energies Article
A Single-Phase Multilevel PV Generation System with an Improved Ripple Correlation Control MPPT Algorithm Manel Hammami
ID
and Gabriele Grandi *
ID
Department of Electrical, Electronic, and Information Engineering, University of Bologna, 40136 Bologna, Italy;
[email protected] * Correspondence:
[email protected]; Tel.: +39-051-20-93571 Received: 12 November 2017; Accepted: 30 November 2017; Published: 3 December 2017
Abstract: The implementation of maximum power point tracking (MPPT) schemes by the ripple correlation control (RCC) algorithm is presented in this paper. A reference is made to single-phase single-stage multilevel photovoltaic (PV) generation systems, when the inverter input variables (PV voltage and PV current) have multiple low-frequency (ripple) harmonics. The harmonic analysis is carried out with reference to a multilevel configuration consisting of an H-bridge inverter and level doubling network (LDN) cell, leading to the multilevel inverter having double the output voltage levels as compared to the basic H-bridge inverter topology (i.e., five levels vs. three levels). The LDN cell is basically a half-bridge fed by a floating capacitor, with self-balancing voltage capability. The multilevel configuration introduces additional PV voltage and current low-frequency harmonics, perturbing the basic implementation of the RCC scheme (based on the second harmonic component), leading to malfunctioning. The proposed RCC algorithm employs the PV current and voltage harmonics at a specific frequency for the estimation of the voltage derivative of power dP/dV (or dI/dV), driving the PV operating point toward the maximum power point (MPP) in a faster and more precise manner. The steady-state and transient performances of the proposed RCC-MPPT schemes have been preliminarily tested and compared using MATLAB/Simulink. Results have been verified by experimental tests considering the whole multilevel PV generation system, including real PV modules, multilevel insulated-gate bipolar transistor (IGBT) inverters, and utility grids. Keywords: photovoltaic; multilevel inverter; single-phase converter; maximum power point tracking (MPPT); ripple correlation control (RCC); harmonics
1. Introduction Photovoltaic energy sources play an important role in reducing greenhouse gas emissions and their use has been increasing significantly over the years because of the high cost and environmental impact of conventional energy sources [1]. Maximum power point tracking (MPPT) algorithms are used to maximize the power extracted from a photovoltaic (PV) field. In order to improve efficiency and tracking performance, numerous MPPT algorithms have been published based on many aspects such as complexity, sensors required, cost, and efficiency [2–7]. Several approaches have been discussed in order to eliminate and/or reduce the number of sensing elements [8]. The ripple correlation control (RCC) algorithm [5,9,10] is particularly effective in single-phase single-stage systems, since it exploits the inherent instantaneous power oscillations (second-order harmonics) as perturbations of the working point in order to determine the voltage derivative of power (dP/dV) and drive the operating point to the maximum power point (MPP). RCC is generally simple, fast, and does not require any external action to perturb the PV operating point as compared to other MPPT algorithms. Energies 2017, 10, 2037; doi:10.3390/en10122037
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maximum power point (MPP). RCC is generally simple, fast, and does not require any external action perturb Energiesto 2017, 10, 2037the PV operating point as compared to other MPPT algorithms. 2 of 19 The basic RCC-MPPT algorithm was proposed in [9,11]; two high-pass filters (HPF), and two low-pass filters (LPF) are required for implementation. In order to overcome the problem of Thethe basic RCC-MPPT algorithm was proposed in [9,11]; two high-pass filters and two defining time constant of the filters, modified RCC-MPPT algorithms have been(HPF), introduced in low-pass filters (LPF) are required for implementation. In order to overcome the problem of defining [5,12]. Similar methods are also presented in [5], in which the moving average (MAvg) concept has the time filters, modified RCC-MPPT algorithms have been introduced in [5,12]. been usedconstant insteadofofthehigh/low-pass filters to identify PV current and voltage oscillations Similar methods are also presented in [5], in which the moving average (MAvg) concept has been (second-order harmonic components). Furthermore, only the sign of the product of PV power used and instead ripple of high/low-pass to identify PV current and voltage (second-order harmonic voltage can be usedfilters to drive the operating point toward theoscillations MPP instead of the estimation of components). Furthermore, only the sign of the product of PV power and voltage ripple can be used dP/dV. In [10] a hybrid RCC-MPPT has been proposed to smooth out the instability introduced by to drive the operating point the MPP of instead of thepower estimation dP/dV.(MPPT) In [10] schemes a hybrid fast irradiance transients. Thetoward implementation maximum point of tracking RCC-MPPT has been proposed to smooth out the has instability introducedinbythe fastcase irradiance transients. by ripple correlation control (RCC) algorithms been discussed of multiple PV The implementation of maximum power point tracking (MPPT) schemes by ripple correlation control harmonics [13]. (RCC) hasmultilevel been discussed in the casebecome of multiple harmonics In algorithms recent years, inverters have morePV attractive for[13]. single- and three-phase In recent years, multilevel inverters haveconverter become more attractive for singleand three-phase systems [14–18]. The most common multilevel topologies, presented in literature, are the systems [14–18]. The most common multilevel converter topologies, presented in literature, are the neutral-point-clamped (NPC), flying capacitor (FC) and cascaded H-Bridge (CHB) converters neutral-point-clamped (NPC), flying capacitor (FC) and cascaded H-Bridge (CHB) converters [18–21]. [18–21]. In both NPC and FC configurations, the number of additional components (diodes or In both NPC and FC configurations, the number of additional or capacitors) capacitors) proportionally increases with the number of levels,components leading to (diodes lower reliability and proportionally increases with the number of levels, leading to lower reliability and higher complexity, higher complexity, volume, and cost. Increasing the number of levels using the cascaded H-bridge volume, and cost. Increasing the number of not levels using additional the cascaded H-bridge configuration configuration is a flexible solution; it does require components but it needs is ana flexible solution; it does not require additional components but it needs an isolated dc power source isolated dc power source for each H-bridge cell. Recently, asymmetric cascaded H-bridge multilevel for eachtopology H-bridgehas cell. Recently, asymmetric cascaded H-bridge multilevel inverter topology has inverter gained interest from many researchers for PV system applications [13,22,23]. interest many researchers(two for PV systemcascaded applications It is based on to a modular Itgained is based on a from modular half-bridge switches) to a[13,22,23]. full H-bridge in order double half-bridge (two switches) cascaded to a full H-bridge in order to double the output voltage levels the output voltage levels (also called the level doubling network, LDN). A proper pulse-width (also called (PWM) the level doubling network, A propermechanism pulse-width modulation (PWM) pattern modulation pattern provides for aLDN). self-balancing keeping the floating capacitor providesaround for a self-balancing thevoltage. floating The capacitor around the half of the voltage the half of mechanism the dc-link keeping H-bridge LDNvoltage configuration is becoming dc-link H-bridge LDN configuration is becoming due to its simple, popular due to itsvoltage. simple, The modular, and reliable structure andpopular it can be considered as amodular, retrofit and reliable it canH-bridge be considered as a retrofit whichtocan be added to existing H-bridge which can bestructure added toand existing configurations in order double the output voltage levels configurations in order to double the output voltage levels (Figure 1). (Figure 1). PV field
(a) PV field
(b) Figure 1. Single-phase single-stage photovoltaic (PV) generation system: (a) basic H-bridge Figure 1. Single-phase single-stage photovoltaic (PV) generation system: (a) basic H-bridge configuration; (b) H-bridge plus level doubling network (LDN) multilevel configuration. configuration; (b) H-bridge plus level doubling network (LDN) multilevel configuration.
