**Contents**


## **About the Editor**

**Emilio Figueres** received his M.Sc. degree from the Ecole Nationale Superieure ´ d'Electrotechnique, d'Electronique, d'Informatiqueet d'Hydraulique de Toulouse, Toulouse, France, in1995, and his Dr. Ingeniero Industrial (Ph.D.) degree from the Universidad Politecnica de ´ Valencia (UPV), Valencia, Spain, in 2001. Since 1996, he has been associated with the Electronics Engineering Department, UPV, where he was Head of the Department from 2008 to 2016, and currently works as a Full Professor. His main research interests include modeling and control of power converters, power processing of renewable energy sources, and grid-connected converters for distributed power generation and improvement of power quality. In the above areas, he has co-authored over 100 papers published in Journals Citation Report (JCR) indexed journals and conferences. He habitually collaborates with companies such as Power Electronics, Mahle Electronics, Ingeteam Power Technology, among others, in the development of power converters for processing of renewable energies, onboard chargers for electric vehicles, and auxiliary systems in railway applications. He holds several patents on these topics.

## **Preface to "Photovoltaic and Wind Energy Conversion Systems"**

In the barely two decades since the advent of the 21st century, renewable energies have gone from being one of the great opportunities for the future of humanity to become one of the main players on the world electricity production stage. Except for hydraulics, the two types of renewable energy that have contributed to this radical change the most have been solar and wind energies. Indeed, according to the statistics published by the International Renewable Energy Agency (IRENA) in 2018 (the latest with data available at this time), hydropower produced 4,149,215 GWh (63% of the total production of all energies renewable energy), onshore wind produced 1,194,718 GWh (18.1%), and solar power contributed with 549,833 GWh (8.3%), which clearly shows the importance of both technologies, especially remarkable in their increasing weight in the production of electricity at a global scale. In this context, this Special Issue of Energies highlights some of the latest technological advances in solar photovoltaic and wind technologies, including maximum power point tracking algorithms, parallel connection of central inverters in high power photovoltaic plants, modeling of systems, etc. Likewise, there is also the participation of authors who show a current overview of the state-of-the-art in solar concentrators and also applied case studies.

The authors of the papers and I hope that readers will find the contributions interesting and useful for their personal and professional development, or that the papers simply allow them to delve into some aspects of two technologies that are as exciting as they are relevant to the future of humanity.

> **Emilio Figueres** *Editor*

## *Article* **A Control Scheme without Sensors at the PV Source for Cost and Size Reduction in Two-Stage Grid Connected Inverters**

#### **Raúl González-Medina \*, Marian Liberos, Silvia Marzal, Emilio Figueres and Gabriel Garcerá**

Grupo de Sistemas Electrónicos Industriales del Departamento de Ingeniería Electrónica, Universitat Politècnica de València, Camino de Vera s/n, 46022 Valencia, Spain

**\*** Correspondence: raugonme@upv.es; Tel.: +34-963-879-606

Received: 21 June 2019; Accepted: 23 July 2019; Published: 1 August 2019

**Abstract:** In order to reduce the cost of PV facilities, the market requires low cost and highly reliable PV inverters, which must comply with several regulations. Some research has focused on decreasing the distortion of the current injected into the grid, reducing the size of the DC-link capacitors and removing sensors, while keeping a good performance of the maximum power point tracking (MPPT) algorithms. Although those objectives are different, all of them are linked to the inverter DC-link voltage control loop. Both the reduction of the DC-link capacitance and the use of sensorless MPPT algorithms require a voltage control loop faster than that of conventional implementations in order to perform properly, but the distortion of the current injected into the grid might rise as a result. This research studies a complete solution for two-stage grid-connected PV inverters, based on the features of second-order generalized integrators. The experimental tests show that the proposed implementation has a performance similar to that of the conventional control of two-stage PV inverters but at a much lower cost.

**Keywords:** photovoltaics; two-stage grid-connected PV inverters; reduced DC-link; sensorless MPPT

#### **1. Introduction**

In order to reduce the installation and maintenance costs, the photovoltaic (PV) market requires low cost and reliable systems. Moreover, grid-connected PV inverters must comply with several electromagnetic compatibility (EMC) regulations, some of which limit the distortion of the current injected into the grid (THDi), like IEC EN61000 and IEEE519 [1–4]. Maximum power point tracking (MPPT) algorithms are implemented to optimize the performance of PV systems [5–9]. Conventional MPPT algorithms use current and voltage sensors to calculate the power extracted from the PV source. Several investigations have focused on reducing the number of sensors in PV inverters when implementing MPPT algorithms [10–13], which has a positive impact on cost reduction.

One possible solution to achieve both a good maximum point tracking (MPPT) performance and a reduced THDi is the use of two-stage grid-connected PV inverters, based on a DC-DC converter connected to the PV source, followed by a grid-connected inverter [14–16]. This paper focused on this kind of topologies.

There is a trend to reduce the required capacitance at the DC-link between the DC-DC converter and the inverter stage, which allows replacing electrolytic capacitors by film capacitors, which are more durable [17–22]. Two major effects of the DC-link capacitance reduction are the increase of the voltage ripple at the capacitors and higher transient variations of that voltage under dynamic operation point changes of the inverter. These variations must be bound for the proper operation of the inverter.

In this research, the implementation of control structures based on second-order generalized integrators (SOGI) [23–27] is proposed to support the DC-link capacitance reduction of two-stage grid-connected PV inverters with a small number of sensors. On the one hand, it will be shown that SOGIs can improve the dynamic response of the DC-link voltage. On the other hand, the frequency adaptability of SOGI structures will help to reduce the THDi in the case of variations of the grid frequency, even under highly distorted grid voltage conditions.

A sensorless MPPT algorithm that does not require sensors of the PV panel electrical variables was developed in this study. It is based on the power balance at the DC-link [11] and the improvement of the PV inverter voltage control loop achieved by the use of SOGIs.

The applied techniques allow both a reduced DC-link and a sensorless MPPT algorithm, keeping the performance similar to that of conventional MPPTs, but at a much lower cost.

#### **2. Two-Stage Grid-Connected PV Inverter**

The two-stage grid-connected PV inverter shown in Figure 1 has been used for validating the theoretical study. It is formed by a flyback DC-DC and a single phase inverter. The inverter connects a single 230 W PV panel to the single-phase grid (230 Vrms, 50 Hz).

**Figure 1.** Two-stage grid-connected inverter.

The inverter has a reduced size DC-link. Voltage and current sensors for the measurement of the PV panel voltage, VPV, and current, IPV, are available, in order to compare conventional MPPT algorithms, which make use of those sensors, with sensorless MPPT algorithms under the same conditions.

Costs saving associated with a smaller bulk capacitor and to the absence of MPPT sensors depends on several characteristics, but the estimation done for the implemented prototype is detailed in Table A1 of Appendix A.

#### *2.1. Grid-tied VSI*

The voltage source inverter (VSI) is formed by the DC-link capacitance (CDC), a full bridge of IGBTs and an LCL grid filter. The full-bridge is commutated by means of unipolar sinusoidal pulse width modulation (SPWM). The grid filter was designed following the guidelines of Reference [28]. The values of the inverter are shown in Table 1.

The current ICDC has high-frequency current components (IDC\_SW) due to the switching of the transistors and a low-frequency current component (IDC\_AC). The frequency of IDC\_AC is twice the grid frequency and causes a voltage ripple at the DC-link (VDC\_R). In Reference [18] it was shown that the minimum value of CDC is determined by the maximum and minimum permissible voltage at the DC-link caused by VDC\_R. However, this criterion does not consider the dynamics of the DC-link [19] and transient voltage variations, that are due to changes in the operation point. The proposed value of the capacitance, CDC, to study the effects of a reduced DC-link is designed to limit the peak to peak value of VDC\_R to 10% of VDC at nominal power (PG = 230 W), thus the value of the capacitance CDC is calculated following Equation (1).

$$\mathbf{C\_{DC}} \ge \sqrt{2} \cdot \frac{\mathbf{V\_{G\_{RMS}}} \mathbf{I\_{G\_{RMS}}}}{\mathbf{V\_{DC\_{AV}}} \cdot \pi \cdot \mathbf{F\_{RDC}} \cdot \mathbf{V\_{DC\_{R}}} \text{ PP}} = \frac{230 \mathbf{V} \cdot \mathbf{1A}}{380 \mathbf{V} \cdot \pi \cdot 100 \mathbf{H} \mathbf{z} \cdot 38 \mathbf{V}} = 50.7 \text{ } \mu\text{F} \tag{1}$$

**Table 1.** Values of the Voltage Source Inverter (VSI) stage.


#### *2.2. Step Up DC-DC Converter*

The DC-DC stage, shown in Figure 1, is a Flyback converter designed for boosting the voltage from the PV panel (VPV) up to the voltage of the DC-link (VDC) and providing high-frequency galvanic isolation between the PV panel and the grid. The converter is designed to work in discontinuous conduction mode (DCM) because the value of the transformer magnetizing inductance (LM) and the physical size of the transformer become smaller [15]. The MPPT algorithm establishes the operation point of this stage since the PV panel voltage is at the input of the DC-DC converter. It is worth pointing out that in the two-stage PV inverter structure the output voltage of the DC-DC converter is regulated by the inverter stage, whereas the panel voltage is controlled by the DC-DC converter following the reference value provided by the MPPT algorithm.

The current ISW, through the switch of the DC-DC converter, is composed by Equation (2) an average value equal to the PV panel current, IPV, and a high-frequency component, ICIN, provided by a low voltage input capacitance, CIN, following Equation (2).

$$\mathbf{I}\_{\rm SW} = \mathbf{I}\_{\rm PV} - \mathbf{I}\_{\rm CIN} \,\mathrm{.}\tag{2}$$

The size of CIN depends on the high-frequency current components at the input of the DC-DC converter. Besides, the value of the output capacitance, CDC, has an influence on the MPPT performance, since VPV is susceptible to the low-frequency voltage ripple at the DC-link voltage, VDC\_R. It is worth pointing out that the implementation of a peak current control (PCC) is highly desirable for protecting the power switches from transient overcurrents. The values of the DC-DC converter are detailed in Table 2.


**Table 2.** Values of the DC-DC stage.

#### **3. Control**

The control of the two-stage inverter has been implemented digitally in a Texas Instruments TMS320F28335 [29] microcontroller with digital signal processor (DSP) extensions at a sampling frequency (FS) of 40 kHz. The controllers have been calculated in the continuous domain, having taken into account the digital delays, and then discretized using the bilinear "Tustin" transform. The delay between the sampling and the update of the reference inside the DSP has been done by using a second-order Padé approximation.

It is worth pointing out that the dynamic models used in this control study result from perturbing the averaged variables of the DC-DC converter or of the inverter stage around an operation point, as expressed by Equation (3). In Equation (3), X and xˆ denote the operation point value and the small-signal term of the averaged variable, x, respectively. The averaging is done in every cycle of the switching frequency.

$$\mathbf{x} = \mathbf{X} + \mathbf{\hat{x}} \tag{3}$$

#### *3.1. Control Scheme of the VSI Stage*

The complete control structure of the VSI is shown in Figure 2.

**Figure 2.** VSI control scheme.

#### 3.1.1. Synchronization with the Grid

The synchronization with the grid voltage (VG) has been implemented by means of an SOGI based Frequency Locked Loop (FLL-SOGI) [27], which provides the calculation of the grid frequency in rad/s, ωG\_FLL, a sinusoidal signal in phase with the grid, v', and a sinusoidal signal in quadrature, qv'. The amplitude of the fundamental harmonic of VG, v'pk, is calculated following Equation (4), and a normalized sinusoidal signal in phase with the grid, v'n, is obtained in Equation (5).

$$\mathbf{v}'\_{\rm pk} = \sqrt{\mathbf{v}'^2 + \mathbf{q}\mathbf{v}'^2} \tag{4}$$

$$\mathbf{v}'\_{\mathbf{n}} = \frac{\mathbf{v}'}{\mathbf{v}'\_{\mathbf{pk}}} \to \mathbf{v}'\_{\mathbf{n}}(\mathbf{t}) \equiv \cos(\omega \mathbf{q}\_{\text{-FLL}} \cdot \mathbf{t})\tag{5}$$

#### 3.1.2. Control of the Current Injected into the Grid

The control of the current injected into the grid, IG, is indirectly performed by controlling the current through the inductance Lf of the filter, ILf, because the control of ILf is less sensitive to grid impedance variations [30].

A proportional + resonant controller + a harmonics compensator (P + R + HC) current regulator, GILf(s), expressed by Equation (6), has been designed following Reference [26] and References [31–34] for tracking the sinusoidal reference of ILf, ILf Ref. Both the resonant and the harmonics compensator have been implemented by means of second-order generalized integrators (SOGI).

$$\mathbf{C\_{IL}(s) = P\_{\rm Lf} + R\_{\rm Lf}(s) + HC(s) = K\_{\rm PLf} + \sum\_{i=1,3,5,7} K\_{\rm RLf[i]} \frac{\mathbf{K\_{\rm WPLf[i]}} \cdot (\boldsymbol{\varpi\_{G\_{\rm FL}}} \cdot \mathbf{i}) \cdot \mathbf{s}}{\mathbf{s^2} + K\_{\rm WWRL[i]} \cdot (\boldsymbol{\varpi\_{G\_{\rm FL}}} \cdot \mathbf{i}) \cdot \mathbf{s} + \left(\boldsymbol{\varpi\_{G\_{\rm FL}}} \cdot \mathbf{i}\right)^2} \tag{6}$$

Taking into account that the value of ωG\_FLL used in Equation (6), provided by the FLL-SOGI, it can be concluded that GILf is adaptive in frequency, allowing high performance even under large variations of the grid frequency. The index 'i' in Equation (6) represents the corresponding harmonic. The gains of GILf are shown in Table 3.


**Table 3.** Constants of the regulator P + R + HC.

#### 3.1.3. Control of the DC-link Voltage (VDC)

A reduced DC-link capacitance leads to the fast dynamics of the VDC control loop at the expense of an increase of the THDi of IG [16,17]. In Reference [16] a notch filter in the DC-link voltage control loop was implemented to reduce the low-frequency harmonics of IG. In the current study, the notch filter is implemented by means of SOGIs, achieving adaptation to grid frequency variations. This implementation allows an increase of the crossover frequency of the VDC control loop without increasing the distortion of IG, even with a high low voltage ripple at the DC-link and under large grid frequency variations. The control scheme of VDC is shown in Figure 3.

**Figure 3.** Control loop of the DC-link voltage (VDC) with second order generalized integrator (SOGI) notch filter: (**a**) detailed model, (**b**) equivalent model.

