voltage Referred to Primary Side(Vdc′)

The average DC-link voltage referred to primary side can be calculated from (16) [73]:
(16)Vdc’=ηDC·Vin=0.97·263⇒Vdc’=255V.The voltage presented in (16) is raised to Vdc = 816 V by n=3.2. This turns ratio keeps the DC-link voltage above 600 V for all MPPT ranges.


### 5.6. Converter Losses Resistance(Rloss)

In nominal conditions, to sinusoidal resonant currents, the losses are found aroundRloss = 0.32 Ω, according to (7).


### 5.7. Main Data of the Laboratory Prototype

A SK20GD065 Semikron IGBT module was used. The diode bridge was made with six ultrafast recovery rectifiers FFPF05U120S, of 1200 V and 5 A. A top view of the prototype is depicted in Figure18.Figure 18
Laboratory prototype of the modified dual-stage inverter.


## 5.1. PV Array Parallel Capacitor (Cf1)

The PV array parallel capacitor is obtained from (12), where ΔVin is the peak-to-peak ripple on the DC-DC stage input voltage. The adopted value is ΔVin = 1.0% Vin = 2.63 V:
(12)Cf1≥Iin215·fDC·ΔVin=15.2215·40k·2.63≥672.03nF.A polyester capacitor of 680 nF is used.


## 5.2. DC-AC Stage Input Capacitor (Cf2)

According to (13) [73], the input capacitor of the DC-AC stage is obtained. The ripple ΔVCf2 needs to be a magnitude perturbation’s percentage involved in the P and O algorithm. The adopted value is ΔVCf2 = 0.2 V, while the magnitude perturbation is 4 V:
(13)Cf2≥Pdc12·2·Vg·ΔVCf2·fAC=3880122·220·0.2·20k≥259.81μF.The adopted value of Cf2 is 333μF, obtained by a series association of three electrolytic capacitors of 1000 μF.


## 5.3. Grid Line Inductors(L=L1=L2=L3)

The grid line inductors are calculated by (15), assuming phase current ripple equal to ΔIL = 0.42 A. This value corresponds to 5% of the peak line current, obtained by (14) and is sufficient to keep the THD below 5% [73]:
(14)ILpk=2·Pdc3·Vg=2·38803·220=8.31A,(15)L=2·Vg4·fAC·ΔIL=2·2204·20k·0.42⇒L=9.3mH.


## 5.4. Resonant Capacitor (Cr)

The three-phase transformer is constructed with three single-phase transformers, in wye connection. The turns ratio is defined byn=N2/N1=3.2. The leakage inductances have been measured to Ld = 3.6 μH. Applying (4), this results in a resonant capacitor of Cr = 4.4 μF, for ZCS operation. Polypropylene capacitors were adopted.


## 5.5. Average DC-Link Voltage Referred to Primary Side(Vdc′)

The average DC-link voltage referred to primary side can be calculated from (16) [73]:
(16)Vdc’=ηDC·Vin=0.97·263⇒Vdc’=255V.The voltage presented in (16) is raised to Vdc = 816 V by n=3.2. This turns ratio keeps the DC-link voltage above 600 V for all MPPT ranges.


## 5.6. Converter Losses Resistance(Rloss)

In nominal conditions, to sinusoidal resonant currents, the losses are found aroundRloss = 0.32 Ω, according to (7).


## 5.7. Main Data of the Laboratory Prototype

A SK20GD065 Semikron IGBT module was used. The diode bridge was made with six ultrafast recovery rectifiers FFPF05U120S, of 1200 V and 5 A. A top view of the prototype is depicted in Figure18.Figure 18
Laboratory prototype of the modified dual-stage inverter.


## 6. Simulation and Experimental Results

The specifications of the whole system are given in Section5.