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The LDN topology was presented in [22] and the concept of self-balancing capability of the capacitor was analyzed. In [22,23], only nearest voltage-level control (staircase modulation) was developed to support the LDN operation. At present, the multilevel PWM strategy for the considered LDN inverter has not yet been clearly reported in the literature. An analysis of the PV current and voltage ripples in single-phase H-bridge inverters has been presented in [24,25]. With reference to the LDN configuration, the literature does not provide analysis of the dc voltage and current harmonics. In this paper, the modulation strategy of the proposed LDN multilevel inverter is introduced. A simplified efficiency estimation of the whole multilevel conversion system is given as well, in comparison to the basic H-bridge configuration, taking into account also the ac-link (grid) inductor losses. Modified RCC-MPPT algorithms are proposed and compared, taking into consideration the existence of multiple PV harmonics on the basis of the concept introduced in [13]. A detailed analysis of the PV current and voltage harmonics is proposed. In particular, the amplitudes of PV current and voltage harmonics have been analytically determined in order to design the proposed RCC-MPPT scheme in the case of multiple PV harmonics. Preliminary numerical simulations and comprehensive experimental results are presented to prove the effectiveness and the feasibility of the proposed multilevel PV generation systems including the improved RCC-MPPT scheme in both steady-state and transient working conditions. 2. Multilevel Modulation Principle and Analysis of Output Voltages 2.1. Modulation Principle The multilevel PWM principle of the proposed LDN configuration is identified and analyzed with reference to Figure 1. Introducing the dc component of the H-bridge dc-link voltage, V, the instantaneous output voltage, v ac , normalized by V and averaged over the switching period (Tsw = 1/fsw ), is determined within the linear modulation range as: u ac ∼ = u ac = m sin ϑ,
(1)
with u ac being the normalized reference output voltage (sinusoidal), ϑ = ωt the phase angle, ω the fundamental angular frequency (ω = 2π/T), T the fundamental period, and m the modulation index. Introducing subscripts L and H for LDN and H-bridge cells, respectively, the LDN modulating signal is proposed in order to obtain a proper PWM multilevel output voltage waveform with self-balancing capability (normalized LDN voltage equal to 1/2) as: ( L u ac =
m |sin ϑ|, 1 − m |sin ϑ|,
m |sin ϑ| ≤ 0.5 . m |sin ϑ| ≥ 0.5
(2)
For the H-bridge, the modulating signal can be obtained as: H L u ac = u ac − u ac .
(3)
leading to the following original compact form: ( H u ac
=
m sin ϑ − m |sin ϑ|, m sin ϑ + m |sin ϑ| − 1,
m |sin ϑ| ≤ 0.5 . m |sin ϑ| ≥ 0.5
(4)
The multilevel logic is implemented by carrier-based phase disposition PWM, considering two carriers for driving the two H-bridge legs, and one carrier for driving the LDN leg, according to modulation scheme shown in Figure 2.
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As an example of application of the above modulation principle, simulation results are given in case of m = 0.5 and m = 1. In particular, Figure 3 shows the modulating signals (1)–(4) and normalized instantaneous output voltages. Energies 2017, 10, 2037 4 of 19 Energies 2017, 10, 2037
1 1½
H-bridge
m = 0.5
H-bridge
m = 0.5
½0 −0 ½
½0
1½
H-bridge H-bridge
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m=1 m=1
½0 −0 ½
−1 −½ − 1½
1
−1 −½ m = 0.5
LDN
m = 0.5
LDN
LDN
−½ 1
½0
LDN
m=1 m=1
0 Figure2.2.Modulation Modulation principle multilevel pulse-width modulation: carriers and modulating 0 modulation: Figure principle for for multilevel pulse-width carriers and modulating signals signals for H-bridge (top) and LDN (bottom) in the case of m = 0.5 (left) and m = 1 (right). for H-bridge (top) and LDN (bottom) in the case of m = 0.5 (left) and m = 1 (right). Figure 2. Modulation principle for multilevel pulse-width modulation: carriers and modulating
signals for H-bridge (top) and LDN (bottom) in the case of m = 0.5 (left) and m = 1 (right).
Time (s)
Time (s)
(s) Time kHz): (s) Figure 3. Simulation results (fsw = 2.5 modulating signals and instantaneous outputTime voltages normalized by V: total (top), H-bridge (middle), and LDN (bottom) in the case of m = 0.5 (left) and Figure Figure3.3.Simulation Simulationresults results(f(fswsw= =2.5 2.5kHz): kHz):modulating modulatingsignals signalsand andinstantaneous instantaneousoutput outputvoltages voltages m = 1 (right). normalized by V: total (top), H-bridge (middle), and LDN (bottom) in the case of m = 0.5 (left) normalized by V: total (top), H-bridge (middle), and LDN (bottom) in the case of m = 0.5 (left)and and mm==1 1(right). (right).