In Figure 3, two small-signal transfer functions play an important role. The first one is the transfer function ˆiLf/ˆiLfRef(s) in Figure 3a, which is obtained by closing the control loop of ILf. This transfer function can be approximated by (7) in Figure 3b, where ωCi is the crossover frequency of the VSI current loop.

$$\frac{\hat{\mathbf{i}}\_{\text{Lf\\_pk}}}{\hat{\mathbf{i}}\_{\text{Lf\\_Ref\\_pk}}}(\mathbf{s}) \approx \frac{1}{1 + \frac{\mathbf{s}}{\omega \nu\_{\text{Ci}}}} \tag{7}$$

In the case of a three-phase grid connected inverter, the derivation of the transfer function from the AC side active current to the DC-link voltage and the adjustment of the voltage loop PI regulator can be found in Reference [35], pages 210–219. In the case of the single-phase inverter under study, an analogous Equation (8) can be derived, based on the power balance and the power perturbation at the DC and AC sides. The transfer function from the peak value of the inverter output current at the AC side to the DC-link voltage can be expressed by Equation (8) after some derivation. Note that Equation (8) consists of a first-order transfer function with an (unstable) right half plane (RHP) pole, ωP\_RHP, whose value depends on the operation point values VDC and IDC.

$$\frac{\text{\reflectbox{ $V\_{DC}$ }}}{\text{\reflectbox{ $V\_{Lf\_{-pk}}$ }}(\text{s})} = \frac{\frac{\text{V\_G}}{\sqrt{2}\text{I}\_{\text{DC}}}}{1 - \frac{\text{s}}{\left(\frac{\text{I\_{DC}}}{\text{C}\_{\text{DC}} \cdot \text{V\_{DC}}}\right)}} = \frac{\frac{\text{V\_G}}{\sqrt{2}\text{I}\_{\text{DC}}}}{1 - \frac{\text{s}}{\text{a}\nu\_{\text{-}RHP}}}; \text{ } \text{w}\_{\text{-}RHP} = \frac{\text{I\_{DC}}}{\text{C}\_{\text{DC}} \cdot \text{V\_{DC}}}\tag{8}$$

The loop gain TVDC(s) of Figure 3 is tuned by means of the PI regulator GVDC(s). Equation (9) provides the crossover frequency, Fc\_VDC, one decade higher than ωP\_RHP/(2 π). Note that an open loop unstable system can be stabilized by feedback only if the loop gain has a gain crossover frequency much higher than the maximum possible value of the unstable open loop pole, Fc\_VDC >> ωP\_RHP/(2 π) in this case. Besides, the value of Fc\_VDC must be much lower than twice the grid frequency (FG), to

reduce the effect of the low-frequency voltage ripple at the DC-link (fripple = 2 Fc\_VDC) in the current reference signal iLf\_ref, which could produce an unacceptable distortion of the grid injected current.

$$\text{Gv}\_{\text{DC}}(\text{s}) = -0.03902 \cdot \left(\frac{\text{s} + 0.6283}{\text{s}}\right) \tag{9}$$

It can be observed from Figure 3 that a notch filter, FNS(s), is placed in series with the PI controller GvDC(s). The expression of the notch filter transfer function is given by Equation (10). The center frequency of FNS(s) is twice the grid frequency (ωNS = 2 ωG\_FLL) in order to filter the ripple at fripple coming from the sensed DC-link voltage. The tuning of the notch filter is provided by the FLL-SOGI previously described. The constant KNS is used to adjust the bandwidth of the filter, BWNS, as shown in Equation (11). The notch filter allows getting a high enough crossover frequency of the voltage loop with no distortion of the grid injected current. Note that a fast enough DC-link voltage loop is crucial to keep the DC-link voltage within safe values in reduced size DC-links with low capacitance.

$$F\_{\rm NS}(\mathbf{s}) = \frac{\mathbf{s}^2 + \omega\_{\rm NS}\mathbf{s}^2}{\mathbf{s}^2 + \mathbf{K}\_{\rm NS} \cdot \omega\_{\rm NS} \cdot \mathbf{s} + \omega\_{\rm NS}\mathbf{s}^2} \tag{10}$$

$$\text{BW}\_{\text{NS}} \cdot 2\pi = \text{K}\_{\text{NS}} \cdot \omega\_{\text{NS}} \to \text{K}\_{\text{NS}} = \frac{2\pi \cdot \text{BW}\_{\text{NS}}}{\omega\_{\text{NS}}} = \frac{100 \cdot 2\pi}{2 \cdot \text{F}\_{\text{G}} \cdot 2\pi} = 1\tag{11}$$

The Bode plots of TVDC(s) depicted in Figure 4a are those obtained when the notch filter, FNS(s), placed in series with GvDC(s) isn´t used. The PI regulator GVDC(s) (9) has been tuned in order to achieve a crossover frequency FC\_VDC = 53 Hz, with a phase margin higher than 82◦ (PM > 82◦) and a gain margin higher than 70 dB (GM > 70 dB). The system is stable but the attenuation at 100 Hz is just 5.5 dB. Therefore, the output of the voltage regulator has a remarkable low-frequency voltage ripple due to VDC\_R, thus producing a high distortion of the grid current.

**Figure 4.** Loop gain frequency response of TvDC(s) @ IDC = [0.2, 0.4, 0.6] A. CDC = 50 μF: (**a**) without notch SOGI filter, (**b**) with notch SOGI filter.

Figure 4b shows the Bode plots of TVDC(s) when the notch filter is used in series with GVDC(s). In that case the value of FC\_VDC has slightly decreased (FC\_VDC = 46 Hz), getting high stability margins: PM > 52◦ and GM > 70 dB. The system is also stable, but the attenuation at 100 Hz is higher than 100 dB.

#### *3.2. Control Scheme of the DC-DC Stage*

The control structure of the DC-DC stage is depicted in Figure 5. It is composed by an outer digital voltage loop, regulating VPV, in cascade with an analog peak current control (PCC) circuit, which sets the peak value of the current, ISW, through the Flyback converter power transistor.

**Figure 5.** Flyback DC-DC PCC control scheme.

#### 3.2.1. Peak Current Control of DC-DC Stage

The PCC control scheme shown in Figure 5 and has been designed following Reference [15]. This control structure is based on the cycle-by-cycle measurement of the current through the transistor, ISW, of the DC-DC converter. The peak value of ISW is limited by the control signal VC. An external stabilization ramp signal, VSe, is added to the sensed current signal, VSn. This method also provides protection for both the HF transformer and the power transistors against an eventual overcurrent.

The modulation index of the PCC, mc = 1 + Se/Sn (Equation (13)), is tuned by means of the slope Se of the external ramp VSe. The value Se = 110 V/ms accomplishes a dynamic behaviour of vˆ VPV / ˆvC(s) close to that of a first-order system, as it can be observed from Figure 6a. The high VDC\_R ripple value produced by the low size of CDC can change the operation point along the I–V curve of the PV source, degrading the MPPT performance. Therefore, a low susceptibility of VPV to the ripple VDC\_R is required. The open loop susceptibility of VPV to variations of VDC (Equation (14)) at 100 Hz is lower than −41.5dB as shown in Figure 6b, therefore VDC\_R, that is 10% of VDC (40 Vpp), causes a VPV voltage ripple of 340 mVpp. It can be concluded that the 100 Hz ripple at VDC has a low influence on the PV voltage. Therefore, the sensing of VPV could be avoided and still a good MPPT would be obtained.

$$\mathbf{S\_N} = \mathbf{R\_i} \frac{\mathbf{V\_{PV}}}{\mathbf{L\_m}} \tag{12}$$

$$\mathbf{F\_M} = \frac{1}{(\mathbf{S\_n} + \mathbf{S\_e}) \cdot \mathbf{T\_{SW\_F}}} = \frac{1}{\mathbf{m\_c} \cdot \mathbf{S\_n} \cdot \mathbf{T\_{SW\_F}}} \tag{13}$$

$$\mathbf{A(s)} = \left. \frac{\mathfrak{v}\_{\rm PV}(s)}{\mathfrak{v}\_{\rm DC}(s)} \right|\_{\mathfrak{h}\_{\rm C} = 0} \tag{14}$$

#### 3.2.2. PV Panel Voltage (VPV) Control Loop in the Conventional MPPT

The reference of VPV (VPVRef) is updated at the sampling frequency of the MPPT, FMPPT. The PV panel voltage control loop is implemented digitally and its control scheme is depicted in Figure 7. The sampling frequency of the control loop (Fs) is 40 kHz. This control loop is adjusted by means the PI regulator GVPV(s), whose values are shown by Equation (15). The crossover frequency, FcVPV, of the loop gain TVPV(s) must be much higher than FMPPT so that VPV can track VPVref. The transfer function vˆ VPV / ˆvC(s) is the closed loop of the PCC and vˆ VPV / ˆvC(s) is the open loop susceptibility of VPV to the variations of VDC.

As it can be observed from the Bode plots of the loop TVPV(s) in Figure 8a, the crossover frequency FcVPV achieved by GVPV(s) is higher than 100 Hz. Therefore, an MPPT algorithm running at FMPPT = 10 Hz is suitable Figure 8b shows that the presence of this control loop reduces the susceptibility vˆ VPV / ˆvDC(s) (16) at 100 Hz down to −55 dB, therefore the voltage ripple in VPV caused by VDC\_R is 71 mVpp. Note that the use of a control loop of VPV reduces the sensitivity of VPV to the 100 Hz ripple

in VDC in a factor of around 5: The ripple in VPV is 71 mVpp with voltage loop compared with 340 mVpp with only PCC loop.

$$\text{G}\_{\text{VPV}}(\text{s}) = -1.8345 \boxed{\frac{\text{s} + 350}{\text{s}}} \tag{15}$$

$$\mathbf{A\_{CL}(s)} = \frac{\boldsymbol{\uppsi\_{PV}(s)}}{\boldsymbol{\uppsi\_{DC}(s)}} \bigg|\_{\boldsymbol{\upphi\_{PVRa}}=\boldsymbol{0}} \tag{16}$$

**Figure 6.** Frequency response of the PCC: (**a**) closed loop vˆ VPV / ˆvC(s) response and (**b**) open loop susceptibility: A(s). VPV∈[24, 30, 35] V, PPV = 230 W, Se = 110 V/ms.

**Figure 7.** VPV control loop.

**Figure 8.** Frequency response of VPV control loop: (**a**) TVPV(s) loop gain of the VPV control loop and (**b**) closed loop susceptibility, ACL(s). VPV = [24, 30, 35] V, PPV = 230 W, Se = 110 V/ms.

#### **4. MPPT Implementation without VPV and IPV Sensors**

The conventional implementations of MMPT algorithms use current and voltage sensors to measure the voltage (VPV) and the current (IPV) of the PV source as it is shown in Figure 9. The use of voltage and current sensors increases the cost of the power converter. Sensorless MPPT algorithms have been developed [10–13] in order to reduce the number of sensors, yielding a cost reduction. In [11] a sensorless MPPT implementation based on the power balance at the DC-link was presented, which relies on the fact that the power injected into the grid (PG) can be considered almost equal to the power extracted from the PV panels, PG ≈ PPV. In that implementation, the reference of the current injected into the grid (IG) is used as an estimation of PG. This reference current depends on the control loop of VDC so that the sampling frequency of the MPPT algorithm (FMPPT) is limited by the dynamics of that loop. Besides, the reference of IG is sensitive to the variations of the grid voltage and has a low-frequency ripple due to VDC\_R. Moreover, the method explained in Reference [11] is based on the assumption that the amplitude of the grid voltage is stable.

**Figure 9.** Conventional MPPT implementation.

A sensorless MPPT implementation shown in Figure 10 is proposed in this work. In this implementation, there is neither VPV nor IPV sensors and it is assumed that the power injected into the grid is almost equal to the power extracted from the PV panels, PG ≈ PPV, as in Reference [11]. A novelty of this research is that it takes de advantages of the PCC of the DC-DC stage and some SOGI based enhancements applied to the control of the VSI stage to improve the performance of the MPPT implementation.

**Figure 10.** Proposed sensorless MPPT implementation.

A perturb and observe (P and O) algorithm [36] has been programmed in both the conventional (Figure 10) and the proposed sensorless (Figure 11) MPPT algorithms to compare the performance of both implementations. Both implementations use the grid frequency as a time-base to execute the MPPT algorithm. This technique increases the rejection of the disturbances caused by VDC\_R. In previous applications of a similar technique [12], the MPPT algorithm was executed at twice the grid frequency, but that sampling frequency is too fast for the control loops implemented in the proposed sensorless algorithm. In the present study, the MPPT algorithm was executed once every five cycles of the grid, yielding FMPPT = 10 Hz.

**Figure 11.** Experimental setup.

In the conventional implementation, the MPPT algorithm provides the reference of VPV (VPV\_Ref) to the VPV control loop. In the proposed sensorless MPPT, both the VPV sensor and the VPV control loop have been removed, so that the MPPT provides the reference Vc to the analog PCC.

#### *4.1. Estimation of the Power Injected into the Grid in the Sensorless MPPT*

The reference ILf\_Ref\_PK is the peak value of the current injected into the grid, being proportional to PG when the amplitude of the grid voltage is a static value. The signal ILf\_Ref\_PK may have a remarkable low-frequency voltage ripple due to VDC\_R, thus producing a high distortion of the grid injected current along with a disturbance in the estimation of PG. The SOGI based notch filter FNS(s) in series with the regulator GVDC(s) shown in Figure 3 is used to filter out that ripple. The use of the notch filter also enables a high crossover frequency of TVDC(s) without increasing the ripple in ILf\_Ref\_PK, which is useful to implement a fast MPPT algorithm. The crossover frequency of TVDC(s) is FC\_VDC = 45 Hz so that an MPPT of FMPPT = 10 Hz can be implemented.

The estimation of PG, PG\_est, depends on IG and on the grid voltage RMS value (VG) so that variations of VG perturb the calculations of PG\_est. To overcome this issue, the value of the signal v'pk is used to calculate the estimation of PG as it is shown in Equation (17). The signal v'pk is the amplitude of the fundamental of VG and is provided by the FLL-SOGI, not requiring additional computational resources. The signal v'pk has very low sensitivity to the distortion of the grid voltage because it is naturally filtered by the FLL-SOGI.

$$\mathbf{P}\_{\rm G\\_est} = \frac{\mathbf{v}\_{\rm pk}^{\prime}}{\sqrt{2}} \cdot \frac{\mathbf{I}\_{\rm L\,Ref\\_PK}}{\sqrt{2}} = \frac{\mathbf{v}\_{\rm pk}^{\prime} \cdot \mathbf{I}\_{\rm L\,Ref\\_PK}}{2} \approx \mathbf{P}\_{\rm PV} \tag{17}$$

#### *4.2. Implementation of the Perturb and Observe (P&O) Algorithm*

The conventional MPPT algorithm uses the measurements of VPV and IPV to set the operation point of the PV source. The algorithm increases or decreases VPV\_Ref in perturbation steps of a value ΔVPV\_Ref to move the operation point along the I–V curve. In the sensorless implementation shown in Figure 10, instead of the measurements of VPV and IPV, the value of PG\_est expressed by Equation (17) was used. In Reference [11] it was proposed to manage the duty cycle of the switches (DF) of the DC-DC to move the operation point along the I–V curve. In inverters with a reduced DC-link, the high susceptibility of VPV to the ripple VDC\_R disturbs the operation point in the PV panel.

In the proposed sensorless MPPT, the value of IPV is indirectly set by means of the reference signal of the PCC loop, Vc. The use of PCC has two functions: reducing the susceptibility of VPV to the ripple VDC\_R and protecting the DC-DC converter from overcurrents.

The variable Vc is increased or decreased in small steps of a value ΔVc. It is worth pointing out that an increase of Vc causes an increase of IPV, moving the PV operation point to the left of the I–V curve, whereas a decrease of Vc moves the PV operation point to the right of the I–V curve.