### 6.1. DC-DC Three-Phase Series Resonant Converter (SRC3) Performance

A compare unit pertained to a peripheral of the Texas TMS320F2812 controller was configured to generate six gate pulses for SRC3, with duty cycle of 50% and deadtime of 640 ns. The same DSP carried out the grid-current control and the MPPT.Figure19 shows the resonant currents. Obviously, the leakage inductances of each phase are not exactly the same. These asymmetries cause small differences in resonant currents amplitudes. However, this has a negligible impact on the SRC3 operation.Figure 19
Resonant currents under nominal conditions.Figures20 and 21 present the input and output currents of the DC-DC stage for different power conditions. These currents have a low ripple and a frequency of six times that of the switching frequency, resulting in a continuous power flux. These features are not common among three-phase DC-DC converters and lead to reduced filter devices. The PV array parallel capacitor is of only 680 nF, for example. The voltage ripple on it is showed in Figure 22. This ripple was produced by the current Iin of Figure 21. It has a negligible impact above PV array efficiency.Figure 20
Input and output currents of the DC-DC stage and collector-emitter voltage, under nominal conditions.Figure 21
Input and output currents of the DC-DC stage and collector-emitter voltage, withPdc = 500 W.Figure 22
PV array output voltage forPdc = 500W (400 mV/div).The ZCS commutation is shown in detail in Figure23. Unfortunately, the commutation losses drastically increase at low power levels, according to Figure 24. Figure 25 shows the SRC3 efficiency.Figure 23
Resonant current and collector-emitter voltage, under nominal conditions.Figure 24
Resonant current and collector-emitter voltage, forPdc = 500 W.Figure 25
SRC3 efficiency.


### 6.2. Behavior Matching

This subsection presents experimental results that validate the Behavior Matching technique.Figure26 shows the harmonic spectrum of the voltage across the resonant circuit [42]. The fundamental frequency is 40 kHz, that is, the resonant frequency.Figure 26
Harmonic spectrum of the voltage across the resonant circuit.The impact of the voltage harmonic components on the resonant current depends on the resonant circuit quality factor, presented as follows:(17)Q=2·π·fr·LdRloss.The requirement of large impedance for frequencies that are different of the resonant frequency is fulfilled when the quality factorQ is high. The value of Q is high in nominal conditions, such as Figure 23. On the other hand, for low SRC3 output power rate, the Q value is low too, according to Figure 24. This occurs due increasing of Rloss for output powers below of approximately 1700 W. Figure 25 points to decreasing of efficiency below this power. Larger Ld could reduce the power from which Rloss rises. The efficiency would be more flat.Figure27 depicts the consequence of the resonant current distortion. For high rated power, the ratio Iin/Idc = 3.2 = n. For low rated power, the ratio transformer is affected.Figure 27
Input and output currents ratio in the DC-DC stage.In conclusion, whenRloss increases, the equivalent ratio transformer (Iin/Idc) also rises.Reporting to (8), it is verified that Vin could be continuously reduced with the DC-DC stage output power reduction. This effect is depicted in Figure 28. Thus, variations in Rloss are compensated by variations in the equivalent ratio transformer. Moreover, they have the same cause.Figure 28
DC-DC converter inputI–V characteristics.The inclination of the SRC3 inputI–V characteristic, for all Vdc operation range, is given by (18), obtained from (5). Figure 28 shows experimentally measured points of SRC3 prototype input I-V characteristic
(18)dIndVin=6π2·Rloss=2.3A/V.The DC-DC converter’s input characteristic has a propitious behavior to MPPT algorithm. Putting Figure28  (Vdc=816V) on the PV array I–V characteristic curves, the proximity between the DC-DC converter’s input characteristic and the MPP loci can be established, as shown in Figure 29. This collaborates with the MPPT performance for fast changes in atmospheric conditions.Figure 29
Crossing between SRC3 and PV array characteristic curves.