2.2. H-Bridge and LDN Output Voltage Harmonics
2.2. and Voltage Harmonics In order toLDN determine the individual output voltage harmonics of both the H-bridge and LDN 2.2.H-Bridge H-Bridge and LDNOutput Output Voltage Harmonics cells, the modulation symmetry of the LDN can be exploited. In particular, according to the InInorder to the individual individual outputvoltage voltageharmonics harmonics both H-bridge LDN order to determine determine the of of both thethe H-bridge andand LDN waveform symmetry shown in Figure 3output (bottom traces), the harmonic spectrum contains only cells, even cells, the modulation symmetry thecan LDN can be exploited. In particular, to the the modulation of (the the of LDN be exploited. particular, accordingaccording to the waveform harmonics with symmetry cosine terms LDN modulating signal In is an even function). waveform symmetry Figure 3traces), (bottom the harmonic contains spectrum contains even symmetry shown in shown Figure 3in(bottom thetraces), harmonic even only harmonics The normalized LDN output voltage averaged over thespectrum switching period only (Equation (2)), can be harmonics with cosine terms (the LDN modulating an even function). with cosine terms LDN modulating signal is ansignal even is function). written in terms of(the harmonics as: The normalized LDN output voltage averaged over the The normalized LDN output voltage averaged over theswitching switchingperiod period(Equation (Equation(2)), (2)),can canbe be ∞ ∞ written in terms of harmonics as: written in terms of harmonics as: L L u ac = U 0 + uk = U 0 + U k cos(k ϑ) . (5) ∞ ∞k =2
∞k =2 ∞
k ϑ) . (5)(5) uukLkL==UU0 0++∑U kUcos( ∑ k cos( k ϑ). =2 k| is the amplitude k =2 of the kth harmonic component, with where U0 is the average component andkk|U =2 k =2 u ac==U U 0++ L u ac 0 L
k being evenaverage number. The amplitude of |these arethe calculated analytically as: with is the component and |U is thecomponents amplitude of kth harmonic component, where U0an where U0 is the average component and |Ukk | is the amplitude of the kth harmonic component, with k kbeing beingan aneven evennumber. number.The Theamplitude amplitude ofthese thesecomponents componentsare arecalculated calculatedanalytically analyticallyas: as: 2 of for m ≤ 0.5 πm , U 0 (m () =2 2 2 (6) mm + π − arcsin (1 / 2m) − 4m 2 − 1 forfor mfor ≤m0.5 m ≤ 0.5 ≥ 0.5 πm h i π √ U0 (m , , U)0 (=m) = 2 2 π ππ 2 (6)(6) + − arcsin (1/2m) − 4m2 − 1 m for m ≥ 0.5 π m + 2 − arcsin (1 / 2m) − 4m 2 − 1 for m ≥ 0.5 π m 2 4 for m ≤ 0.5 − π ( k 2 − 1) m . U k ( m) = 4 (7) for m ≤ 0.5 − 4 π ([kA2k − ( m1) + B k ( m)] for m ≥ 0.5 . U k ( m) = π (7) 4 [A k ( m) + B k ( m)] for m ≥ 0.5 π The expressions of the coefficients Ak (m) and Bk (m) in Equation (7) are given in the Appendix A. The expressions of the coefficients Ak (m) and Bk (m) in Equation (7) are given in the
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(
− π4
Uk (m) =
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4 π
m ( k 2 −1)
for m ≤ 0.5
[ Ak (m) + Bk (m)] for m ≥ 0.5
.
of(7) 19 55 of 19
The expressions of the coefficients Ak (m) andof (m) in Equation (7) are given in harmonics the Appendix A. Figure shows the the normalized amplitudes ofBklow-order low-order LDN output output voltage harmonics over Figure 44 shows normalized amplitudes LDN voltage over Figure 4 shows the normalized amplitudes of low-order LDN output voltage harmonics over the the whole modulation index range. the whole modulation index range. whole modulation index range. 0.5 0.5
|Uk|| |U k 0.4 0.4
dc dc 2nd 2nd 4th 4th 6th 6th 8th 8th
0.3 0.3 0.2 0.2 0.1 0.1 0 0
0 0
0.1 0.1
0.2 0.2
0.3 0.3
0.4 0.4
0.5 0.5
0.6 0.6
0.7 0.7
0.8 0.8
0.9 0.9
m m
1 1
Figure 4. 4. Normalized amplitudes amplitudes of low-order low-order LDN output output voltage harmonics harmonics as aa function function of the the Figure Figure 4. Normalized Normalized amplitudesof of low-orderLDN LDN outputvoltage voltage harmonicsas as a function of of the modulation index. index. modulation modulation index.
as: as:
According to to Equations Equations (3) (3) and and (5), (5), the the normalized normalized output output voltage voltage of of the the H-bridge H-bridge is is written written According According to Equations (3) and (5), the normalized output voltage of the H-bridge is written as: ∞∞ ∞
H HH u ac coskk(ϑ U +mm msin sin(((ϑϑ ϑ))) − cos( ϑk))ϑ..). 0+ ∑UUUk kcos( 0+ uuac===−−−UU sin −− ac
kk==22
0
k =2
(8) (8) (8)
k
The amplitudes amplitudes of harmonics the harmonics in Equation can be considering determinedEquations considering The amplitudes of the the harmonics in Equation Equation (8) can can be be(8) determined considering Equations (6) The of in (8) determined (6) and (7). Comparing Equation (8) with Equation (5) it is evident that, compared to the LDN, the Equations (6) and (7). Comparing Equation (8) with Equation (5) it is evident that, compared to and (7). Comparing Equation (8) with Equation (5) it is evident that, compared to the LDN, the the LDN, the H-bridge has an additional first harmonic component U = m. Figure 5 shows the = m. Figure 5 shows the normalized H-bridge has an additional first harmonic component U H-bridge has an additional first harmonic component U11 = m. Figure1 5 shows the normalized normalized amplitudes of H-bridge low-order output H-bridge outputharmonics voltage harmonics the modulation whole modulation amplitudes of low-order low-order voltage over the theover whole index amplitudes of H-bridge output voltage harmonics over whole modulation index index range. range. range. |Uk|| |U k
1 1
0.8 0.8
dc dc 1st 1st 2nd 2nd 4th 4th 6th 6th 8th 8th
0.6 0.6 0.4 0.4 0.2 0.2 0 0
0 0
0.1 0.1
0.2 0.2
0.3 0.3
0.4 0.4
0.5 0.5
0.6 0.6
0.7 0.7
0.8 0.8
0.9 0.9
m m
1 1
Figure 5. Normalized amplitudes of of the thelow-order low-orderH-bridge H-bridgeoutput outputvoltage voltageharmonics harmonics as function Figure 5. 5. Normalized Normalized amplitudes asas a function of Figure the low-order H-bridge output voltage harmonics aa function of the modulation index. thethe modulation index. of modulation index.