#### **5. Results**

Figure 11 depicts the experimental setup. The laboratory tests have been performed using the two-stage inverter presented in Figure 1. The inverter under test was designed for connecting a single PV panel of 230 W to the single phase grid (230 VRMS @ 50 Hz) and has a DC-link with the capacitance calculated in (1), CDC = 50 μF. The challenge of using such a small value of CDC is to keep a low distortion of the grid current and small transient overvoltages at the DC-link. The control of the two-stage inverter has been implemented digitally in a Texas Instruments TMS320F28335 DSP [30] at a sampling frequency (FS) of 40 kHz.

The grid was emulated by means of a Cinergia GE&EL 50 grid emulator and electronic load. The voltage waveform was programmed according to the test waveform described in the international standard IEC-61000-4-7 [37], which has a value: THDV = 1.2%. The PV panel has been emulated by means of an AMETEK TerraSAS ETS1000X10D PV simulator.

#### *5.1. Control of the VSI*

#### 5.1.1. Transients of the DC-link Voltage

Figure 12 shows the transient response of the DC-link voltage (VDC) and the current injected into the grid (IG) when the PV power steps from 150 W up to 200 W. The results shown in Figure 12a were obtained with a crossover frequency Fc\_VDC = 10 Hz, yielding an overvoltage in VDC of 53V from its steady-state value. The response in Figure 12b is obtained with a crossover frequency Fc\_VDC = 45 Hz. The response in Figure 12b was close to five times faster and the overvoltage is only 15 V, which represents a reduction of 72%.

**Figure 12.** Power step from 150 W to 200 W. CDC: 50 μF. (**a**): FcTVDC: 10 Hz. (**b**): FcTVDC: 45 Hz.

5.1.2. Influence of the SOGI Notch in the Distortion of the Current Injected to the Grid

The increment of the crossover frequency Fc\_VDC involves a higher susceptibility to VDC\_R, which increases the harmonic distortion of the current injected into the grid (THDi). Figure 13 depicts the current injected into the grid (IG) using two different regulators for controlling VDC. In Figure 13b, the regulator GVDC(s) of the VDC control loop is the PI shown in Equation (9) with a crossover frequency Fc\_VDC = 53 Hz. The waveform of IG in Figure 13b was obtained using the same PI regulator, but in series with the SOGI notch filter FNS(s) shown in Equation (10), centered at 100 Hz. The crossover frequency has been slightly reduced, Fc\_VDC = 45 Hz, taking into account the addition of FNS(s).

**Figure 13.** THDi of the current injected into the grid at PG = 200 W. Grid: EN-61000-4-7 (1.2% THDv). Fc\_VDC = 50 Hz. (**a**): PI Reg.: GvDC(s). (**b**): PI Reg. + Notch filter: GvDC(s) + FNS(s).

Both tests have been performed injecting 200 W to the grid. The controller of the current injected into the grid (IG) is formed by the P+R regulator in series with the HC expressed by Equation (6). The resulting values of the THDi are 21.51% and 0.96%, respectively. This result indicates the effectiveness of the notch filter for reducing the THDi in spite of the small DC-link.

The following tests of the THDi have been performed for values of PG in the range PG = [40, 180] W. The results are shown in Figure 14. The purple trace represents the values of THDi obtained without the SOGI notch FNS(s) and the green trace represents the values obtained with FNS(s). The THDi without notch varies from 21.51% to 22.43%, clearly exceeding the limits of the IEEE519 standard [4] (5%), shown by the red line. The values of the THDi obtained with FNS(s) vary from 0.96% to 3.14%, widely complying with IEEE519 in the whole range of PG values. In all the measurements the distortion of the grid voltage is THDv = 1.2%.

**Figure 14.** THD of IG. Grid: EN-61000-4-7 (1.2% THDv). Linear sweep of power injected into the grid (PG) from 40 W to 180 W at grid frequency FG = 50 Hz.

#### 5.1.3. Loop Gain Measurement

The loop gain TVDC(s) has been validated by means of loop gain measurement procedures [38–40]. The setup of the test is shown in Figure 15. An NF FRA5097 frequency response analyzer (FRA) is configured to perform an AC sweep from 2 Hz to 20 kHz. The signal generated by the oscillator of the FRA (vOSC\_A) is acquired by the DSP, which carries out the control of the inverter through an internal 12-bit ADC. The acquired signal (vOSC) was digitally injected into the control loop as a perturbation. Both, vDC and vDC+vOSC signals were adapted digitally to be loaded into the pulse width modulation (PWM) unit of the DSP. The offset of the signals was removed through digital high pass filters, and then the amplitudes are digitally adjusted to maximize the resolution of the PWM. The PWM signals (vDC)PWM and (vDC + vOSC)PWM were measured by the FRA. The PWM signals were filtered by the FRA through its internal tracking filter. The loop gain measurement of TVDC(s), shown in Figure 16,was performed at PG = 230 W. The results show the similarity between the experimental and theoretical Bode plots of TVDC(s).

#### *5.2. MPPT*

The performance of the sensorless "perturb and observe" (P and O) MPPT algorithm presented in this study has been compared to a conventional implementation, which uses sensors to measure VPV and IPV. The experimental tests have carried out to measure the start-up time until the maximum power point (MPP) was reached, and the performance at the MPPT under irradiation transients. All the tests were performed with an MPPT sampling frequency, FMPPT = 10 Hz. The perturbation step value programmed in the conventional implementation was ΔVPV\_Ref = 300 mV. In the sensorless MPPT the perturbation step was ΔVc = 12.5 mV, which corresponds to a 250 mA step in IPV in the operation region close to the MPP. It is worth pointing out that different values are perturbed in both MPPTs (either VPV\_Ref or ΔVc) and that the DC-DC converter operates in discontinuous conduction mode. Both facts prevent finding an equivalent value of the perturbation step in both MPPTs.

**Figure 15.** Loop gain measurement setup.

**Figure 16.** Loop gain frequency response of TvDC(s). PG = 230 W, CDC = 50 μF. Blue: theoretical, red: experimental.

#### 5.2.1. Start-up Time to Reach the MPP

Figure 17 shows the evolution of the operation point at the PV source (VPV, IPV and PPV) from the start-up until the MPP was reached. The conventional implementation was faster (2.75 s) than the sensorless (12.6 s) due to the differences in the perturbation step, but once the MPP is achieved, both implementations continue at the MPP.

**Figure 17.** MPPT response from the start-up until the MPP is reached. FMPPT: 10 Hz. (**a**): Conventional, VPV step: ΔVPV\_Ref = 300 mV. (**b**): Sensorless, PCC Vc step: ΔVc = 12.5 mV (≈250 mA close to the MPP).

#### 5.2.2. MPPT Performance Close to the MPP

A key factor in the MPPT performance is the accuracy of the PV estimation and the dispersion of the operation point from the MPP along the I–V curve. The results shown in Figure 18 were obtained tracking the MPP of the 230 W PV panel at constant irradiation of 1000 W/m<sup>2</sup> during 50 s. The results in Figure 18a correspond to the conventional MPPT algorithm and the results in Figure 18b correspond to the sensorless algorithm. In both implementations, the operation points are very close to the MPP. The power extracted from the PV panel during the tests is shown in Table 4. In order to calculate the MPP tracking performance, the quotient between the average power and the peak power measured in each experiment has been used as a reference, as it is expressed by Equation (18). Although the sensorless algorithm presents a very slight disadvantage (0.06%) in terms of the tracking efficiency, the performance of both methods is almost the same.

**Figure 18.** Dispersion of the operation point near the MPP. Total test time: 50 s. FMPPT = 10 Hz. (**a**): Conventional, VPV step: ΔVPV\_Ref = 300 mV. (**b**): Sensorless, PCC Vc step: ΔVc = 12.5 mV (≈250 mA near the MPP).


**Table 4.** Performance of the MPPT algorithms at the MPP under constant irradiation (1000 W/m2).

The tracking of the MPP under heavy irradiation transients is shown in Figure 19. The tests have been performed for 50 s following the stages shown in Table 5. The black traces are the IV curve of the PV panel at 1000 W/m<sup>2</sup> and 600 W/m2. The blue traces represent the evolution in stage 2, during the reduction of the irradiation, and the red trace represents the evolution in stage 4, during the increase of irradiation.

**Figure 19.** Irradiation transients test. Total time: 50 s. FMPPT = 10 Hz. (**a**) Conventional, VPV step: ΔVPV\_Re = 300 mV. (**b**) Sensorless, PCC Vc step: ΔVc = 12.5 mV (≈250 mA near the MPP).

**Table 5.** Stages of the transient radiation test.


The power extracted from the PV panel during the tests is shown in Table 6. During stages 1, 3 and 5, both MPPTs tracked the MPP as was expected from the results shown in Table 4. During stage 2, the sensorless MPPT was not as accurate as the conventional one, but during stage 4 the sensorless MPPT exhibited a smaller dispersion of the operation point and, thus, a higher accuracy than the conventional one. It can be stated that the MPPT performance of both MPPTs was highly similar. Although the conventional MPPT had a faster start-up and was slightly more accurate in some tests, the performance of the proposed sensorless MPPT near the MPP was almost equal to the conventional implementation, but at a lower cost.


**Table 6.** Transient radiation test results.

#### **6. Conclusions**

This paper focused on control techniques which help to reduce the cost of two-stage grid-connected PV inverters. In previous works, sensorless algorithms and techniques to reduce the capacitance of the DC-link have been proposed separately. The present study attempts to integrate both trends in a single implementation. The reduction of the DC-link capacitance requires a faster control loop to keep the DC-link voltage within safe values. However, this practice usually increases the THD of the current injected into the grid. The DC-link can be reduced in a factor of ten compared to standard values of the DC-link capacitance, yet with good values of the THDi, by increasing the speed of the voltage control loop and using a frequency adaptive notch filter tuned at twice the grid frequency.

SOGI structures are an effective way to implement tuned filters both in the inverter voltage loop and in its current loop. It is shown that the crossover frequency of the DC-link voltage control loop can be increased from typical values of 10 Hz to a value around 50 Hz, yet getting a THDi value lower than 1%. The combination of a fast voltage loop with a SOGI notch filter allows the reduction of the DC-link capacitance. An FLL-SOGI is used to get the value of the grid frequency to tune the SOGI controllers in the inverter control loops.

Summing up, this study proposes the implementation of the MPPT with no sensors at the PV source side, which takes advantages of the SOGI based enhancements implemented in the control of the converter. The high dynamics achieved by the inverter controllers yield a performance of the sensorless MPPT very similar to that of conventional MPPT implementations, but at a lower cost.

**Author Contributions:** R.G.-M., G.G. and E.F. proposed the main idea, performed the investigation and designed the experiments; R.G.-M. and G.G. developed the software, performed the experiments, and wrote the paper. M.L. and S.M. processed the data from the experimental results and reviewed the paper. G.G. and E.F. lead the project, acquired the funds for research.

**Funding:** This work has been co-financed by the Spanish Ministry of Economy and Competitiveness (MINECO) and by the European Regional Development Fund (ERDF) under Grant ENE2015-64087-C2-2-R and by the Spanish Ministry of Science, Innovation and University (MICINN) and the European Regional Development Fund (ERDF) under Grant RTI2018-100732-B-C21. The European Regional Development Fund (ERDF) and the Generalitat Valenciana (GVA) financed the purchase of the Cinergia GE&EL 50 grid emulator and electronic load used during the experimental part of this work under Grant IDIFEDER/2018/036.

**Conflicts of Interest:** The authors declare no conflict of interest. The funders had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript, or in the decision to publish the results.

#### **Appendix A**

Table A1 shows the approximated costs of the prototype built for this study at the present time. It is important to note that a 6% cost saving is estimated.


**Table A1.** Approximate costs of the prototype.

#### **References**


© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## *Article* **A Novel Fast MPPT Strategy for High Efficiency PV Battery Chargers**

#### **Jose Miguel Espi \* and Jaime Castello**

Department of Electrical Engineering, University of Valencia, Avd. Universitat S/N, 46100 Burjassot-Valencia, Spain; jaime.castello@uv.es

**\*** Correspondence: jose.m.espi@uv.es; Tel.: +34-963-543-450

Received: 26 February 2019; Accepted: 19 March 2019; Published: 25 March 2019

**Abstract:** The paper presents a new maximum power point tracking (MPPT) method for photovoltaic (PV) battery chargers. It consists of adding a low frequency modulation to the duty-cycle and then multiplying the ac components of the panel voltage and power. The obtained parameter, proportional to the conductance error, is used as a gain for the integral action in the charging current control. The resulting maximum power point (MPP) is very still, since the integral gain tends to zero at the MPP, yielding PV efficiencies above 99%. Nevertheless, when the operating point is not the MPP, the integral gain is large enough to provide a fast convergence to the MPP. Furthermore, a fast power regulation on the right side of the MPP is achieved in case the demanded power is lower than the available maximum PV power. In addition, the MPPT is compatible with the control of a parallel arrangement of converters by means of a droop law. The MPPT algorithm gives an averaged duty-cycle, and the droop compensation allows duty-cycles to be distributed to all active converters to control their currents individually. Moreover, the droop strategy allows activation and deactivation of converters without affecting the MPP and battery charging operation. The proposed control has been assayed in a battery charger formed by three step-down converters in parallel using synchronous rectification, and is solved in a microcontroller at a sampling frequency of 4 kHz. Experimental results show that, in the worst case, the MPPT converges in 50 ms against irradiance changes and in 100 ms in case of power reference changes.

**Keywords:** photovoltaic (PV); maximum power point (MPP); maximum power point tracking (MPPT); perturbe and observe (P&O); incremental conductance (IC)

#### **1. Introduction**

Photovoltaic (PV) battery chargers are designed to maximize the energy extracted from solar panels. This requires the maximization of both electronic and PV efficiencies. The electronic efficiency is increased by a parallel configuration of multiple power converters and a synchronous rectification implemented in each one. Paralleling permits activating/deactivating the converters so that each active converter works near its nominal power, thus saving the conduction and switching losses of the inactive converters. Moreover parallelized solutions allows power scaling and increases reliability. On the other hand, the PV efficiency is defined as the ratio between the average power extracted from the panels and the maximum power that can be extracted at a given irradiance. Maximum power point tracking (MPPT) algorithms automatically adjust the PV voltage at the converter input to get the maximum power for each present irradiance level. When a change in irradiance occurs, an ideal MPPT algorithm should reach the new maximum power point (MPP) as fast as possible, and then remain at the MPP without fluctuations. However, in practice, the MPPT algorithms exhibit oscillations around the MPP and take a certain time to converge, penalizing the PV efficiency.

The MPPT strategies can be classified into two main categories: the stand-alone MPPTs [1–5] and the converter-embedded MPPTs [6–10]. A stand-alone MPPT is an independent module that uses the PV voltage and current to determine the input voltage reference to be transmitted to all converters installed. This is typically implemented using perturb and observe (P&O) algorithms. The advantage of the stand-alone method is that it can be used to manage parallelized converters without having to modify their respective controls. In contrast, a converter-embedded MPPT is programmed in the converter control to determine directly the duty-cycle that maximizes the PV power. It is usually implemented using incremental conductance (IC) algorithms [6–8]. Converter-embedded strategies are much faster than stand-alone strategies, thus presenting a higher PV dynamic efficiency, which makes them more suitable in applications where irradiance changes are fast and frequent [11]. However, the parallel multi-stage arrangement becomes difficult to control using a converter-embedded MPPT, as it calculates a single duty-cycle that maximizes the PV power.