### 6.3. MPPT Performance

This subsection presents simulation results that reflect the possibility to perform the MPPT using the direct axis current(Id).The mathematical model of a 4 kWp PV array is accurately implemented in the form of a current source controlled by voltage, sensible to two input variables, that is, temperature and solar irradiation power. This model simulates the physical PV array.The MPOP’s shift is an almost horizontal line when the temperature changes, see Figure12. The intersection between the characteristics of the PV array and DC-DC converter can deviates from the MPOP. If Vdc is keeping constant, thus Vin stays, approximately, constant too. This implies that Ipv and the power from PV array (PMPPT) will decrease. Figure 30 shows the consequences of cell’s temperature changes from 25°C to 35°C. The irradiance is 1 kW/m2 and the air mass is of 1.5.Temperature variation and consequences.
(a)
(b)
(c)In order to solve this problem, the P and O method is put in action in the DC-AC stage, as shown in Figure31. The P and O parameters are Ta = 50 ms and ΔV = 4 V (ΔV is the perturbation magnitude and Ta is the sampling interval). Thus, in order to maximize the Id current, each 50 ms, the Vdc voltage is perturbed by 4 V on its magnitude. This perturbation can be added or subtracted of Vdc. The MPPT algorithm resolves based on the variation of Id. The system uses a turns ratio transformer of 1 : 3.2. Figure 32 shows a zoom on the PV array characteristic curves with the positions (initial, intermediate, and final) of the operation point.Execution of theVdc perturbation and Id observation algorithm.
(a)
(b)Figure 32
I–V characteristic curves of the PV array.Figure33 depicts the trajectory of the PV array output power (PMPPT) as a consequence of the Vdc adjustments, drawn from Figure 31. The relative tracking error, εR, practically achieves zero in steady state conditions. This occurs due to the shifting of the SRC3 input I–V characteristic close to the MPOP.MPPT performance.
(a)
(b)


### 6.4. Performance of the DC-AC Three-Phase Stage

The DC-AC stage is performed by a three-phase PWM voltage source inverter (VSI3) controlled by current, whom power topology is presented in Figure14. The space vector modulation (SVM) is used in order to minimize the THD of the grid current.Figure34 depicts the utility voltage and the current transferred to the main power supply, both in phase a, for rated power condition. The THD of the voltage and current is shown in Figures 35 and 36, respectively. An efficiency of 95% was obtained for the DC-AC stage, and 92% for the whole system, including the DC-DC stage.Figure 34
Voltage and current in the main power supply (phase a)-100 V/div; 4 A/div; 2 ms/div.Figure 35
Voltage harmonic analysis of the main power supply-THD = 3.69%.Figure 36
Current harmonic analysis of the main power supply-THD = 4.89%, phase angle = −175°.Figure37 presents the performance of the utility voltage and utility current (THD of current 4.98%, phase angle = 172°), with 32% of the full power delivery to the grid (around 1300 W). For this power conditions, the whole system efficiency is 90%. Below this power the efficiency decreases drastically.Figure 37
Utility voltage and utility current with 32% of the full power-100 V/div; 2 A/div; 2 ms/div.


## 6.1. DC-DC Three-Phase Series Resonant Converter (SRC3) Performance

A compare unit pertained to a peripheral of the Texas TMS320F2812 controller was configured to generate six gate pulses for SRC3, with duty cycle of 50% and deadtime of 640 ns. The same DSP carried out the grid-current control and the MPPT.Figure19 shows the resonant currents. Obviously, the leakage inductances of each phase are not exactly the same. These asymmetries cause small differences in resonant currents amplitudes. However, this has a negligible impact on the SRC3 operation.Figure 19
Resonant currents under nominal conditions.Figures20 and 21 present the input and output currents of the DC-DC stage for different power conditions. These currents have a low ripple and a frequency of six times that of the switching frequency, resulting in a continuous power flux. These features are not common among three-phase DC-DC converters and lead to reduced filter devices. The PV array parallel capacitor is of only 680 nF, for example. The voltage ripple on it is showed in Figure 22. This ripple was produced by the current Iin of Figure 21. It has a negligible impact above PV array efficiency.Figure 20
Input and output currents of the DC-DC stage and collector-emitter voltage, under nominal conditions.Figure 21
Input and output currents of the DC-DC stage and collector-emitter voltage, withPdc = 500 W.Figure 22
PV array output voltage forPdc = 500W (400 mV/div).The ZCS commutation is shown in detail in Figure23. Unfortunately, the commutation losses drastically increase at low power levels, according to Figure 24. Figure 25 shows the SRC3 efficiency.Figure 23
Resonant current and collector-emitter voltage, under nominal conditions.Figure 24
Resonant current and collector-emitter voltage, forPdc = 500 W.Figure 25
SRC3 efficiency.