3. PV Voltage and Current Harmonics 3. 3. PV PVVoltage Voltage and and Current Current Harmonics
In the considered single-phase single-stage PV generation system (Figure 1), the PV field is In In the the considered considered single-phase single-phase single-stage single-stage PV PV generation generation system system (Figure (Figure 1), 1), the the PV PV field field is is directly connected to the input side of the H-bridge, so the PV field voltage corresponds to the directly directly connected connected to to the the input input side side of of the the H-bridge, H-bridge, so so the the PV PV field field voltage voltage corresponds corresponds to to the the H-bridge input voltage, whereas the PV current can be calculated on the basis of H-bridge input H-bridge H-bridge input input voltage, voltage, whereas whereas the the PV PV current current can can be be calculated calculated on on the the basis basis of of H-bridge H-bridge input input dc-link current and dc-link impedance, as shown shown in in the the following. following. dc-link dc-link current current and and dc-link dc-link impedance, impedance, as 3.1. H-Bridge H-Bridge Input Input Current Current Harmonics Harmonics 3.1. of the the H-bridge H-bridge is is composed composed by by its its averaged averaged value value The instantaneous instantaneous input input dc-link dc-link current current iiHH of The H H and the the instantaneous instantaneous switching switching ripple ripple Δ Introducing also also the the over the the switching switching period period ii H ,, and ΔiiH .. Introducing over
(dc component) component) and and the the alternating alternating low-frequency low-frequency average over over the the fundamental fundamental period period IIHH (dc average ~ H ~ harmonic component component ii H ,, harmonic ~ H H (9) ΔiiHH == II HH ++ ~ii HH ++ Δ ΔiiHH ,, (9) iiH == ii H ++ Δ
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3.1. H-Bridge Input Current Harmonics The instantaneous input dc-link current iH of the H-bridge is composed by its averaged value over H
the switching period i , and the instantaneous switching ripple ∆i H . Introducing also the average over the fundamental period IH (dc component) and the alternating low-frequency harmonic component ei H , Energies 2017, 10, 2037
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H
H
H
H
i = i + ∆i = I + ei H + ∆i H , (9) Considering the input–output H-bridge power balance for quantities averaged over the Considering the normalizing input–output power balance for quantities averaged over the switching period, and by H-bridge V, the input current is expressed as: switching period, and normalizing by V, the input current is expressed as: i H = uacH i ac , (10) H
i = u H i ac , (10) with i ac being the output current averaged overacthe switching period. Supposing unity power
factor and almost sinusoidal current, which is the the switching basic requirement for most unity of grid-connected with i ac being the output current averaged over period. Supposing power factor applications, it can be written as: and almost sinusoidal current, which is the basic requirement for most of grid-connected applications, it can be written as:
i ac ≅ i ac = I ac sin ϑ , (11) i ac ∼ (11) = i ac = Iac sin ϑ, where I ac is the amplitude of the output current. where Iac is the amplitude of the output current. Replacing Equations (8) and (11) in Equation (10), the harmonic spectrum of the averaged Replacing Equations (8) and (11) in Equation (10), the harmonic spectrum of the averaged H-bridge H-bridge input current is obtained as (see Appendix A): input current is obtained as (see Appendix A): ∞ # "1 H ∞(U n + 1 − U n −1 ) sin nϑ I ac . i m ( 2 U U ) sin m cos 2 = + − ϑ − ϑ − (12) 0 2 1 H 2 + (2U0 − U2 ) sin ϑ − m cos 2ϑ −n =∑ i = m (12) nϑ Iac . 3 (Un+1 − Un−1 ) sin 2 n =3 where n is an odd number, n ≥ 3. Figure 6 shows the harmonic amplitudes appearing in Equation where an odd number, n ≥ 3. Figure 6 shows the harmonic amplitudes in Equation (12) (12) as na isfunction of the modulation index. It should be noted that the appearing dc and second harmonic as a function of the modulation index. It should be noted that the dc and second harmonic components components have the same amplitude, proportional to the modulation index, as in the case of the have the same amplitude, proportional the modulation index, as in the case of thecomparable H-bridge without H-bridge without LDN (Figure 1a). Thetofirst harmonic has a relevant amplitude, to the LDN (Figure 1a). The first harmonic has a relevant amplitude, comparable to the amplitude of the amplitude of the second harmonic and aside from this in the third harmonic (that can be noticeable second harmonic andmaside this in theharmonics third harmonic (that negligible can be noticeable aroundmodulation m = 0.5 and around m = 0.5 and = 1),from higher-order are quite in the whole m = 1), higher-order harmonics are quite negligible in the whole modulation range. range. 0.5 H
Ik
0.4 dc,2nd 0.3
1st
(A)
0.2
3rd 5th
0.1
7th 0.0 0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
m
1
9th
Figure Figure 6. 6. Amplitudes Amplitudesof oflow-order low-orderH-bridge H-bridgeinput input current current harmonics harmonics for for sinusoidal sinusoidal unity unity output output current as a function of the modulation index. current as a function of the modulation index.
3.2. 3.2.PV PVVoltage VoltageHarmonics Harmonics As in Equation Equation (9), (9), the theinstantaneous instantaneousH-bridge H-bridgedc-link dc-linkvoltage voltagev,v,corresponding corresponding As for for the the current current in to to the PV voltage in the considered single-stage PV generation scheme, is composed of the averaged the PV voltage in the considered single-stage PV generation scheme, is composed of the averaged value and the instantaneous switching ripple Δv. Introducing the value over the switching over the switching period vperiod and thev instantaneous switching ripple ∆v. Introducing the average over average over theperiod fundamental period Vand (dcthecomponent) and the alternating low-frequency the fundamental (dc component) alternating low-frequency harmonic component ve, ~V v , it becomes: harmonic component it becomes:
~ve++Δv∆v≅∼ v v==vv++∆v V ++v =VV+ + Δv = =V v~ .ve.
(13) (13)
The switching frequency component Δv is strongly smoothed by the dc-link capacitor, and it is usually negligible for switching frequencies starting from 5 to 10 kHz. It is assumed here that Δv ≅ 0. The amplitude of the kth order harmonic component V k can be calculated as:
V = Zk I kH ,
(14)
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The switching frequency component ∆v is strongly smoothed by the dc-link capacitor, and it is usually negligible for switching frequencies starting from 5 to 10 kHz. It is assumed here that ∆v ∼ = 0. Energies 2017, 10, 2037 7 of 19 The amplitude of the kth order harmonic component Vk can be calculated as:
with Zk being the total dc-link impedance for the kth harmonic, given by the parallel between the Vk = Zk IkH , (14) reactance of the dc-link capacitor (CH), and the equivalent resistance of the PV field (RPV ≅ VMPP/IMPP with thethe total dc-link impedance for the kth harmonic, given by the parallel between the in theZvicinity MPP): k being of reactance of the dc-link capacitor (CH ), and the equivalent resistance of the PV field (RPV ∼ = VMPP /IMPP RPV , Z = in the vicinity of the MPP): k (15) (RPV kRωPVC H )2 + 1 , (15) Zk = q 2 ( R k ω C ) + 1 Considering realistic parameter values forPVPV fields and dc-link capacitors, the assumption H 1/(kωC RPV >> H) is generally satisfied, and Equation (15) becomes: Considering realistic parameter values for PV fields and dc-link capacitors, the assumption RPV >> 1/(kωCH ) is generally satisfied, and Equation1 (15) becomes: Zk = . k ω CH 1 . Zk = kωcan CHbe written as: Using the previous assumption, Equation (14)
(16) (16)
Using the previous assumption, Equation (14) can 1 beH written as: Vk = Ik (17) k ω CH 1 Vk = IkH (17) kω CH and Introducing Equation (12) into Equation (17) considering only the first and second dominating harmonic components, gives: Introducing Equation (12) into Equation (17) and considering only the first and second dominating harmonic components, gives: [U ( m) − 2U 0 ( m)] I , V1 ( m) = 2 (18) −C2U (m)] ac [U2 (m2) ω H0 Iac , (18) V1 (m) = 2ω CH m V ( m) = m I ac . (19) V2 (2m) = 4 ω C Iac . (19) 4ω CH H As an example, Figure 7 shows the amplitudes of the first first and and second second PV PV voltage voltage harmonics harmonics as a function of modulation index m, in the case that dc-link capacitor C = 1 mF, and there is unity function modulation case that dc-link capacitor HH = 1 sinusoidal output output current. current. 2
Vk 1.5
(V)
1st
1
0.5 2nd 0 0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
m
1
Figure 7. 7. Amplitude function of of the the Figure Amplitude of of dominating dominating low-frequency low-frequency PV PV voltage voltage harmonics harmonics as as aa function modulation index index in in case case C CHH == 11 mH = 1 A. modulation mH and and IIac ac = 1 A.