In recent research [6,7], new converter-embedded MPPT strategies based on IC have been presented for a step-up converter that combines a fast convergence with a small fluctuation around the MPP. In [7], the static *g*dc and dynamic *g*ac PV conductances are explicitly calculated using a moving average filter (MAF) and a lock-in amplifier (LIA) respectively, and then compared and regulated to be equal using an integrator. The ac components used to calculate *g*ac are the switching ripple components of the PV voltage and current. The MPPT converges to the MPP in approximately 400 ms. As the method requires a small input capacitance to measure *g*ac, the current ripple is present in the PV current and the MPP fluctuates at the switching frequency. To minimize this effect, a large inductance was utilized to achieve a PV efficiency of 99%. More recently, in [6], the IC is solved in a traditional way by incrementing or decrementing the duty-cycle depending on the sign of the conductance error *g*ac − *g*dc. The ac components used to evaluate the incremental PV magnitudes are the natural oscillations of the input filter. The MPPT settling time was around 300 ms, and the MPP oscillates at the natural frequency of the filter, resulting in a PV efficiency of 97.5%. In both papers, the high frequency used to calculate the PV AC components makes high frequency sampling rates of above 100 kHz necessary, which increases hardware cost and complexity.

This paper presents a new MPPT strategy for step-down battery chargers that combines the benefits of both stand-alone and converter-embedded methods. The proposed MPPT is integrated with the proportional-integral (PI) current regulator, offering a fast convergence to the MPP in less than 100 ms when the demanded power is higher than available maximum PV power, with smooth transitions to regulation when the required power is lower than the available PV power. The basic idea is to insert the conductance error as an additional gain for the PI's integrator when tracking the MPP. This leads to a fast MPPT but a still MPP with PV efficiency higher than 99%, since the conductance error is null at the MPP. A low frequency modulation of 40 Hz is added to the duty-cycle to get the conductance error by multiplying the AC components of PV power and voltage. As a consequence of the low frequency modulation, the proposed MPPT can be solved at 4 kHz sampling rate by a low-cost microcontroller. Additionally, a droop law is proposed to solve the multiple control of parallel converters, proving that converters' currents can be controlled individually without affecting the MPP operation or battery charge. The proposed MPPT with droop has been assayed in the battery charger shown in Figure 1, where electronic efficiency was improved by means of active rectification using *QRj*<sup>1</sup> and *QRj*<sup>2</sup> transistors and blocking transistors *QB*, instead of using Schottky diodes.

**Figure 1.** Photovoltaic (PV) battery charger using three parallelized step-down converters with synchronous rectification.

#### **2. Small-Signal Modeling**

Figure 2 shows a single step-down converter with synchronous rectification, thus operating always in continuous conduction mode. The circuit shows averaged values of transistors currents, *d* · *iL* and (1 − *d*) · *iL*, where *d* is the converter duty-cycle and *r* stands for the inductor series resistance. All relationships can be gathered into the block diagram shown in Figure 3, which constitutes a large-signal model. The function *f*pv solves the current *i*pv of the PV panel, using the characteristic I-V curve for a given irradiance and voltage *v*pv. In case of a parallelized step-down converters, *d* and *iL* are vectors containing all duty-cycles and inductor currents, *d* · *iL* is a dot product and *d* · *v*pv is a vector. Notice that, if all converters use the same filtering inductance and receive approximately the same duty-cycle, the presented model is valid just considering that *iL* = ∑ *iLj* ≡ *i*bat is the battery charging current, and *L* = *Lj*/*n*, *r* = *rj*/*n*, where *n* is the number of active parallelized converters, since all active inductors can be considered as operating in parallel.

**Figure 2.** Large-signal averaged circuit of a PV step-down converter with active rectification.

**Figure 3.** Block diagram of the large-signal model.

As the battery voltage *V*o changes much slower than all other circuit variables, it can be assumed constant and the small-signal block diagram results as depicted in Figure 4, where the PV panel voltage and current are related through the incremental conductance *<sup>g</sup>*ac <sup>=</sup> <sup>−</sup> <sup>d</sup>*i*pv <sup>d</sup>*v*pv . From this figure, the duty-to-voltage and duty-to-current small-signal transfer functions are deduced

$$G\_{\rm{\nu}}(s) \equiv \frac{\vec{v}\_{\rm{pv}}}{\vec{d}}(s) = \frac{G\_{\rm{cap}}(s) \cdot (I\_L + DV\_{\rm{pv}} G\_{\rm{ind}}(s))}{1 - D^2 G\_{\rm{cap}}(s) G\_{\rm{ind}}(s)},\tag{1}$$

$$\mathcal{G}\_{l}(s) \equiv \frac{\tilde{l}\_{L}}{d}(s) = \frac{\mathcal{G}\_{\rm ind}(s) \cdot \left(V\_{\rm PV} + D I\_{L} \mathcal{G}\_{\rm cap}(s)\right)}{1 - D^{2} \mathcal{G}\_{\rm cap}(s) \mathcal{G}\_{\rm ind}(s)},\tag{2}$$

where *G*cap(*s*) = −1/(*Cs* + *g*ac) and *G*ind(*s*) = 1/(*Ls* + *r*).

**Figure 4.** Small-signal model of the power converter and PV panel.

Using the steady-state relationships *DV*pv = *V*<sup>o</sup> and *D IL* = *I*pv, basic manipulations reveal that the small-signal transfer functions (1) and (2) can be expressed as

$$\mathcal{G}\_{\upsilon}(s) = \frac{-k\_{\upsilon}(\frac{s}{\omega\_{\upsilon}} + 1)}{(\frac{s}{\omega\_{\upsilon}})^2 + 2\mathcal{J}(\frac{s}{\omega\_{\upsilon}}) + 1},\tag{3}$$

$$G\_{l}(s) = \frac{k\_{l}(\frac{s}{\omega\_{z\_{l}}} + 1)}{(\frac{s}{\omega\_{n}})^{2} + 2\zeta(\frac{s}{\omega\_{n}}) + 1},\tag{4}$$

where the natural frequency *ω<sup>n</sup>* and damping factor *ζ* are

$$
\omega\_{\rm tr} = \sqrt{\frac{g\_{\rm ac}r + D^2}{L\mathcal{C}}},\tag{5}
$$

$$\zeta = \frac{1}{2} \cdot \frac{(r/Z\_b + \text{g}\_{\text{ac}} Z\_b)}{\sqrt{\text{g}\_{\text{ac}} r + D^2}},\tag{6}$$

being *Zb* <sup>=</sup> <sup>√</sup>*L*/*C*. The frequencies of the zeros *<sup>ω</sup>zv* and *<sup>ω</sup>zi* are

$$
\omega\_{z\_{\nu}} = \frac{r + DV\_{\boldsymbol{\theta}}/I\_{\text{PV}}}{L} \approx \frac{DV\_{\boldsymbol{\theta}}}{LI\_{\text{PV}}} = \frac{V\_{\text{o}}^2}{P\_{\text{PV}}L} \tag{7}
$$

$$
\omega\_{z\_i} = \frac{\mathcal{g}\_{\rm ac} - \mathcal{g}\_{\rm dc}}{\mathcal{C}}\_{\prime} \tag{8}
$$

where *<sup>g</sup>*dc <sup>=</sup> *<sup>I</sup>*pv *<sup>V</sup>*pv is the static conductance, and the dc-gains are

$$k\_{\upsilon} = \frac{I\_{\text{pv}}r/D + V\_{\text{o}}}{g\_{\text{ac}}r + D^2} \approx \frac{V\_{\text{o}}}{g\_{\text{ac}}r + D^2} \tag{9}$$

$$k\_{\rm i} = \frac{V\_{\rm PV}(g\_{\rm ac} - g\_{\rm dc})}{g\_{\rm ac}r + D^2}. \tag{10}$$

Equation (8) indicates that *Gi* presents a non-minimum phase zero when operating at the left of the MPP, where *g*ac < *g*dc. In addition, Equation (10) shows that *Gi* presents null dc-gain when operating exactly at the MPP.

#### **3. Working Principles of the Proposed MPPT**

This paper proposes to embed an MPPT strategy in a PI current controller, so that it behaves as a normal PI when the power demanded by *i* ∗ bat ≡ *i* ∗ *<sup>L</sup>* is smaller than the present PV power *p*pv = *i*pv*v*pv, that is, when *i* ∗ bat < *iL*, but it opens the current regulation loop and starts MPP tracking when *i* ∗ bat > *iL*.

Regarding the MPPT strategy, the basic idea is to detect the slope of the P–V curve and use it as a gain for the integral action of the PI. This slope is detected by adding a small amplitude modulation *dm* to the duty-cycle

$$d\_m(t) = d\_{m\_{pk}} \cdot \cos(\omega\_m t),\tag{11}$$

where *<sup>ω</sup><sup>m</sup>* = <sup>2</sup>*<sup>π</sup>* · 40 rad/s and *dm*pk = <sup>5</sup> · <sup>10</sup>−<sup>3</sup> ≡ 0.5% have been used. According to the duty-to-voltage transfer function *Gv*, this produces a voltage modulation *vm* in the PV panel. As the modulation frequency *ω<sup>m</sup>* is much smaller than *ωzv* , it holds that *vm* = −*kv* · *dm*. The voltage modulation in turn generates a power modulation *pm* as shown in Figure 5.

**Figure 5.** Detection of the PV panel operation: to the left of the MPP (red, with *pm* and *vm* in-phase) and to the right of the MPP (green, with *pm* and *vm* in anti-phase).

At a given operating point determined by the PV voltage and current levels (*V*pv, *I*pv), the differential increment of the power is

$$\mathrm{d}p\_{\mathrm{PV}} = l\_{\mathrm{PV}} \cdot \mathrm{d}v\_{\mathrm{PV}} + V\_{\mathrm{PV}} \cdot \mathrm{d}i\_{\mathrm{PV}} \tag{12}$$

and therefore the power slope is

$$\frac{\mathrm{d}p\_{\mathrm{PV}}}{\mathrm{d}v\_{\mathrm{PV}}} = I\_{\mathrm{PV}} + V\_{\mathrm{PV}} \frac{\mathrm{d}i\_{\mathrm{PV}}}{\mathrm{d}v\_{\mathrm{PV}}} = V\_{\mathrm{PV}}(\mathrm{g}\_{\mathrm{dc}} - \mathrm{g}\_{\mathrm{ac}}),\tag{13}$$

which reveals the well known incremental conductance condition *g*dc = *g*ac for any local MPP.

In order for the MPPT algorithm to get the value of this slope, a parameter *δ* is calculated as

$$\delta(t) \equiv -k\_m \cdot p\_m(t) \cdot \upsilon\_m(t). \tag{14}$$

Taking into account that *pm*(*t*) ≈ ( d*p*pv <sup>d</sup>*v*pv ) · *vm*(*t*) and Equation (13), we get

$$\delta(t) = k\_{\rm m} V\_{\rm PV} k\_v^2 (\mathcal{g}\_{\rm ac} - \mathcal{g}\_{\rm dc}) \cdot d\_m^2(t) \tag{15}$$

and, using Equation (11),

$$\delta(t) = \frac{1}{2} k\_{\text{\textquotedblleft}V} V\_{\text{PV}} (k\_{\text{\textquotedblleft}d\_{\text{\textquotedblleft}pk})})^2 (\text{g}\_{\text{ac}} - \text{g}\_{\text{dc}}) (1 + \cos(2\omega\_{\text{\textquotedblleft}t})),\tag{16}$$

which can be separated as *δ*(*t*) = *δ*dc + *δ*ac(*t*), where *δ*ac = *δ*dc · cos(2*ωmt*) and

$$\delta\_{\rm dc} = \frac{1}{2} k\_{\rm m} V\_{\rm PV} (k\_{\rm v} d\_{\rm m\_{pk}})^2 (\mathcal{g}\_{\rm ac} - \mathcal{g}\_{\rm dc}). \tag{17}$$

The strategy of the proposed MPPT method is to insert *δ* as a multiplying factor between the current error *e* and the PI controller, as shown in Figure 6. When the current error becomes positive, the MPPT is activated by adding the duty-cycle modulation *dm*, and the error is multiplied by *δ*. Only the dc value of *δ* (17) generates a dc value for the PI to increase or to decrease the duty-cycle. On the left side of the MPP (point 1 in Figure 7a), it holds that *δ*dc < 0 and therefore the duty-cycle *d* decreases and *v*pv increases (Figure 6b). On the right side of the MPP (point 2 in Figure 7a), *δ*dc > 0 and *v*pv decrease (Figure 6c). Hence, when the MPPT is activated, the operating point climbs power automatically to a local MPP (yellow point in Figure 7a). The speed of convergence to the MPP is determined by the integrator gain *kiI* and the maximum values of *δ* and *e*. In the proposed implementation, *δ* is constrained to the interval [−1, 1] and the error is upper-limited to 1A.

Since |*δ*| < 2|*δ*dc|, and *δ*dc tends to zero as the operating point approaches the MPP, the power at the MPP is quiescent without oscillations, even if the integrator is set for a fast MPPT, resulting in an excellent static efficiency.

**Figure 6.** Control block diagram (**a**) and MPP tracking situations: (**b**) to the left of the MPP; and (**c**) to the right of the MPP.

When the current error becomes negative due to an increase in irradiance or a decrease in battery power demand (points 3 or 4 in Figure 7b), *dm* is set to zero and *δ* is set to 1, so the MPPT is transformed into a normal PI current regulation (Figure 6a with *δ*dc = 1). In this situation, the only possible stable point is on the right side of the MPP (yellow point in Figure 7b).

**Figure 7.** Control performance against the two possible scenarios: (**a**) requested power is higher than available PV power; (**b**) requested power is lower than available PV power.

#### **4. Description of the Implemented Solution**

The proposed strategy has been carried out as depicted in Figures 8–11.

Figure 8 shows the typical PI-based control to regulate the battery voltage. The voltage reference *v*∗ bat is around 14.7 V/battery during the absorption charge stage, and it is around 13.6 V/battery temperature-compensated float voltage during the float charge stage. The PI determines the charging current *i*bat, which is limited to *i*max = 60 A. If the battery SOC is below 80%, this results in a constant current charge at *i*max and battery voltage below *v*∗ bat, while at a higher SOC the battery is charged at a constant voltage *v*∗ bat with current below *i*max. The voltage reference is changed to the temperature-compensated float voltage when charging current is below 10−3*C*<sup>10</sup> or absorption time exceeds the 8-hour limit.