## 6.2. Behavior Matching

This subsection presents experimental results that validate the Behavior Matching technique.Figure26 shows the harmonic spectrum of the voltage across the resonant circuit [42]. The fundamental frequency is 40 kHz, that is, the resonant frequency.Figure 26
Harmonic spectrum of the voltage across the resonant circuit.The impact of the voltage harmonic components on the resonant current depends on the resonant circuit quality factor, presented as follows:(17)Q=2·π·fr·LdRloss.The requirement of large impedance for frequencies that are different of the resonant frequency is fulfilled when the quality factorQ is high. The value of Q is high in nominal conditions, such as Figure 23. On the other hand, for low SRC3 output power rate, the Q value is low too, according to Figure 24. This occurs due increasing of Rloss for output powers below of approximately 1700 W. Figure 25 points to decreasing of efficiency below this power. Larger Ld could reduce the power from which Rloss rises. The efficiency would be more flat.Figure27 depicts the consequence of the resonant current distortion. For high rated power, the ratio Iin/Idc = 3.2 = n. For low rated power, the ratio transformer is affected.Figure 27
Input and output currents ratio in the DC-DC stage.In conclusion, whenRloss increases, the equivalent ratio transformer (Iin/Idc) also rises.Reporting to (8), it is verified that Vin could be continuously reduced with the DC-DC stage output power reduction. This effect is depicted in Figure 28. Thus, variations in Rloss are compensated by variations in the equivalent ratio transformer. Moreover, they have the same cause.Figure 28
DC-DC converter inputI–V characteristics.The inclination of the SRC3 inputI–V characteristic, for all Vdc operation range, is given by (18), obtained from (5). Figure 28 shows experimentally measured points of SRC3 prototype input I-V characteristic
(18)dIndVin=6π2·Rloss=2.3A/V.The DC-DC converter’s input characteristic has a propitious behavior to MPPT algorithm. Putting Figure28  (Vdc=816V) on the PV array I–V characteristic curves, the proximity between the DC-DC converter’s input characteristic and the MPP loci can be established, as shown in Figure 29. This collaborates with the MPPT performance for fast changes in atmospheric conditions.Figure 29
Crossing between SRC3 and PV array characteristic curves.


## 6.3. MPPT Performance

This subsection presents simulation results that reflect the possibility to perform the MPPT using the direct axis current(Id).The mathematical model of a 4 kWp PV array is accurately implemented in the form of a current source controlled by voltage, sensible to two input variables, that is, temperature and solar irradiation power. This model simulates the physical PV array.The MPOP’s shift is an almost horizontal line when the temperature changes, see Figure12. The intersection between the characteristics of the PV array and DC-DC converter can deviates from the MPOP. If Vdc is keeping constant, thus Vin stays, approximately, constant too. This implies that Ipv and the power from PV array (PMPPT) will decrease. Figure 30 shows the consequences of cell’s temperature changes from 25°C to 35°C. The irradiance is 1 kW/m2 and the air mass is of 1.5.Temperature variation and consequences.
(a)
(b)
(c)In order to solve this problem, the P and O method is put in action in the DC-AC stage, as shown in Figure31. The P and O parameters are Ta = 50 ms and ΔV = 4 V (ΔV is the perturbation magnitude and Ta is the sampling interval). Thus, in order to maximize the Id current, each 50 ms, the Vdc voltage is perturbed by 4 V on its magnitude. This perturbation can be added or subtracted of Vdc. The MPPT algorithm resolves based on the variation of Id. The system uses a turns ratio transformer of 1 : 3.2. Figure 32 shows a zoom on the PV array characteristic curves with the positions (initial, intermediate, and final) of the operation point.Execution of theVdc perturbation and Id observation algorithm.
(a)
(b)Figure 32
I–V characteristic curves of the PV array.Figure33 depicts the trajectory of the PV array output power (PMPPT) as a consequence of the Vdc adjustments, drawn from Figure 31. The relative tracking error, εR, practically achieves zero in steady state conditions. This occurs due to the shifting of the SRC3 input I–V characteristic close to the MPOP.MPPT performance.
(a)
(b)