3.3. PV PV Current 3.3. Current Harmonics Harmonics As for for the the PV PV voltage voltage v, v, the the instantaneous instantaneous PV PV current current ii can can be be written written in in terms terms of of components, components, As corresponding to to Equation Equation (13). (13). Neglecting Neglecting the the switching switching frequency frequency component, component, itit becomes: becomes: corresponding ~ i = I+i i = I + ei
(20) (20)
~ with I being the average over the fundamental period (dc component) and i = v~ / RPV the with I being the average over the fundamental period (dc component) and ei = ve/R PV the alternating alternating low-frequency harmonic component. Note due tocapacitor, the dc-link I = IH. low-frequency harmonic component. Note that, due to that, the dc-link I =capacitor, IH . Based on the amplitudes of PV voltage harmonics given by Equations (18) and (19), the amplitudes of the dominating first and second PV current harmonics can be written as: I 1 ( m) =
V1 ( m) R PV
=
U 2 ( m) − 2U 0 ( m) I ac , 2 RPV ωC H
(21)
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Based on the amplitudes of PV voltage harmonics given by Equations (18) and (19), the amplitudes of the dominating first and second PV current harmonics can be written as: Energies 2017, 10, 2037
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V (m) U (m) − 2U0 (m) = 2 Iac , (21) I1 (m) = 1 R PV ωC V 2 ( m) 2R PV m H I 2 ( m) = = I ac . (22) ωC H V2R(PV m) 4 RPVm I2 (m) = = Iac . (22) R PV 4 R PV ωCH As an example, Figure 8 shows the amplitudes of the first and second PV current harmonics as As an example, Figureindex 8 shows of the capacitor first and second harmonics mF current and there is unity a function of modulation m, inthe theamplitudes case the dc-link CH = 1 PV as a function of modulation index m, in the case the dc-link capacitor C = 1 mF and there is unity the sinusoidal output current, considering PV series equivalent resistance RHPV = 40 Ω, representing sinusoidal output current, considering PV series equivalent resistance RPV = 40 Ω, representing the scale factor with Figure 7. scale factor with Figure 7. According to Equations (18), (19), (21), and (22), it is clear that the amplitudes of the PV voltage According to Equations (18), (19), (21), and (22), it is clear that the amplitudes of the PV and current harmonics are inversely proportional to the dc-link capacitor CH since the dc-link voltage and current harmonics are inversely proportional to the dc-link capacitor CH since the dc-link capacitor reactance dominates the dc-link impedance. However, PV current harmonics also depend capacitor reactance dominates the dc-link impedance. However, PV current harmonics also depend Energies 2017, 10, 2037 8 of 19 on the PV field on the PV series-equivalent resistance RPV, that is a function of the operating point on the PV series-equivalent resistance RPV , that is a function of the operating point on the PV field V ( mkW ) m characteristic. Considering Considering aa PV PV plant plant of of a)few few (single-phase), ranges between between few Ω Ω near near PV ranges I 2 (a m = 2 kW = (single-phase), I ac . (22) characteristic. RRPV aa few RPV 4 RPV ωC H the open-circuit open-circuit point, point, passing passing to to around around aa few few tens tens of of Ω Ω near near the the MPP MPP and and up up to to hundreds hundreds or or even even the As an example, Figure 8 shows the amplitudes of the first and second PV current harmonics as thousands of Ω toward the short-circuit point [26]. These considerations can be readily exploited to thousands of Ωa function towardof the short-circuit point [26]. Thesecapacitor considerations can beisreadily exploited to there unity modulation index m, in the case the dc-link CH = 1 mF and in order to obtain PV voltage and current harmonics amplitude in design the dc-link capacitor C representing the sinusoidal output current, PV series PV equivalent resistance design the dc-link capacitor CHHin considering order to obtain voltage and Rcurrent amplitude in the PV = 40 Ω,harmonics scale factor with Figure 7. to the order of a of few percent compared MPPMPP values. the order a few percent compared torated the rated values.
According to Equations (18), (19), (21), and (22), it is clear that the amplitudes of the PV voltage and current harmonics are inversely proportional to the dc-link capacitor CH since the dc-link 0.05
capacitor reactance dominates the dc-link impedance. However, PV current harmonics also depend I on the PVk series-equivalent resistance RPV, that is a function of the operating point on the PV field 0.04 characteristic. Considering a PV plant of a few kW (single-phase), RPV ranges between a few Ω near 0.03
the open-circuit point, passing to around a few tens of Ω near the MPP and up to hundreds1stor even (A) Ω toward the short-circuit point [26]. These considerations can be readily exploited to thousands of 0.02 design the dc-link capacitor CH in order to obtain PV voltage and current harmonics amplitude in the order of0.01 a few percent compared to the rated MPP values. 0
2nd
0.05
Ik 0.1
0
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
m
0.04 0.03
1
1st
(A) Figure 8. 8. Amplitude low-frequency function of of the the Figure Amplitude of of dominating dominating low-frequency PV PV current current harmonics harmonics as as aa function 0.02 modulation index in the case C = 1 A, and R = 40 Ω. = 1 mH, I modulation index in the case0.01CHH = 1 mH, Iacac = 1 A, and RPVPV = 40 Ω. 2nd 0 0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
m 4. RCC-MPPT RCC-MPPT Algorithms Algorithms in in the the Case Case of of Single Single and and Multiple Multiple PV PV Harmonics 4. Harmonics
Figure 8. Amplitude of dominating low-frequency PV current harmonics as a function of the
modulation of index in the case C = PV R = 40 Ω. considered in this paper is depicted in 1 mH, control I = 1 A, andscheme The block block diagram diagram the whole The of the whole PV control scheme considered in this paper is depicted in Figure9.9.The TheRCC RCC algorithm used estimate the voltage of the power, dP/dV, driving 4. RCC-MPPT Algorithms in the Case of Single Multiple PVderivative Harmonics Figure algorithm isisused toto estimate theand voltage derivative of the power, dP/dV, driving the the working point toward the MPP and determining the reference grid current amplitude I g*. In The block diagram of the whole PV control scheme considered in this paper is depicted in working point toward the MPP and determining the reference grid current amplitude Ig *. In particular, Figure 9. The RCC algorithm is usedv* to estimate the voltage derivative the power, dP/dV, particular, the reference dc-link voltage is simply obtained by thedriving estimated dP/dV, Ig* the reference dc-link voltage v* is simply obtained by integrating theofintegrating estimated dP/dV, I * is determined the working point toward the MPP and determining the reference grid current amplitude Ig*. Ing is determined by a PI voltage regulator, and the dq current controller has been implemented by a PI voltageparticular, regulator, and the dqvoltage current controller been implemented inIgorder to inject in a * the reference dc-link v* is simply obtainedhas by integrating the estimated dP/dV, order to inject a sinusoidal current with unity power is determined by grid a PI voltage regulator, and the dq currentfactor. controller has been implemented in sinusoidal grid current with unity power factor. H
ac
PV
order to inject a sinusoidal grid current with unity power factor.