**Figure 8.** Block diagram of the battery voltage regulation to calculate the battery current reference.

Figure 9 illustrates the implemented strategy to detect the converter working point position relative to the MPP. If the current error is positive and the PV current is higher than *i*start = 50 mA, the MPPT is started by setting MPPT\_ON = 1, and *δ* is calculated as shown in Equation (14). The voltage *vm* and power *pm* modulations are extracted from the measured *v*pv and *p*pv, respectively, by means of band-pass digital filters *G*BP(*z*). These filters were implemented using second-order all-pass filters *G*AP(*z*) as

$$\mathcal{G}\_{\rm BP}(z) = \frac{1}{2} \left[ 1 - \mathcal{G}\_{\rm AP}(z) \right] \tag{18}$$

and

$$G\_{\rm AP}(z) = \frac{k\_2 z^2 + k\_1 (1 + k\_2) z + 1}{z^2 + k\_1 (1 + k\_2) z + k\_2},\tag{19}$$

whose coefficients are calculated as

$$\begin{aligned} k\_1 &= -\cos(\omega\_0 T), \\ k\_2 &= \frac{1 - \tan(\omega\_{\text{BW}} T/2)}{1 + \tan(\omega\_{\text{BW}} T/2)}, \end{aligned} \tag{20}$$

for a given center-frequency *ω*0, bandwidth frequency *ω*BW and sampling period *T* = 1/ *fs*. Using *ω*<sup>0</sup> = *ωm*, *ω*BW = 2*π* · 80 rad/s and *fs* = 4 kHz, the programmed all-pass filter resulted in

$$G\_{\rm AP}(z) = \frac{0.8816 \, z^2 - 1.8779 \, z + 1}{z^2 - 1.8779 \, z + 0.8816}. \tag{21}$$

Gains *kpm* and *kvm* must fit the power and voltage oscillations to the interval [−1,1], resulting *km* ≡ *kpm* · *kvm* in Equation (14). Values *kpm* = 0.5 and *kvm* = 2 are used to set a high *δ* sensitivity, while ensuring the non-saturation of *δ* when operating at the neighbourhoods of the MPP.

The condition *i*pv ≤ *i*start in Figure 9 inhibits the MPPT at the start-up, where the duty-cycle is small and operation is in open-circuit with *i*pv = 0, and hence without any chance to get information by power modulation. Thus, the converter starts on the right side of the MPP with *δ* = 1, i.e., with a conventional PI action.

On the other hand, when the current error becomes negative, MPPT\_ON is set to zero and *δ* = 1.

**Figure 9.** Detection of the operating point position relative to the MPP using the parameter *δ*.

Figure 10 shows the proposed modified PI current control including MPPT action. As mentioned, if *e* < 0, then *δ* = 1 is applied, the duty-cycle modulation stops, and thus the current control is a conventional PI control. On the contrary, when *e* > 0, the duty-cycle modulation is initiated and *δ* is calculated. The current error *e* is limited to 1 A, so that the speed of convergence to the MPP is given by the integrator gain *kiI* and the value of *δ*, but it is not dependent on the current reference level. As the converter approaches the MPP, *δ* tends to zero and the PI slows down the duty-cycle variation to finally get a still MPP operation.

**Figure 10.** Proposed proportional-integral (PI) current control modification to achieve In-Cond MPPT.

The proposed modified PI has to ensure the stability and fast response of the control shown in Figure 6a for all operating points on the right side of the MPP. The design worst case is with *δ*dc = 1, that is, when MPPT action is inhibited and the controller behaves as a PI compensator. Expressing the PI in its continuous form

$$\text{PI}(\text{s}) = k\_{p\text{I}} + \frac{k\_{i\_l}}{\text{s}} = k\_{p\text{I}} \cdot \frac{\text{s} + \omega\_{\text{z}}}{\text{s}},\tag{22}$$

the zero *ω<sup>z</sup>* is designed at the minimum value of the natural frequency

$$
\omega\_{\overline{z}} = \omega\_{\text{\tiny min}} \approx \frac{V\_{\text{o}}}{V\_{\text{MPP}} \sqrt{LC}} \tag{23}
$$

in order to get the maximum phase margin as possible.

The gain *kpI* is designed to achieve a high control bandwidth by setting the loop-gain crossover frequency *ω<sup>c</sup>* at *ω*nyq/6 = *π*/(6*T*). At these high frequencies, the duty-to-current transfer function (4) can be approximated by

$$G\_l(s) \approx \frac{k\_i \omega\_n^2}{\omega\_{z\_i s}} = \frac{V\_{\rm PV}}{Ls} \tag{24}$$

that exhibits the maximum gain (worst case) when operating at the open-circuit voltage *V*pv = *V*oc and with all three converters working in parallel *L* = *Lj*/3. Hence, a simple design equation for *kpI* yields 1 = |PI(*jωc*)| · *V*oc/(*Lωc*), or

$$k\_{pl} = \frac{L\omega\_c^2}{V\_{\infty} \cdot |j\omega\_c + \omega\_z|}\tag{25}$$

and

$$k\_{l\_l} = k\_{p\_l} \cdot \omega\_z \tag{26}$$

The designed values for *kpI* and *kiI* are given in Table 1.

#### **Table 1.** Control parameters.


Figure 12 shows the resulting Bode diagrams of the open-loop gain at the two ending points of the stable region (*V*pv = *V*MPP and *V*pv = *V*oc), where the gain crossover frequency *ω<sup>c</sup>* and both phase and gain margins are detailed. In Figure 12a, the converter operates at the open-circuit voltage with a control bandwidth *ω<sup>c</sup>* = 973 rad/s that ensures a fast response. However, when the converter reaches the MPP in Figure 12b, the control bandwidth is strongly reduced to *ω<sup>c</sup>* = 0.1 rad/s, much lower than the modulation frequency *ωm*, and therefore the MPP operation is not affected by the modulation and remains constant without oscillations. The gain and phase margins are large enough to ensure a robust stability in the whole operating range *V*MPP ≤ *V*pv ≤ *V*oc.

**Figure 11.** Droop correction to equalize and control output currents of all converters.

**Figure 12.** Open-loop Bode diagrams when operating at: (**a**) *V*pv = *V*oc; and (**b**) *V*pv = *V*MPP.

Simulated results are presented in Figure 13, obtained using PSIM©, where steps in the charging current reference are applied from 0 A to 20 A. The converter moves from the open-circuit voltage to the MPP in less than 100 ms. The calculated PV efficiency is *η* = 100 · 422.6/422.9 = 99.9%. The ac components *pm* and *vm* extracted by the band-pass filters and the parameter *δ* are also shown. The power *pm* oscillates at frequency *ω<sup>m</sup>* when operating out of the MPP, but at 2*ω<sup>m</sup>* when operation is at the MPP.

Despite a duty-cycle, *d* is calculated to maximize the extracted PV power if needed, it is not advisable to directly apply it to all parallelized converters, since they have slight differences in the inductors series resistances *rj* and turn-on/off delays, which may cause the currents to unbalance. Instead, a droop strategy is proposed as shown in Figure 11 to distribute duty-cycles *dj* to the converters. If the *j*-th converter is active, its duty-cycle is calculated as

$$d\_{\dot{j}} = d + \Delta d\_{\dot{j}} ; \quad \Delta d\_{\dot{j}} = k\_{dr} \cdot (i\_{L\_{\dot{j}}}^{\*} - i\_{L\_{\dot{j}}}) ; \tag{27}$$

where *i* ∗ *Lj* = *<sup>i</sup>*bat/*<sup>n</sup>* is the current reference, *<sup>n</sup>* is the number of active converters and *<sup>i</sup>*bat = <sup>∑</sup> *iLk* . On the contrary, when a converter is not active, all transistors *Qj*, *QRj*<sup>1</sup> and *QRj*<sup>2</sup> in Figure 1 are switched off and therefore *iLj* is zero. It can be noticed that the averaged value of all applied duty-cycles to the active converters is *<sup>d</sup> <sup>n</sup>*

$$\sum\_{j}^{n} \Delta d\_{j} = 0 \; ; \quad \sum\_{j}^{n} d\_{j} = nd. \tag{28}$$

Each active converter current satisfies

$$\dot{a}\_{L\_j} = (d\_j v\_{\text{PV}} - V\_{\text{o}}) G\_{\text{ind}} \tag{29}$$

and adding for all active converters

$$i\_{\rm bat} = \sum\_{j}^{n} i\_{L\_{j}} = (\sum\_{j}^{n} d\_{j} v\_{\rm PV} - nV\_{\rm o})G\_{\rm ind} = (dv\_{\rm PV} - V\_{\rm o})nG\_{\rm Ind} \tag{30}$$

which shows, as mentioned before, that the parallel configuration behaves as a single stage with an averaged duty-cycle *d* and with all inductances *G*ind in parallel.

**Figure 13.** Simulated transient response against current reference steps. The power alternates between zero and the MPP—from top to bottom: *i*pv, *v*pv, *p*pv, *pm*, *vm* and *δ*.

Equation (29) in steady-state results *iLj* = (*djv*pv − *V*o)/*rj*, and therefore a variation in the duty-cycle Δ*dj* generates a variation in the converter current Δ*iLj* = Δ*djv*pv/*rj*, which gives an estimation for the droop gain *kdr* in (27). In order to guarantee the currents compensation, the droop gain is designed as

$$k\_{dr} \gtrsim \frac{\max r\_j}{V\_{\text{MPP}}}.\tag{31}$$

The proposed droop strategy allows for controlling each converter current individually to optimize the overall efficiency. For instance, when the power is lower than one third of the total installed power, only one converter is active and the other two are kept off, so that switching losses are minimized. When power is between one third and two thirds of the total power, two converters are active and share the power from 50% to 100% of their rated power. Finally, when the power is higher than two thirds of the total, all three converters are activated and share the power from 66% to 100% of their rated power. Moreover, a rotation strategy is also implemented to alternate the active and inactive converters to equalize transistors aging and to minimize thermal cycling. It will be shown in the experimental results that the converters' activation and deactivation for losses rotation does not have a transient effect on the MPP operation, and hence it can be done without affecting the photovoltaic efficiency.

#### **5. Experimental Results**

The presented MPPT strategy with droop was assayed in the battery charger shown in Figure 1, whose main parameters are specified in Table 2. Transistors *Qj* (*j* = 1, 2, 3) are fired at *fsw* = 40 kHz with complementary drive for the rectification transistors *QRj*<sup>1</sup> and *QRj*<sup>2</sup> . Schottky diodes *Dj* drive only during the PWM dead-time. The anti-series transistors *QRj*<sup>2</sup> impede the conduction of the lossy body-diodes of *QRj*<sup>1</sup> during the dead-time. The blocking transistors *QB* are always on, except if the input voltage gets close to the battery voltage or a if a panel reverse current is detected. The converter was designed to charge lead acid batteries with voltage ranging from 12 V to 48 V and charging current up to 60 A. Though the converter is 3 kW rated, presented experimental results were obtained at a lower power, using the 480 W E4350B solar array simulator from Agilent/Keysight Technologies©.

The control is resolved in a RX630 microcontroller from Renesas Electronics© at 4 kHz, using three independent PWM outputs to drive the three converters, and six analog input channels for the PV voltage and current, the battery voltage and the three output currents. A human interface device (HID) class USB communication, readable with computers, tablets, etc., has also been implemented to send internal data at a speed of 2 kB/s.

**Table 2.** Power converter parameters.


Figure 14 shows the P–V irradiance curves programmed in the E4350B to test the performance of the proposed MPPT method. Initially, the converter is turned on with irradiance level corresponding to curve PV1, which is maintained for five seconds. Then, the irradiance is suddenly changed to the curve PV2 and is maintained for another five seconds, and so on with curves PV3 and PV4. Finally, the converter is turned off with irradiance PV4. Red circles indicate the converter operating point motion and were obtained in real time via USB with a 2 ms sampling period. It can be seen that there is only one red circle in the transition between two consecutive MPPs, which indicates that the MPPT takes less than 4 ms to converge when the irradiance decreases.

**Figure 14.** Operating point shift during irradiance step changes from PV1 to PV4 (samp. time = 2 ms).

In more detail, Figures 15 and 16 show transients produced by a sudden increase and decrease in irradiance, respectively. In Figure 15, the irradiance is changed from PV4 to PV1. The new MPP is reached in approximately 50 ms. The 40 Hz oscillation is barely distinguishable in the PV current or voltage, and it cannot be observed in the PV power. In Figure 16, the irradiance is decreased from PV1 to PV4, and, as mentioned before, most of the power transition takes less than 4 ms.

**Figure 15.** Transient response against an irradiance step from PV4 to PV1. Ch1: PV current (2 A/div). Ch2: PV voltage (10 V/div). Math: PV power (100 W/div).

**Figure 16.** Transient response against an irradiance step from PV1 to PV4. Ch1: PV current (2 A/div). Ch2: PV voltage (10 V/div). Math: PV power (100 W/div).

An irradiance change is not the most challenging case in terms of speed response, since the PV voltage and duty-cycle variations are not wide. However, a large step change in the demanded battery charging power requires a significant variation of the duty-cycle and PV voltage, and this is the worst case for the converter settling time. In this sense, Figure 17 shows the transients produced by steps in *i* ∗ bat between 0 A and 20 A when the converter operates at the irradiance PV1 curve. The PV power changes from zero to the MPP, showing a rising time of approximately 100 ms and a falling time around 20 ms. It can be noticed again that the power at the MPP is constant without oscillations.

**Figure 17.** Transient response against a current reference *i* ∗ bat steps between 0 A and 20 A. Ch1: PV current (2 A/div). Ch2: PV voltage (10 V/div). CH4: current reference synchronism digital output. Math: PV power (100 W/div).

Finally, Figures 18 and 19 are intended to show the performance of the droop compensation and the on/off switching of parallelized converters. These figures were obtained when charging batteries at 17 A and 28 V giving the maximum 480 W of the solar simulator. At the beginning of Figure 18, the charge current shown in Ch3 is equally shared by converters 1 and 2 by means of droop action. Then, the current reference of converter 1 is set to zero and the reference of converter 2 is set to the total charge current. After around 2 ms, all of the charge current is provided by converter 2, and the converter 1 is turned off. Moreover, Figure 19 shows the effect of a converter activation. Initially, converter 1 is off and all charge currents are provided by converter 2. Then, converter 1 is turned on and current references of both converters are set to half the charge current. Finally, after 2 ms, the current is equally shared by converters 1 and 2. It can be seen that activation and deactivation of converters have no effect on the charging current and hence do not affect the MPP operation.

**Figure 18.** Droop equalization and converter 1 shut-down. Ch1: converter 1 output current (5 A/div). Ch2: converter 2 output current (5 A/div). Ch3: total output current (5 A/div). Ch4: synchronism digital output.

**Figure 19.** Converter 1 turn-on and current sharing with converter 2 using droop correction. Ch1: converter 1 output current (5 A/div). Ch2: converter 2 output current (5 A/div). Ch3: total output current (5 A/div). Ch4: synchronism digital output.

#### **6. Conclusions**

This paper presents a new fast MPPT method for step-down photovoltaic (PV) battery chargers. The method adds a low frequency modulation to the duty-cycle to calculate the conductance error, which is used as a gain in the current loop. Therefore, it can be considered as another MPPT variant of the incremental conductance (IC) type. This produces a quiescent maximum power point (MPP) since the control bandwidth tends to zero at the MPP, yielding PV efficiencies higher than 99%. However, when the operating point is not close to the MPP, the bandwidth of the current control is around 1 krad/s, resulting in a fast convergence to the MPP.

Furthermore, if demanded power is lower than the available maximum PV power, the proposed design ensures a fast regulation on the right side of the MPP. The presented MPPT exhibits, in the worst case, settling times of 50 ms against irradiance changes, and 100 ms against power reference changes.

In addition, the control problem of a parallel arrangement of converters is solved by means of a droop law. The MPPT algorithm gives an averaged duty-cycle for all active converters, and the droop compensation allows duty-cycles to be distributed to all active converters to control their currents individually. Moreover, the droop strategy allows activation and deactivation of converters without affecting the MPP and battery charging operation.

Finally, it is worth noticing that the proposed battery charger control can be solved at low sampling rates using a low-cost microcontroller.