## 6.4. Performance of the DC-AC Three-Phase Stage

The DC-AC stage is performed by a three-phase PWM voltage source inverter (VSI3) controlled by current, whom power topology is presented in Figure14. The space vector modulation (SVM) is used in order to minimize the THD of the grid current.Figure34 depicts the utility voltage and the current transferred to the main power supply, both in phase a, for rated power condition. The THD of the voltage and current is shown in Figures 35 and 36, respectively. An efficiency of 95% was obtained for the DC-AC stage, and 92% for the whole system, including the DC-DC stage.Figure 34
Voltage and current in the main power supply (phase a)-100 V/div; 4 A/div; 2 ms/div.Figure 35
Voltage harmonic analysis of the main power supply-THD = 3.69%.Figure 36
Current harmonic analysis of the main power supply-THD = 4.89%, phase angle = −175°.Figure37 presents the performance of the utility voltage and utility current (THD of current 4.98%, phase angle = 172°), with 32% of the full power delivery to the grid (around 1300 W). For this power conditions, the whole system efficiency is 90%. Below this power the efficiency decreases drastically.Figure 37
Utility voltage and utility current with 32% of the full power-100 V/div; 2 A/div; 2 ms/div.


## 7. Conclusion

In this paper, a modified dual-stage inverter applied to grid-connected photovoltaic systems performed for high power applications has been studied. The modified dual-stage inverter contains DC-DC stage and DC-AC stage. Through the Behavior Matching, the DC-DC stage operates with constant frequency and duty cycle and the DC-AC stage becomes responsible for the maximum power point tracking and grid-current control.I–V characteristic of the PV array was reproduced in the output of the DC-DC stage, without any control, which was defined as Behavior Matching. Some sensors could be avoided because the grid-current control apparatus produces the variables needed for the MPPT. In addition, only one digital controller can generate gate pulses for all transistors of the PV system, which results in simpler and cheaper topology. The Behavior Matching technique allows the construction of the Modified Dual-Stage Inverter.The three-phase DC-DC series resonant converter switching with constant frequency keeps the structure operating with the maximum efficiency and optimized transformer utilization. The topologies chosen for the converters qualify the modified dual-stage inverter for high power operation. The transformer size can be minimized by the increase of the DC-DC converter switching frequency, which is independent of the DC-AC stage. The filter elements size is also reduced due to the low ripple of DC-DC converter input and output current. As a consequence, an insignificant capacitor was used on implementation of the PV array parallel filter. Two factors mainly contributed with this advantage: the continuous current flux with low ripple and the barrier formed by the resonant circuit to electrical perturbations on DC link that did not affect the primary side voltage bus. Besides, the Series Resonant Converter features a robust operation under unbalanced conditions.The periodic action of the inverter prevents losses and instability, inherent problems of the P and O technique. When the MPOP is found, the voltageVdc is clamped, avoiding voltage ripple in the photovoltaic array terminals. Moreover, it can be opted for a small perturbation magnitude, minimizing the steady-state error, since the DC-DC converter is that will response for the fast atmospheric condition changes. As was seen, it possesses an extremely fast dynamics. Although P and O method was used, other techniques are encouraged.Finally, the author expects that the efficiency and the implementation simplicity of the whole system are advantages that promote its use.



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*Source: 406312-2013-01-31.xml*