PV
v
PV
i
v
i RCC
RCC
v* v
PI
I*g
I*g
current controller
* vac
v g current ig controller
vac
* vac
PW M
I
PI
PW M
dP dV
voltage controller
dP v* voltage controller dV I v
vac
Figure 9. Block diagram of the whole PV control scheme. RCC: ripple correlation control.
Figure 9. Block diagram of the whole PV control scheme. v i RCC: ripple correlation control. g g and current oscillations in case of In order to provide for a comparative example of PV voltage basic (H-bridge) and multilevel (H-bridge with LDN) inverter configurations, Figure 10 shows the
Figure 9. Block diagram of the whole PV control scheme. RCC: ripple correlation control.
In order to provide for a comparative example of PV voltage and current oscillations in case of basic (H-bridge) and multilevel (H-bridge with LDN) inverter configurations, Figure 10 shows the
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In order provide for a comparative example of PV voltage and current oscillations in case Energies 2017, 10,to2037 9 of 19 of basic (H-bridge) and multilevel (H-bridge with LDN) inverter configurations, Figure 10 shows results of of a anumerical the same sameworking working the results numericalsimulation simulationconsidering consideringthe thetwo two conversion conversion schemes schemes in the conditions.ItItisisclearly clearly visible that voltage current contain the second harmonic (100 conditions. visible that PVPV voltage andand current contain onlyonly the second harmonic (100 Hz) theof case of H-bridge without LDNcolumn). (left column). Increasing the number the output voltage inHz) thein case H-bridge without LDN (left Increasing the number of the of output voltage level level3 from 3 to the 5 with the considered LDN configuration, relevant first(50 harmonic (50 Hz) clearly from to 5 with considered LDN configuration, a relevantafirst harmonic Hz) clearly appears in appears to in the addition the second in bothand PV voltage addition secondtoharmonic in harmonic both PV voltage current. and current. vac
vac iac
iac
(a)
v
v (b)
i
i (c)
H-bridge
H-bridge with LDN
Figure10. 10.Simulation Simulationexamples examplesof: of:(a) (a)grid gridcurrent currentand andinverter inverteroutput outputvoltage; voltage;(b) (b)PV PVvoltage; voltage;and and Figure (c) PV current, in case of H-bridge without LDN (left column), and in case of H-bridge with LDN (c) PV current, in case of H-bridge without LDN (left column), and in case of H-bridge with LDN (rightcolumn). column). (right
4.1. Basic RCC-MPPT Algorithm in Case of Single PV Harmonic 4.1. Basic RCC-MPPT Algorithm in Case of Single PV Harmonic In case of single-phase single-stage PV systems, with the inverter being directly connected to In case of single-phase single-stage PV systems, with the inverter being directly connected to the PV field, inherent second-order harmonics of instantaneous power appear in the PV current and the PV field, inherent second-order harmonics of instantaneous power appear in the PV current and PV voltage. As is known, the ripple correlation control algorithm uses these oscillations for PV voltage. As is known, the ripple correlation control algorithm uses these oscillations for providing providing information about the operating point of the PV field. For the considered working point information about the operating point of the PV field. For the considered working point Q, the voltage Q, the voltage derivative of the PV power dP/dV is written as: derivative of the PV power dP/dV is written as: dP dI dP = I + dI V (23) dV Q= I + dV Q V (23) dV Q dV Q where dI/dV defines the relation between PV current and PV voltage harmonics as: where dI/dV defines the relation between PV current and PV voltage harmonics as: ~ dI ~ i = v (24) dV Qe ei = dI v (24) dV Q Generally speaking, the most popular RCC-MPPT implementation consists in multiplying the Generally speaking, most harmonic, popular RCC-MPPT implementation consists in multiplying theof voltage harmonic by thethe current and integrating over the harmonic period, i.e., half voltage harmonic byperiod the current fundamental (grid) T/2, as:harmonic, and integrating over the harmonic period, i.e., half of fundamental (grid) period T/2, as: t ~~ i v dt Rt e I V2 I dI t T − / i2 ve dt = − I2 V = − 2I (25) dI dV =t−T/2t 22 2 =Q t ~ 2 = − V 22 = −V 22 (25) R v dt dV Q V2 V2 2 ve dt
t −T/2 t− T/2
Assuming the fundamental output period T (frequency 50 Hz), the concept of moving average (MAvg) is introducing over T/2 (100 Hz) in Equation (25), and Equation (23) can be rewritten in ~ terms of block diagram representing the estimation of dP/dV, with v~ and i being calculated by Equation (13) and Equation (20), respectively, as shown in Figure 11.
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Assuming the fundamental output period T (frequency 50 Hz), the concept of moving average (MAvg) is introducing over T/2 (100 Hz) in Equation (25), and Equation (23) can be rewritten in terms of block diagram representing the estimation of dP/dV, with ve and ei being calculated by Equation (13) Energies 2017, 10, 2037 10 of 19 and Equation (20), respectively, as shown in Figure 11. Energies 2017, 10, 2037 10 of 19
I I
~ ~i i
MAvg MAvg 100 Hz
100 Hz
MAvg MAvg 100 Hz 100 Hz
100 Hz V V
− +− +
v v
I I
÷ ÷
v~ v~
LPF LPF
MAvg MAvg 100 Hz
dI dI dV dV
− + +−+ +
MAvg MAvg 100 Hz
− +− +
i i
dP dP dV dV
V V
100 Hz
Figure 11. Block diagram of the basic RCC algorithm to estimate dP/dV in the case of existence of a Figure 11. Block diagram the RCC to dP/dV the Figure 11. Block component diagram of of (second the basic basicharmonic) RCC algorithm algorithm to estimate estimate dP/dV inHz). the case case of of existence existence of of aa single harmonic using MAvg over T/2 (100in single harmonic component (second harmonic) using MAvg over T/2 (100 Hz). single harmonic component (second harmonic) using MAvg over T/2 (100 Hz).