**Author Contributions:** J.M.E. proposed the main idea, performed the theoretical analysis and wrote the paper. All authors contributed to the practical implementation and experimental validation, paper review and editing. All authors have read and approved the final manuscript.

**Funding:** This research was funded by the Centro para el Desarrollo Technological Industrial (CDTI) under the project grant number IDI-20160123.

**Acknowledgments:** We thank the company Dismuntel S.A.L. for its help in building the PV charger prototype.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **References**


© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## *Article* **A Method to Enhance the Global E**ffi**ciency of High-Power Photovoltaic Inverters Connected in Parallel**

#### **Marian Liberos \*, Raúl González-Medina, Gabriel Garcerá and Emilio Figueres**

Grupo de Sistemas Electrónicos Industriales del Departamento de Ingeniería Electrónica, Universitat Politècnica de València, Camino de Vera s/n, 46022 Valencia, Spain; raugonme@upv.es (R.G.-M.); ggarcera@eln.upv.es (G.G.); efiguere@eln.upv.es (E.F.) **\*** Correspondence: malimas@upv.es; Tel.: +34-963-879-606

Received: 15 May 2019; Accepted: 6 June 2019; Published: 11 June 2019

**Abstract:** Central inverters are usually employed in large photovoltaic farms because they offer a good compromise between costs and efficiency. However, inverters based on a single power stage have poor efficiency in the low power range, when the irradiation conditions are low. For that reason, an extended solution has been the parallel connection of several inverter modules that manage a fraction of the full power. Besides other benefits, this power architecture can improve the efficiency of the whole system by connecting or disconnecting the modules depending on the amount of managed power. In this work, a control technique is proposed that maximizes the global efficiency of this kind of systems. The developed algorithm uses a functional model of the inverters' efficiency to decide the number of modules on stream. This model takes into account both the power that is instantaneously processed and the maximum power point tracking (MPPT) voltage that is applied to the photovoltaic field. A comparative study of several models of efficiency for photovoltaic inverters is carried out, showing that bidimensional models are the best choice for this kind of systems. The proposed algorithm has been evaluated by considering the real characteristics of commercial inverters, showing that a significant improvement of the global efficiency is obtained at the low power range in the case of sunny days. Moreover, the proposed technique dramatically improves the global efficiency in cloudy days.

**Keywords:** efficiency improvement; photovoltaic inverters; parallel inverters

#### **1. Introduction**

Photovoltaic (PV) generation has had rapid growth in the last years and is now a significant contribution to the renewable sources of electricity [1–5]. With the purpose of improving the profitability of photovoltaic systems, large-scale PV plants are being installed [6]. In large PV plants, low-power decentralized architectures based on string inverters are usually avoided due to their high cost. Therefore, photovoltaic farms are usually connected to the grid through central inverters that manage the whole power of the system, since they offer a good compromise between costs and efficiency [7].

Figure 1 shows two alternatives to build up a centralized inverter that connects a large PV field to the distribution grid. The efficiency of central inverters composed by a single power stage, Figure 1a, is poor in the low power range when the radiation conditions are low. In the range of MWs, the scheme showed by Figure 1b is preferred, as the parallel connection of several modules offers redundancy, scalability, and a certain degree of fault tolerance. A popular technique to manage the connection of paralleled inverters is average current-sharing (CS). CS offers several advantages, such as a good power sharing among the modules and simplicity of implementation. However, the efficiency of the

whole system with this technique at low power is not improved with regard to the single-module configuration [6,8].

**Figure 1.** Topologies of high power central inverters. (**a**) Centralized inverter composed by a single power stage; (**b**) centralized inverter composed by n parallel modules.

To overcome this problem, the inverter modules can be connected and disconnected depending on the global delivered power. The concept of connecting/disconnecting the phases of a converter depending on the load-current level has been also applied in several works to low power converters [9–13]. In [9] the phase shedding points are calculated based on a lookup table defined by the junction temperature and on-resistance of MOSFETs. In [10] a multiphase buck converter with a rotating phase-shedding scheme has been presented. In [11] a method is proposed that linearly increases/reduces the power delivered by some channels when the demanded power changes. In [13] a time optimal digital controller for the phase shedding in multiphase buck converters has been developed. Although all these methods are based on the connection/disconnection of phases (or modules) to improve the efficiency of the global system, they cannot be directly extrapolated to high-power inverters.

Some studies regarding the connection/disconnection of paralleled inverters have been developed in the past. In [14] a methodology based on unidimensional efficiency curves and a genetic algorithm was presented. In that work, a unidimensional model is used, so the changes in the maximum power point tracking (MPPT) voltage are not considered. Therefore, this algorithm could be inappropriate in most of the photovoltaic applications. Moreover, the stochastic nature of the genetic algorithm requires the implantation of complex processes to obtain a useful result. This fact could impede the real-time implementation of the technique. In [15], a piecewise curve fitting is used to define the efficiency function and an artificial-intelligence-based algorithm is implemented to obtain the optimized current-sharing. As in [15], a unidimensional model is used to calculate the efficiency of the inverter, and the algorithm proposes a random initialization that requires many iterations to obtain a valid result. Finally, in [16], each converter regulates its respective output power following an algorithm of prioritization. However, the algorithm improves the efficiency in the low range of power but worsens the efficiency in the middle and high power range.

Regarding the efficiency of inverters, there are several efficiency models for photovoltaic inverters that have been proposed in the literature. These models can be classified as unidimensional and bidimensional, depending on whether they only take into account the generated power, or if they consider both the power generation and the DC voltage at the input of the inverter. Some unidimensional and bidimensional models were studied in [17–25].

In this work, a control technique is proposed that decides, in real time conditions, the proper number of inverters that should be on stream to improve the global efficiency in the whole power range. The developed algorithm is based on a bidimensional model of the inverters' efficiency, which takes into account not only the amount of delivered power but also the value of the DC voltage at the input of the inverters, which is continuously changing to achieve the maximum power point (MPP) of the PV field. The algorithm calculates the efficiency of the whole power system by taking into account various scenarios and selects the one that offers the best instantaneous efficiency. A comparative study of the model's accuracy has also been carried out by considering data of several commercial inverters. Data have been obtained from the "Grid Support Inverters List" published by the California Energy Commission [26]. However, it is worth noting that the parameters could be easily extracted from the datasheet of manufacturers, or obtained by means of a reduced number of efficiency measurements on the inverter.

The main contributions of this paper are the following:


#### **2. E**ffi**ciency Modeling of PV Inverters**

As it has been pointed out in the previous section, two kinds of functional models have been proposed in the literature to evaluate the efficiency of power inverters: unidimensional, in which only the generated power is considered, and bidimensional, in which takes into account both the generated power and the DC voltage at the input of the inverter, which agrees with the MPPT voltage in the case of central inverters.

#### *2.1. Unidimensional Models*

The unidimensional model of Jantsch [17–19] is expressed by (1), where *k*0, *k*1, and *k*<sup>2</sup> are the coefficients to be calculated for any inverter, and *c* is the load factor, which is defined as the ratio between the power that is processed at a certain instant and the nominal power of the inverter. In (1), the part of the losses that are independent of the generated power (constant losses) are weighted by *k*0, the losses that linearly depend on the load factor are weighted by *k*<sup>1</sup> and the losses with a quadratic dependence on the load factor are weighted by *k*2.

$$\eta(c) = \frac{c}{c + (k\_0 + k\_1 c + k\_2 c^2)}\tag{1}$$

Dupont [20] indicates that the efficiency of power inverters can be approximated by the second order function (2), being α1, α0, β1, and β<sup>0</sup> coefficients that can be obtained by applying curve fitting algorithms over experimental measurements and *c* is the load factor of the inverter.

$$\eta(c) = \frac{\infty\_1 c + \infty\_0}{c^2 + \beta\_1 c + \beta\_0} \tag{2}$$

#### *2.2. Bidimensional Models*

In photovoltaic inverters that are directly connected to the PV field, as is the case of central inverters, the DC input voltage of the inverters is continuously following the operation point that is calculated by the maximum power point tracking (MPPT) algorithm. Therefore, the use of unidimensional models that consider the DC voltage as constant is inappropriate to properly evaluate the actual efficiency of

the inverter in the whole operation range. To overcome this problem, several bidimensional models have been proposed.

Rampinelli [23] modifies the model represented by (1), which only considers the efficiency as a function of the delivered power, by taking also into account the influence of the input voltage on the predicted efficiency. To achieve this, the coefficients *k*0, *k*1, and *k*<sup>2</sup> are expressed as a function of the input voltage being modified as *k*<sup>0</sup> (*vin*), *k*<sup>1</sup> (*vin*), and *k*<sup>2</sup> (*vin*). The expressions of these coefficients are defined as (3)–(5), by assuming that the coefficients have a linear dependency with the input voltage and *k*0,0, *k*0,1, *k*1,0, *k*1,1, *k*2,0, and *k*2,1 are the coefficients to be calculated. This modeling approach is described by (6).

$$k\_0'(v\_{in}) = K\_{0,0} + k\_{0,1}v\_{in} \tag{3}$$

$$k\_1'(v\_{in}) = K\_{1,0} + k\_{1,1}v\_{in} \tag{4}$$

$$k\_2'(v\_{in}) = K\_{2,0} + k\_{2,1}v\_{in} \tag{5}$$

$$\eta(c, v\_{in}) = \frac{c}{c + \left(k\_0'(v\_{in}) + k\_1'(v\_{in})c + k\_2'(v\_{in})c^2\right)}\tag{6}$$

Similarly, if it is assumed that the efficiency can vary in a quadratic way regarding both the delivered power and the DC voltage, the coefficients can be calculated as (7)–(9). As a result, Equation (10) describes a model of the inverter that considers a nonlinear dependency with the input voltage.

$$k\_0^{\prime\prime}(v\_{\rm in}) = K\_{0,0} + k\_{0,1}v\_{\rm in} + k\_{0,2}v\_{\rm in}^2 \tag{7}$$

$$k\_{1}^{\prime\prime}(v\_{\rm in}) = K\_{1,0} + k\_{1,1}v\_{\rm in} + k\_{1,2}v\_{\rm in}^2 \tag{8}$$

$$k\_2^{\prime\prime}(v\_{\rm in}) = K\_{2,0} + k\_{2,1}v\_{\rm in} + k\_{2,2}v\_{\rm in}^2 \tag{9}$$

$$\eta(c, v\_{\rm in}) = \frac{c}{c + \left(k\_0''(v\_{\rm in}) + k\_1''(v\_{\rm in})c + k\_2''(v\_{\rm in})c^2\right)}\tag{10}$$

In [24] Sandia Laboratories propose a mathematical model that describes the performance of inverters. The Sandia model is represented in (11)–(14), where *pac* is the output power of the inverter; *pac\_o* is the nominal AC power rating, *pdc* is the input power of the inverter, *vdc* is the input voltage of the inverter, *vdc\_o* is the nominal DC voltage, *pdc\_o* is the nominal input power, *pso* is the minimum considered DC power; *co* is a parameter that defines the curvature of the relationship between AC power and DC power at nominal voltage. Finally, *c*1, *c*2, and *c*3 are coefficients that represent the linear relationship between *pdc\_o*, *pso*, and *co*, respectively, and the DC input voltage.

$$A = p\_{d\mathfrak{c\\_o}}(1 + c\_1(v\_{d\mathfrak{c\\_o}})) \tag{11}$$

$$B = p\_{\ast 0} (1 + c\_2 (v\_{dc} - v\_{dc\\_o})) \tag{12}$$

$$\mathcal{C} = \mathfrak{c}\_{\mathcal{o}} (1 + \mathfrak{c} (\upsilon\_{dc} - \upsilon\_{dc\\_o})) \tag{13}$$

$$p\_{ac} = \left(\frac{p\_{ac\\_o}}{A - B} - \mathbb{C}(A - B)\right)(p\_{dc} - B) + \mathbb{C}(p\_{dc} - B)^2\tag{14}$$

$$
\eta = \frac{p\_{ac}}{p\_{dc}} \tag{15}
$$

Finally, Driesse [25] proposes a model defined as (16)–(19), with *b*0,0 ... 2, *b*<sup>10</sup> ... 2, *b*2,0 ... <sup>2</sup> being the coefficients to be fitted.

$$b\_0(v\_{in}) = b\_{0,0} + b\_{0,1}(v\_{in} - 1) + b\_{0,2} \left(\frac{1}{v\_{in}} - 1\right) \tag{16}$$

$$b\_1(v\_{in}) = b\_{1,0} + b\_{1,1}(v\_{in} - 1) + b\_{1,2} \left(\frac{1}{v\_{in}} - 1\right) \tag{17}$$

$$b\_2(v\_{in}) = b\_{2,0} + b\_{2,1}(v\_{in} - 1) + b\_{2,2} \left(\frac{1}{v\_{in}} - 1\right) \tag{18}$$

$$\eta(c\_\prime \,\,\upsilon\_{\rm in}) = \frac{c}{c + b\_0(\upsilon\_{\rm in}) + b\_1(\upsilon\_{\rm in})c + b\_2(\upsilon\_{\rm in})c^2} \tag{19}$$

#### **3. Description of the Proposed E**ffi**ciency Oriented Algorithm**

As it was described in the introduction section, the proposed algorithm calculates the optimal number of parallel modules of a central inverter that should be on stream to maximize the global efficiency in the whole power range.

The algorithm is initially based on the calculation of the local maxima by applying the second derivative test to the function that predicts the efficiency of the whole system, *e*ff (*ci*, *vin*). This function (20) computes the efficiency of the whole system starting from the efficiency of each one of the modules, η(*ci*, *vin*) that can be obtained by means of one of the functional models that were described in the previous section. In (20), *ci (i* = 1, 2, ... , *n)* is the load factor of each parallel inverter, i.e., the ratio between the power that is actually managed by each module and its nominal power. The DC voltage of the central inverter is represented by *vin*.

$$eff(c\_{i\prime}, v\_{\rm in}) = \sum\_{i=1}^{n} \frac{c\_i}{c\_1 + c\_2 + \dots + c\_n} \cdot \eta(c\_{i\prime}, v\_{\rm in}) \tag{20}$$

To apply the second derivative test, the critical points of the function (20) can be calculated by solving the equations' system obtained from the first partial derivatives. From the second derivative, the Hessian matrix (21) can be obtained [27]. Finally, if *H* in a critical point is negative definite, that critical point is a local maximum. Therefore, the optimal load factor for each one of the modules on stream that maximizes the global efficiency can be obtained by solving (20) and (21) for a certain operating point described by both the DC voltage and the supplied power. It is worth pointing out that the second derivative test does not calculate the maxima points when some of *ci* = 0. To solve this issue, *n* different *e*ff*j*(*ci*, *vin*) can be defined, being *j* = 1 ... *n* and *i* = 1 ... *n*. Following this procedure, *j* relative maximums are obtained, one for each *e*ff*j*(*ci*, *vin*), with the searched maximum being the highest of these.

$$H = \begin{bmatrix} \frac{\partial^2 \ell f f}{\partial c\_1^2} & \dots & \frac{\partial^2 \ell f}{\partial c\_n c\_1} \\ \vdots & \ddots & \vdots \\ \frac{\partial^2 \ell f f}{\partial c\_1 c\_n} & \dots & \frac{\partial^2 \ell f}{\partial c\_n c\_n} \end{bmatrix} \tag{21}$$

One disadvantage of this procedure is the very high computation time needed to calculate all the critical points for a large set of *ci* values. However, all the local maximums for a certain *e*ff*j*(*ci*, *vin*) are produced when the power is equally shared among the modules, i.e., *c*<sup>1</sup> = *c*<sup>2</sup> ... = *cn*. Therefore, the method can be simplified since only the number of modules on stream that maximizes the global efficiency should be calculated. By applying this condition, a practical implementation of the proposed method can be obtained, which is shown in the following section.

#### *Practical Implementation of the Proposed E*ffi*ciency Oriented Algorithm*

Starting from the predictions of a functional model of inverters efficiency, the algorithm calculates the efficiency of the central inverter by considering all possible combinations of modules on stream and chooses the result that offers the maximum efficiency. As it has been highlighted before, the relative maximum values of efficiency are achieved when the power is equally shared among the modules, so it should be calculated only the value of n that maximizes the global efficiency. One of the most important characteristics of the simplified algorithm is the low need for computational resources and its easy implementation to work in real time conditions.

Figure 2 shows the flowchart of the proposed algorithm. In the figure, *vin* and *pin* are the MPPT input voltage and the generated power, respectively, while *vdcmin* and *vdcmax* are the limits of the MPP voltage range of the inverter. The nominal power of the photovoltaic farm and the one of each parallel inverter are represented by *ptot* and *pmod*, respectively; *n* is the total number of parallel modules; *ni, cmod-i*, *ctot-i*, η*mod-I*, and η*tot-i* (for *i* = 1 to *n)* represent the number of inverters considered in each iteration, the load factor of only a module, the load factor of the whole system, the efficiency of each module and the one of the whole system, respectively, in all cases for the corresponding iteration. Finally, *nON* is the number of modules on stream that achieves the global maximum efficiency η*max*.

**Figure 2.** Efficiency-oriented algorithm.

#### **4. Methods**

#### *4.1. Selection of Inverters for the Study*

Table 1 summarizes the commercial inverters that have been evaluated to validate the proposed concepts. The required data to build up the efficiency models of the inverters have been extracted from the "Grid Support Inverters List" that California Energy Commission (CEC) publishes [26].


**Table 1.** List of inverters under study (Source: CEC (California Energy Commission) Grid Support Inverters List).

Although the proposed methods have been applied to all the inverters summarized by Table 1, in the following only a selection of the most representative results is shown. To choose those representative results, 3 inverters with significant differences in their respective dependence of the MPP voltage on the inverter efficiency have been considered. In Figure 3, the relationship between the input voltage and the efficiency of the listed inverters has been represented. Note that, in some cases, the curves have an ascendant and nonlinear relationship with the input voltage; in some others, they have a descendant and linear relationship with the voltage and, finally, some curves have a descendant and nonlinear relationship with the voltage.

**Figure 3.** Relationship between the input voltage and the efficiency of the inverters extracted from the CEC Grid Support Inverters List.

To consider the three possibilities (categories) of the efficiency dependency regarding the voltage, a sample of each category for the study presented in this paper has been chosen. Thus, the three chosen inverters have been the following: EQX0250UV480TN (Perfect Galaxy International Ltd.), ULTRA-750-TL-OUTD-4-US (Power-One), and FS0900CU (Power Electronics). Tables 2–4 express the data of the selected inverters that have been extracted from the CEC "Grid Support Inverters List". The intermediate value of *vin* will be denoted as nominal in the following.


**Table 2.** Perfect Galaxy International Ltd. EQX0250UV480TN efficiency data.





#### *4.2. Modeling of Inverters*

In this section, the parameters of the efficiency models presented in Section 2 have been calculated. Tables 5–10 show the coefficients of the models for each inverter under study.

#### **Table 5.** Jantsch coefficients.


#### **Table 6.** Dupont coefficients.



#### **Table 7.** Rampinelli coefficients.



#### **Table 9.** Sandia coefficients.




The coefficients that Tables 5–8 show have been calculated by applying fitting algorithms to the data obtained from the CEC for each inverter under study (Tables 2–4). The fitting algorithms have been applied to Equations (1), (2), (6), and (10) for the Jantsch, Dupont, Rampinelli, and Rampinelli nonlinear models, respectively. To apply the fitting algorithms the Statistics and Machine Learning Toolbox of MATLABTM has been employed [28].

In Table 9, the coefficients of the Sandia model have been expressed. To obtain the parameters of the Sandia Model (11)–(14), three separate parabolic fits (2nd order polynomial) have been carried out providing the parameters *pdc\_o*, *pso*, and *co* for each value of DC voltage. The resultant quadratic formula for each voltage value has been used to obtain *pso* by solving the x-intercept when *pac* = 0. In a similar way, *pdc\_o* can be obtained by calculating the x-intercept when *pac* = *pac\_o*. In the model, *pac\_o* is assumed to be equal to the nominal power of each module and the parameter *co* has been considered as the second order coefficient obtained in the polynomial fit. The coefficients *c1, c2,* and *c3* have been determined using the *pdc\_o*, *pso*, and *co* values obtained from the separate parabolic fits. These values are linearly fitted considering their DC voltage dependence. From the resultant equations the coefficients *pdc\_o*, *c*1; *pso*, *c*2, and *co*, *c*<sup>3</sup> have been obtained.

The coefficients of Table 10 have been calculated by applying the fitting algorithms to the data obtained from the CEC and considering Equations (16)–(19).

#### **5. Results**

#### *5.1. Evaluation of the Model's Performance*

A comparative study of the accuracy of the efficiency models is carried out in this section. The results are compared to the actual CEC measurements to evaluate the proper prediction capability of each model.

Figures 4 and 5 show the efficiency curves that are computed by the unidimensional models Jantsch and Dupont when they are applied to the inverters under study. The CEC data around the fitted curve have been highlighted. As expected, with both models the predicted values cannot be accurate in all the range of the DC voltage.

**Figure 4.** Efficiency curves calculated by means of the Jantsch model. (**a**) Perfect Galaxy International Ltd. EQX0250UV480TN. (**b**) Power-One ULTRA-750-TL-OUTD-4-US (**c**) Power Electronics FS0900CU.

**Figure 5.** Efficiency curves calculated by means of the Dupont model. (**a**) Perfect Galaxy International Ltd. EQX0250UV480TN. (**b**) Power-One ULTRA-750-TL-OUTD-4-US. (**c**) Power Electronics FS0900CU.

Figure 6a–c show the efficiency surfaces of the inverters, which have been computed by means of the Rampinelli model in the whole range of MPPT voltages. Figure 6d–f detail these results only for the three values of the DC voltage given by CEC. In Figures 7–9, the same results are depicted, obtained in the same conditions, but in these cases by means of Rampinelli nonlinear model, Sandia model, and Driesse model, respectively. As expected, the results obtained by using bidimensional models significantly improve compared to the ones achieved by means of the unidimensional ones. Regarding the dependence of the efficiency curves with *vin*, note that for the inverter #2 (Power-One ULTRA-750-TL-OUTD-4-US) there are no significant differences between the results offered by the four evaluated bidimensional models. The reason for that is the strong linear dependence with the DC voltage that the efficiency curves of this inverter present. In contrast, in the case of the other two inverters under study, the dependence of the efficiency curves with the DC voltage is not linear and, therefore, the results achieved by means of Rampinelli nonlinear and the Driesse models fit better with the actual CEC data than the Rampinelli and the Sandia models.

**Figure 6.** Efficiency surfaces and detail of curves for three values of the DC voltage calculated by means of the Rampinelli model. (**a**,**d**) Perfect Galaxy International Ltd. EQX0250UV480TN. (**b**,**e**) Power-One ULTRA-750-TL-OUTD-4-US. (**c**,**f**) Power Electronics FS0900CU.

**Figure 7.** Efficiency surfaces and detail of curves for three values of the DC voltage calculated by means of the Rampinelli nonlinear model. (**a**,**d**) Perfect Galaxy International Ltd. EQX0250UV480TN. (**b**,**e**) Power-One ULTRA-750-TL-OUTD-4-US. (**c**,**f**) Power Electronics FS0900CU.

**Figure 8.** Efficiency surfaces and detail of curves for three values of the DC voltage calculated by means of the Sandia model. (**a**,**d**) Perfect Galaxy International Ltd. EQX0250UV480TN. (**b**,**e**) Power-One ULTRA-750-TL-OUTD-4-US. (**c**,**f**) Power Electronics FS0900CU.

**Figure 9.** Efficiency surfaces and detail of curves for three values of the DC voltage calculated by means of the Driesse model. (**a**,**d**) Perfect Galaxy International Ltd. EQX0250UV480TN. (**b**,**e**) Power-One ULTRA-750-TL-OUTD-4-US. (**c**,**f**) Power Electronics FS0900CU.

In summary, it may be concluded that the Rampinelli nonlinear and the Driesse models are the best approaches to predict the performance of photovoltaic inverters in terms of efficiency, independent of the relationship between efficiency and MPP voltage of the inverter.

#### *5.2. Evaluation of the Proposed E*ffi*ciency-Oriented (EO) Algorithm*

The algorithm for the activation/deactivation of the power modules that were described in Section 4 is applied in this section to a central inverter with a nominal power of 3 MW. The algorithm has been tested considering two significant profiles of photovoltaic generation, in sunny and cloudy conditions. Table 11 shows the number of units that are needed to achieve the nominal power with the commercial inverters under study.


**Table 11.** Number of modules to achieve 3 MW with the inverters under study.

As explained in Section 3, for a certain set of values of both the load factor and the MPP voltage, the algorithm calculates the optimal number of connected modules to maximize the efficiency of the whole system. To illustrate by means of an example how the algorithm works, Figure 10 shows the optimal number of modules on stream that are calculated by the proposed EO algorithm for a 3 MW central inverter composed by twelve modules of Perfect Galaxy International Ltd. EQX0250UV480TN. The figure depicts the number of inverters in operation that maximizes the efficiency in the whole range of MPP voltages and power.

**Figure 10.** Number of inverters in operation depending on the power generation and the maximum power point tracking (MPPT) voltage.

The proposed algorithm has been implemented in the TMS320F28379D to evaluate the needed computing resources (execution time and memory). To achieve this, a central inverter composed of twelve modules of the Perfect Galaxy International Ltd. EQX0250UV480TN has been considered. Two options for implementing the algorithm have been evaluated. In the first one, the operation map that Figure 10 shows has been programmed by means of a lookup (LU) table. With this approach, the algorithm equations are not solved in real time, so the execution time of the algorithm is expected to be low. In return, the memory requirements increase due to the need of storing in the DSP all the points of the operation map. The second option to implement the algorithm is directly programming the equations in the DSP and solve them in real time. In this case, lower memory requirements and larger execution time are expected than in the case of using a lookup table.

Figure 11a shows the execution time for option #1. In the case under study (12 modules in parallel), the chosen lookup table has a size 15 × 60 (15 input voltages and 60 power levels), needing 1800 bytes of data memory and 47 words of program memory. With this implementation, the algorithm use and its execution time is 540 ns. Note that the memory resources could vary depending on the resolution of the LU table.

Figure 11b shows the measured execution time when the equations of the algorithm are solved in real time. Note that, in this case, the execution time depends on the number of iterations performed by the algorithm, which are related to the number of modules that compose the central inverter and also on the power generated by the PV field. In the case under study, the execution time at low power is 7.54 μs and at high power is reduced to approximately 3.5 μs. The reason for this difference is that at low power, the efficiency must be calculated considering 1 to *n* inverters on stream. When the power generation increases, the execution time decreases since the algorithm does not calculate the efficiency when the power managed by each module is greater than its nominal power. In other words, the iteration of the loop is not executed when *cmod-i* > 1, as it can be seen in Figure 2. The program memory used in this case is 37 words and the use of data memory is negligible.

**Figure 11.** Execution time of the algorithm (**a**) lookup table implementation. (**b**) equations implemented and solved in real time.

Table 12 summarizes the measured execution times as well as the memory resources for both implementations. The results confirm the expectations about execution time and memory requirements of both kind of implementations, so the choice for a certain application would depend on the need for reducing the implementation time or memory.



5.2.1. Global Efficiency in the Whole Range of Operation of the PV Farm

Figure 12 depicts the efficiency surfaces obtained by applying the conventional average current-sharing control method (CS) and the efficiency-oriented (EO) method algorithm of activation/deactivation to the 3 MW central inverter described before.

**Figure 12.** Efficiency surfaces with average current-sharing control method (**CS**) and efficiency-oriented method (**EO**). (**a**) Perfect Galaxy International Ltd. EQX0250UV480TN. (**b**) Power-One ULTRA-750-TL-OUTD-4-US. (**c**) Power Electronics FS0900CU.

Figure 13a–c depict the detail at 500, 600, and 800 V of the efficiency obtained by both methods applied to the central inverter composed of twelve modules of Perfect Galaxy International Ltd. EQX0250UV480TN. Similarly, Figures 14 and 15 show the results considering the central inverters composed of four modules of Power-One ULTRA-750-TL-OUTD-4-US and three modules of Power Electronics FS0900CU, respectively.

*Energies* **2019**, *12*, 2219

**Figure 13.** Efficiency of Perfect Galaxy International Ltd. EQX0250UV480TN with average current-sharing control method (**CS**) and efficiency-oriented method (**EO**). (**a**) 500 V. (**b**) 600 V. (**c**) 800 V.

**Figure 14.** Efficiency of Power-One ULTRA-750-TL-OUTD-4-US with average current-sharing control method (**CS**) and efficiency-oriented method (**EO**). (**a**) 585 V. (**b**) 746 V. (**c**) 800 V.

**Figure 15.** Efficiency of Power Electronics FS0900CU with average current-sharing control method (**CS**) and efficiency-oriented method (**EO**). (**a**) 552 V. (**b**) 620 V. (**c**) 800 V.

These results show that the efficiency-oriented method achieves the best global efficiency in the whole power range independently of the kind of commercial inverter used to build up the central inverter.

#### 5.2.2. Study for a Typical Daily Power Profile

To evaluate the performance of the proposed EO method in realistic conditions, generation profiles in different scenarios have been considered. Figure 16a shows a typical sunny day generation profile while Figure 16b shows a cloudy day generation profile. In the graphics, both the generated power and the DC voltage vary simultaneously.

Figures 17a, 18a and 19a show the efficiency of the evaluated 3 MW central inverter by considering a typical generation profile on a sunny day. Figures 17b, 18b and 19b shows the efficiency of the inverters by considering a typical generation profile on a cloudy day. It can be noticed that, for the central inverters composed by different commercial inverters, the performance of the proposed EO method is clearly better than the efficiency applying the current-sharing method, CS, considering both the sunny and the cloudy day.

**Figure 16.** Daily power generation and MPPT voltage curves. (**a**) Sunny day. (**b**) Cloudy day.

**Figure 17.** Daily efficiency curves of Perfect Galaxy International Ltd. EQX0250UV480TN with average current-sharing control method (**CS**) and efficiency-oriented method (**EO**). (**a**) Sunny day. (**b**) Cloudy day.

**Figure 18.** Daily efficiency curves of Power-One ULTRA-750-TL-OUTD-4-US with average current-sharing control method (**CS**) and efficiency-oriented method (**EO**). (**a**) Sunny day. (**b**) Cloudy day.