4.2. The Proposed RCC-MPPT Algorithm in the Case of Multiple PV Harmonics 4.2. Case of of Multiple Multiple PV PV Harmonics 4.2. The The Proposed Proposed RCC-MPPT RCC-MPPT Algorithm Algorithm in in the the Case Harmonics In case of multiple PV voltage and current harmonics (multilevel topology), the estimation of In case of multiple PV voltage and current harmonics (multilevel topology), the estimation of dI/dVIn(and dP/dV)PV should beand carried outharmonics by considering a specific harmonic (kth order), case then of multiple voltage current (multilevel topology), the estimation of dI/dV (and then dP/dV) should be carried out by considering a specific harmonic (kth order), preferably with should highestbeamplitude increase the resolution. In this case, (24) can dI/dV (and the thenone dP/dV) carried outtoby considering a specific harmonic (kthEquation order), preferably preferably the highesttoamplitude toresolution. increase the In this case, Equation (24) can be rewritten as:one with the one with highest amplitude increase the In resolution. this case, Equation (24) can be rewritten as: be rewritten as: dI ~ ~ dI vk eik~ik== dI (26) (26) dV Q v~vekk ik = dV (26) Q dV Q By following following the thesame same approachasasininprevious previous sub-section, Equation (25) can be summarized By sub-section, Equation (25) cancan be summarized for By following the sameapproach approach as in previous sub-section, Equation (25) be summarized for each considered harmonic as: each considered harmonic as: as: for each considered harmonic t Rtt ~i v~ dt ~e ikk ~vekk dt k I kI VVk I k Ik dIdI k k t−t −TTik v k dt = = (27) (27) =−−I k kV22k k ==− − Ik . . dI = t −Ttt V R dVdV = − VV (27) k2k = − kV.k QQ = 22 ~ t e dt Vk dV Q vv dt Vk ~ 2kk t− Tv k dt t −T
∫ ∫ ∫ ∫
t −T
Note Note that that integrals integrals in in Equation Equation (27) (27) are are now now calculated calculated over over the the fundamental fundamental period period T T Note that integrals in Equation (27) are now calculated over the fundamental period T (corresponding frequency), as defined by thebyFourier series, to calculate the kth harmonic. (corresponding totothe thegrid grid frequency), as defined the Fourier series, to calculate the kth (corresponding to the grid frequency), as in defined the in Fourier series, to calculate kth The implementation of Equation (27) shown the block diagram of Figure 12, estimating thethe dP/dV harmonic. The implementation of is Equation (27) is by shown the block diagram of Figure 12, harmonic. The implementation of Equation (27) is shown in the block diagram of Figure 12, considering only the kth harmonic component. estimating the dP/dV considering only the kth harmonic component. estimating the dP/dV considering only the kth harmonic component.
I I
~ ~i i
50 Hz
FS FS k th harmonic k th harmonic
I I
Ik Ik ÷
MAvg MAvg 50 Hz 50 Hz
V V
− +− +
v v
v~ v~
÷
FS FS k th harmonic k th harmonic
Vk Vk
LPF LPF
dI dI dV dV
− − +− − +
MAvg MAvg 50 Hz
− +− +
i i
dP dP dV dV
V V
Figure 12. 12. Block Blockdiagram diagramofofproposed proposedRCC RCC estimating dP/dV a specific harmonic (Fourier series, Figure estimating dP/dV by by a specific harmonic (Fourier series, FS). Figure FS). 12. Block diagram of proposed RCC estimating dP/dV by a specific harmonic (Fourier series, FS).
An alternative approach consists in extending the period of the moving average in the basic An alternative approach consists extending the period of the moving average in the basic RCC scheme of Figure 11 from half ofinthe period (T/2) to the whole fundamental period T (from RCC scheme of Figure 11 from half of the period (T/2) to the whole fundamental period T (from 100 Hz to 50 Hz), according to: 100 Hz to 50 Hz), according to:
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Energies 10, 2037 An2017, alternative
of 19 approach consists in extending the period of the moving average in the basic11RCC scheme of Figure 11 from half of the period (T/2) totthe whole fundamental period T (from 100 Hz to ~~ 50 Hz), according to: i v dt t R dI t −T e i ve dt (28) = tT dI dV Q = t− (28) 2 ~ Rtv dt dV Q ve2 dt
∫
∫
t −T t− T
The presentedin inFigure Figure 13. 13. The corresponding corresponding block block diagram diagram to to estimate estimate dP/dV dP/dV isispresented
I
I
~ i
MAvg
50 Hz
50 Hz ÷
50 Hz
V
−
MAvg
+
v
v~
LPF
MAvg
dI dV
− + +
MAvg
− +
i
dP dV
V
50 Hz
Figure 13. 13. Block Block diagram diagram of the modified modified RCC RCC algorithm algorithm to to estimate estimate dP/dV dP/dV extending over T T Figure of the extending MAvg MAvg over (50 Hz). (50 Hz).