**Figure 19.** Daily efficiency curves of Power Electronics FS0900CU with average current-sharing control method (**CS**) and efficiency-oriented method (**EO**). (**a**) Sunny day. (**b**) Cloudy day.

#### **6. Conclusions**

A control technique to activate/deactivate the power modules of high-power central inverters has been proposed in this paper. The proposed method maintains the advantages of conventional current sharing methods that are usually used to manage the parallel connection of power inverters, as the low need for computational resources, easy implementation, and capability to operate in real time conditions.

The proposed efficiency oriented method is based on a functional model of PV inverters that predicts the efficiency of the system starting from the measurements of the processed power and the MPPT voltage and takes decisions about the number of inverters that should be on stream to improve the global efficiency of the whole central inverter.

A comparative study of several kinds of models to calculate the efficiency of PV inverters has been carried out. The models have been tested by using data of commercial inverters from the "Grid Support Inverters List" published by the California Energy Commission. Unidimensional and bidimensional models have been evaluated, showing that bidimensional and nonlinear models fit much better with the available data. It can then be concluded that bidimensional and nonlinear models are the best choice to be implemented in the proposed EO method.

Regarding the implementation of the proposed algorithm, two options have been evaluated, showing that the execution time can be significantly reduced by implementing the operation map in the whole power and voltage ranges by means of a lookup table. On the contrary, the memory requirements are much lower if the equations of the algorithm are implemented and solved in real time. Therefore, it can be concluded that the choice for a certain application would depend on which factor is preferred to be reduced.

The proposed algorithm of activation/deactivation of the power modules has been applied to a PV field with a nominal power of 3 MW. The inverter's efficiency achieved by the proposed EO method has been compared to the one with the current sharing technique in the whole range of operation of the PV farm. Moreover, the algorithm has been tested considering two significant profiles of photovoltaic generation, in sunny and cloudy conditions. At the beginning and the end of the day in both profiles, when the PV generation is low, the global efficiency is clearly improved with regard to the conventional current sharing method. Moreover, in cloudy conditions, the improvement is significant during all day.

**Author Contributions:** M.L., E.F. and G.G. proposed the main idea and performed the investigation; M.L. and R.G.-M. developed the software; M.L., E.F., and G.G. wrote the paper. G.G. and E.F. lead the project and acquired the funds for research.

**Funding:** This work is supported by the Spanish Ministry of Economy and Competitiveness (MINECO), the European Regional Development Fund (ERDF) under Grants ENE2015-64087-C2-2-R and RTI2018-100732-B-C21, and the Spanish Ministry of Education (FPU15/01274).

**Conflicts of Interest:** The authors declare no conflict of interest. The funders had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript, or in the decision to publish the results.

#### **References**


© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## *Communication* **Snapshot of Photovoltaics—February 2019 †**

#### **Arnulf Jäger-Waldau**

European Commission, Joint Research Centre (JRC), Via E. Fermi 2749, I-21027 Ispra (VA), Italy; Arnulf.jaeger-waldau@ec.europa.eu; Tel.: +39-0332-789119

† The scientific output expressed is based on the current information available to the author, and does not imply a policy position of the European Commission.

Received: 8 February 2019; Accepted: 19 February 2019; Published: 26 February 2019

**Abstract:** Over the last two decades, grid-connected solar photovoltaic (PV) systems have increased from a niche market to one of the leading power generation capacity additions annually. In 2018, over 100 GW of new PV power capacity was added. The annual PV capacity addition in 2018 was more than the total cumulative installed PV capacity installed until the mid of 2012. Total installed PV power capacity was in excess of 500 GW at the end of 2018. Despite a 20% decrease in annual installations, China was, again, the largest market with over 44 GW of annual installations. Decentralized PV electricity generation systems combined with local battery storage have substantially increased as well.

**Keywords:** renewable energies; photovoltaic (PV); energy challenge; policy options; technological development; market development; battery storage

#### **1. Introduction**

The urgent need for a de-carbonization of the power sector was stressed again during the 24th session of the Conference of the Parties (COP24) meeting in Katowice, Poland in December 2018. In 2015, the average CO2 emission per kWh of electricity was about 506 g globally [1]. The World Energy Outlook 2017 New Policy Scenario of the International Energy Agency (IEA) predicts that those emissions should decrease to 325 g CO2/kWh by 2040. The situation looks somewhat better in Europe. There, the emissions per kWh of electricity should decrease form 344 g CO2/kWh in 2016 to roughly 150 g CO2/kWh in 2040. However, such a decrease is still not sufficient for the necessary reduction of CO2 emissions. To honor the Paris Agreement, a maximum of 65 g CO2/kWh is allowed [2]. The only scenario that meets this requirement is the Sustainable Development Scenario. Under this scenario the emissions from electricity production in Europe have to decrease to 45 g CO2/kWh.

The crucial role of solar photovoltaics (PVs) to achieve this goal in a cost-effective manner was outlined in a number of 100% renewable energy source (RES) scenarios. Solar PV power generation has the potential to increase from about 600 TWh (2.4%) in 2018 to 6300 TWh (22%) in 2025, and surpassing 40,000 TWh (up to 70%) in 2050 [3]. To achieve such ambitions, the corresponding PV power capacities have to increase from slightly more than 500 GW at the end of 2018 to more than 4 TW by 2025, and to 21.9 TW by 2050 world-wide. This requires a growth of the annual market from slightly over 100 GW in 2018 to a few hundred GW thereafter.

Over the last two decades, the growth dynamics of PV deployment has changed from government driven incentive programs to market driven investment decisions. Besides the financial aspects and changing political framework conditions, the rapid growth was made possible by thorough technology achievements. Major achievements in material and solar cell research, and progress in manufacturing technology, have made the transition possible. Environmental and health concerns over the use of fossil energy sources, the increased volatility and upward pressure of fossil energy prices, and the commitment of many countries to the Paris Agreement are all adding momentum to it.

#### **2. Photovoltaic Solar Cell Production**

Reports of global solar cell production in 2018 vary between 110 GW and 120 GW. As more and more companies go private, manufacturing data collection gets more complicated. In addition, there is no common reporting format. Only a few companies report production figures, whilst others report shipment or sales figures. This explains the considerable uncertainty in this data.

Manufacturing data for this communication were collected through public stock market reports, commercial market reports, as well as through personal contacts. The different data sets were then compared, and this resulted in an estimate of 113 GW produced in 2018 (Figure 1). This corresponds to an annual increase of about 7% compared to 2017.

Production Statistics Uncertainties:


**Figure 1.** World photovoltaic (PV) cell/module production from 2005 to 2018.

Solar system hardware prices have declined by over 80% over the last two decades. Over the last 10 years the Levelized Cost of Electricity (LCoE) benchmark has decreased by over 75% to USD 69/MWh. These developments were made possible by the rapid increase of countries embracing solar energy, and the rapid growth of the PV manufacturing industry in China after 2005. However, the consequence was not only a massive market growth, but a severe price pressure, which resulted in a major consolidation in the PV manufacturing industry [4]. Despite a significant number of bankruptcies and low profit margins in the manufacturing part of the PV value chain, there is a significant number of new market entrants.

One of the fastest growing companies is Tongwei Solar. The company was set up just six years ago in 2013 as part of the Tongwei Group. The latter is a private company that has its core business in agriculture and new energy. In 2011, Tongwei Group and the Xinjiang Government agreed on an integrated PV strategic cooperation project. This project included the setup of a 50,000 ton solar-grade polysilicon plant, a 3 GW manufacturing capacity for solar wafers and solar cells, and last but not least, 5 solar power plants. In 2018, Tongwei Solar reported an increase of its annual production capacity to 80,000 tons of polysilicon, and 12 GW for solar cells and solar modules. With a polysilicon production of about 17,000 tons and solar cell shipments of 3.85 GW, the company has already ranked 6th for both products in 2017 [4].

Overall, there are still new capacity announcements, which will increase the total manufacturing capacity significantly. The rational for these expansion plans are the expectations of an annual 40 to 50 GW market in China, a continuation of market growth in India, and new markets in Africa, the Middle East, and South America. Further manufacturing cost reductions are expected through Manufacturing 4.0 factories, improved solar cell efficiencies, and reduced material consumption along the whole cell and module manufacturing process. PV manufacturers with older production equipment are suffering the most as they have to compete with new entrants, which have the advantage of a lower manufacturing capital expenditure (CAPEX) and higher efficiency products. For example, CAPEX in 2018 for a polysilicon plant with an annual production capacity of 10,000 tons has decreased by 90%, compared to the USD 1.5 billion in 2006–2007. Electricity represents about 20% to 40% of wafer and polysilicon production costs; therefore, the industry is looking for manufacturing sites with the lowest costs. This can be seen in China. The country's northwestern and southwestern regions have attracted major investments in new manufacturing plants for polysilicon and wafers due to the fact that power prices can be up to 90% lower than in the Eastern coastal regions [5].

The overall number of jobs in solar photovoltaic electricity will significantly increase in line with the expanding PV markets. However, most of the jobs will be downstream in the value chain. The main reason for this development is the increasing implementation of manufacturing 4.0 with a low workforce for solar cell and module manufacturing.

#### **3. Solar PV Electricity generation and Markets**

Since 2009, the weighted benchmark levelized costs of electricity (LCOE) for non-tracking crystalline silicon PV systems has decreased from about USD 225/MWh in the first half of 2009 to USD 60/MWh in the second half of 2018 [6,7]. This corresponds to a reduction of almost 75% over the last decade.

Actual electricity generation costs from photovoltaic systems depend on various factors like solar radiation, type of system, fixed operation and maintenance (O&M) costs, as well as finance conditions, which can differ significantly from country to country. In the second half of 2018, the range of LCoE for non-tracking systems varied between USD 38/MWh and USD 147/MWh, and between USD 41/MWh and USD 83/MWh for 1-axis tracking systems [7].

The prices for Power Purchase Agreements (PPA) can be even lower, especially in sunnier regions of the world with low financing costs. An example of the importance of stable and low-cost financing conditions is the result of the PV tender in Senegal, where the projects will be financed under the International Finance Corporation (IFC)-backed Scaling Solar initiative [8]. In April 2018, Senegal's Commission de Régulation du Secteur de l'Electricité (CRSE) announced the tender results to build two 60MW solar PV plants. The winning bids were EUR 38.026/MWh and EUR 39.83/MWh (USD 43.28/MWh and USD 45.33/MWh).

Despite a decrease of investments in solar energy to USD 130.8 billion (−24%), the annual installations modestly increased, by about 10%, to 109 GW in 2018 (Figure 2) [9]. The reasons for the decline were the lower capital costs for solar photovoltaic systems on the one side, and a decline of the PV installations in China by roughly 20%, compared to 2017, on the other side.

Most, but not all, market analysts expect a larger growth rate in 2019. New installations are forecasted to be between 107 GW and 140 GW [9,10]. The IEA's Renewable Energy Market Report 2018 forecasts a new photovoltaic power capacity between 575 and 720 GW that will be installed globally between 2018 and 2023 [11].

Market Statistics Uncertainties:


**Figure 2.** Annual photovoltaic installations from 2010 to 2019 (data source: [12–14], and own analysis).

At the end of 2018, China was home to roughly one-third of global installed PV capacity (about 180 GW). The European Union was second with about 23% (or 117 GW), followed by the United States of America with 12% (or 63 GW) (Figure 3).

**Africa:** Africa has vast solar resources, and the electricity generation from solar photovoltaic systems can be twice as high in large parts of Africa compared to Central Europe. Despite these advantages, solar photovoltaic electricity generation is still limited. Solar home systems (SHS) or solar lanterns were the main applications until the end of the last decade. The statistics for these applications are extremely imprecise, or even non-existent. Major policy changes have occurred since 2012, and the number of utility-scale PV projects, which are in the planning or realization stages, have increased considerably. In 2018 about 1.6 GW of new PV capacity was installed. The main markets were Egypt (>600 MW), Algeria (>200 MW), and Rwanda (180 MW).

Total African (documented) operational PV power capacity was close to 4.5 GW by the end of 2018. For 2020, the targeted capacity is currently in excess of 10 GW.

**Figure 3.** Cumulative photovoltaic installations from 2010 to 2019 (data source: [12–14], and own analysis).

**Asia and Pacific Region:** Despite the 20% decrease in new photovoltaic electricity system installations in China, the market remained almost stable due to significant market increases in Australia, India, and South Korea, as well as market uptakes in a number of countries in the Middle East and Southeast Asia. With over 44 GW, China was, again, the largest market, followed by India with almost 11.7 GW, Japan with over 6.7 GW, and Australia at 3.8 GW. For 2019, a slight increase to about 80 GW could be possible under stable policy conditions.

**European Union:** On 14 June 2018, the negotiators from the Commission, the European Parliament, and the Council reached a political agreement regarding the increase of renewable energy use in the European Union [15]. The new, renewable energy target for the EU for 2030 was set at 32%. However, this target is only binding for the EU as a whole, not on Member State levels. The revised renewable energy directive, which included a review clause by 2023 for an upward revision of the EU-level target, was published on 21 December 2018 [16].

After its peak in 2011, when PV installations in the EU accounted for 70% of worldwide installations, six years of market decreases and stagnation followed [17]. This trend was finally reversed when the PV market in the European Union increased almost 50%, from about 6 GW in 2017 to 8.8 GW in 2018. The increase was due to stronger than expected markets in Germany (3.1 GW), the Netherlands (1.4 GW), France (>1 GW), and Hungary (>0.5 GW).

In November 2018, the European Commission published its Vison for 2050, *A Clean Planet for All*, which outlined that the use of renewable energy sources has to exceed 60% by 2050 to reach an average increase of 1.5 ◦C, or net zero emissions [18]. To meet the EU's new energy and climate targets for 2030, Member States were required to prepare and submit by the end of 2018 a National Energy and Climate Plan (NECP) for the period from 2021 to 2030.

**Americas:** Markets in North and South America increased by over 25% and added about 17.5 GW of new solar photovoltaic power in 2018. The three largest markets were the USA (11.4 GW), Mexico (2.5 GW), and Brazil (1.5 GW). The number of countries embracing solar photovoltaic energy in Central and South America is increasing, and six countries had a PV market larger than 100 MW in 2018.

#### **4. Conclusions**

There is a general consensus amongst investment and energy analysts that solar photovoltaic energy will continue to grow faster than the overall energy demand in the coming years. Various industry associations, as well as Bloomberg New Energy Finance (BNEF), the European Renewable Energy Council (EREC), the Energy Watch Group with Lappeenranta University of Technology (LUT), Greenpeace, and the International Energy Agency have all published scenarios showing the possible growth of PV power capacity [19–23]. In Table 1 the numbers of the different scenario studies are compared. An interesting development can be observed looking at the IEA scenarios, which show a significant increase from the 2016 to 2018 scenario. However, IEA expectations are still at the lower end.

**Table 1.** Projected evolution scenarios of the world-wide cumulative solar electrical capacities through 2040.


Note: \* 2025 value is interpolated, as only 2020 and 2030 values are given.

With forecasted world-wide new installations between 270 and 310 GW in 2019 and 2020, only the 100% RES Power Sector scenario for 2020 is out of reach [9]. Global solar electricity production in 2018 was around 600 TWh and could reach 1000 TWh (or 4%) by 2020.

**Funding:** This research received no external funding.

**Conflicts of Interest:** The author declares no conflict of interest.

#### **References**


© 2019 by the author. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

#### *Article*