The relationship between the single harmonic method given by Equation (27) and the extended The relationship between the single harmonic method given by Equation (27) and the extended moving average method (Equation (28)) can be carried out by considering all the harmonics of the moving average method (Equation (28)) can be carried out by considering all the harmonics of the alternating PV voltage and current components: alternating PV voltage and current components: ~ ~ v~ = ∑k v~ k , i = ∑k ik (29) ve = ∑ vek , ei = ∑ eik (29) k k In fact, considering Parseval’s theorem and introducing Equation (29) into Equation (28), the estimation of considering dI/dV can be written as: theorem and introducing Equation (29) into Equation (28), In fact, Parseval’s the estimation of dI/dV can be written as:t
~
ik ∑ v~k dt ∫R∑ k t k
∑ ∑
I V dI t −T k k k = t ∑k eik ∑k vek dt = − 2 dV dI Q ∑k kIV kk kV ~ 2 dt = t− T v = − k 2
dV
Q
∫Rt(∑k )
t −T
t− T
2
(∑k vek ) dt
∑k Vk
(30) (30)
Equation (30) can be reorganized multiplying and dividing by Vk, as follows: Equation (30) can be reorganized multiplying and dividing by Vk , as follows: k dI 2 Ik k w dI k ∑ k∑kV(Vk k2VVIk ) ∑ ) ∑k k( wk dV dV Q dI dI Q −= − 2 2 kk = = = dV w V ∑ ∑ k k Q k dV Q ∑ k wk ∑ k Vk k
(31) (31)
Equation (31)states statesthat that estimation of dI/dV the modified Equation (31) thethe estimation of dI/dV mademade by theby modified movingmoving average average method method (28) (i.e., MAvg over the fundamental period T) is equivalent to the weighted average of (28) (i.e., MAvg over the fundamental period T) is equivalent to the weighted average of the the estimation of dI/dV obtained by the individual harmonics (Equation (27)), using as weight (w ) the estimation of dI/dV obtained by the individual harmonics (Equation (27)), using as weight (wkk) the square of the kth voltage harmonic amplitude: square of the kth voltage harmonic amplitude: 2 wwk == V Vkk2.. k
(32) (32)
In order order to to preliminarily preliminarilyprove provethe theeffectiveness effectivenessofof modified RCC compared to the basic RCC In modified RCC compared to the basic RCC for for the estimation of dP/dV, a simulation example is given in Figure In particular, considering a the estimation of dP/dV, a simulation example is given in Figure 14. In 14. particular, considering a simple simpleI-V linear I-Vcorresponding curve corresponding to a parabolic P-V curve, the dP/dV is calculated the basic linear curve to a parabolic P-V curve, the dP/dV is calculated by thebybasic RCC RCC scheme of Figure 11 (MAvg over T/2, 100 Hz, top), and the modified RCC scheme of Figure 13 (MAvg over T, 50 Hz, bottom), in the case of H-bridge with LDN (i.e., Figure 1b, and Figure 10, right column). Reference is made to the two steady-state operating points symmetrically placed on
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scheme of Figure 11 (MAvg over T/2, 100 Hz, top), and the modified RCC scheme of Figure 13 (MAvg Energies 2017, 10, 2037 12 of 19 over T, 50 Hz, bottom), in the case of H-bridge with LDN (i.e., Figure 1b, and Figure 10, right column). Reference is made toright-side the two steady-state points to symmetrically the left-inand on the left- and on the of the MPP,operating corresponding dP/dV ≅ ±0.25placed A, as on depicted Figure ∼ the right-side of the MPP, corresponding to dP/dV ± 0.25 A, as depicted in Figure 14. It is evident = 14. It is evident that applying the MAvg at 100 Hz results in an oscillating estimation of dP/dV, that applying MAvg atis100 Hz results an oscillating whereas the estimation whereas the the estimation precise and in stable applyingestimation the MAvgofatdP/dV, 50 Hz, as a consequence of isEquation precise and stable applying the MAvg at 50 Hz, as a consequence of Equation (31). (31). \
dP/dV (V1*)
P
-0.2
0.2 0 0.6
-0.4 0.62
0.66
0.68
0.7
the left- and on the right-side of the MPP, corresponding to dP/dV ≅ ±0.25 A, as depicted in Figure -0.4 14. evident0.68that applying the MAvg at 100 Hz results in an oscillating estimation of dP/dV, 0.7 0.64 It is 0.66 0.6 0.62 0.64 0.66 Time (s) Time (s) V1* applying VMPP V2* the MAvg V at 50 Hz, as a consequence whereas the estimation is precise and stable of Equation (31).
0.68
0.7
0.64
0.6
0.7
0.68
0.66
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0.62
0.64
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0
0.4
-0.2
0.2 0 0.6
dP/dV (V2*)
0
0.4
0.62
Figure14. 14.Simulation Simulationresults: results:steady-state steady-stateestimation estimationof ofdP/dV dP/dV by by RCC RCC with with moving moving average average over over Figure dP/dV (V1*) dP/dV (V2*) different periods and working points onP the left (V1* = 45 V) and on the right (V2* = 55 V) of the MPP different periods and working points on the left (V 1 * = 45 V) and on the right (V 2 * = 55 V) of the MPP (VMPP == 50 V). Top traces: MAvg over T/2 (100 Hz, basic RCC), and bottom traces: MAvg over T (50 (V MPP 50 V). Top traces: MAvg over T/2 (100 Hz, basic RCC), and bottom traces: MAvg over T (50 Hz, Hz, modified modified RCC).RCC). \
0
0.4
-0.2
0.2
0 0.6
-0.4
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0.68
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V 5. Implementation and Experimental Results V V V 5. Implementation and Experimental Results Figure 14. Simulation results: steady-state estimation of dP/dV by RCC with moving average over In order to different verifyperiods the effectiveness ofthe the multilevel and PVon inverter with the the proposed RCC-MPPT and working points themultilevel left (V1* = 45 V) PV the right (V V) of proposed MPP In order to verify the effectiveness of on inverter with RCC-MPPT 2* = 55 the algorithms in both steady-state and transient conditions, a grid-connected PV generation system (VMPP = 50 V). Top traces: MAvg over T/2 (100 Hz, basic RCC), and bottom traces: MAvg over T (50 algorithms in both steady-state and transient conditions, a grid-connected PV generation system with Hz, modified RCC). with H-bridge and LDN has been implemented, according to the circuit scheme of Figure 15. H-bridge and LDN has been implemented, according to the circuit scheme of Figure 15. The picture of the corresponding experimental setup is shown in Figure 16. It consists of two 5. Implementation and Experimental Results The picture of the corresponding experimental setup is shown in Figure 16. It consists of two power boards (H-bridge andtheLDN) based the onmultilevel Mitsubishi insulated-gate bipolar transistor (IGBT) In order toand verifyLDN) effectiveness PV inverter with the proposed RCC-MPPT(IGBT) smart power boards (H-bridge based onofMitsubishi insulated-gate bipolar transistor smart modules, IPMin PS22A76 (1200 25 A,conditions, Mitsubishi Electric PV Corporation, Tokyo, Japan), algorithms both steady-state andV, transient a grid-connected generation system modules, IPM PS22A76 (1200 V, 25 A, Mitsubishi Electric Corporation, Tokyo, Japan), Yokogawa DLM with H-bridge and LDN has been implemented,Electric according Corporation, to the circuit scheme of Figure 15. Yokogawa DLM 2024 oscilloscope (Yokogawa Tokyo, Japan) with the PICO 2024 oscilloscopeThe (Yokogawa Corporation, Japan) picture of theElectric corresponding experimentalTokyo, setup is shown in with Figure the 16. ItPICO consistsTA057 of two differential TA057 differentialboards voltage probe (25 MHz, ±1400 V, ±2%, Pico Technology, Tyler, TX, USA), and LDN) based on Mitsubishi insulated-gate bipolar (IGBT) voltage probepower (25 MHz, ±(H-bridge 1400 V, and ±2%, Pico Technology, Tyler, TX, USA), andtransistor LEM PR30 current probe LEM PR30 current probeIPM (dcPS22A76 to 20 kHz, LEMElectric Europe GmbH, Tokyo, Fribourg, smart modules, (1200 ±20 V, 25A, A, ±1%, Mitsubishi Corporation, Japan),Switzerland). (dc to 20 kHz, ±20 A, ±1%, LEM Europe GmbH, Fribourg, Switzerland). Additionally, two current Yokogawa DLM 2024 oscilloscope (Yokogawa Electric Tokyo, Japan) the PICO ® Company, Additionally, two current transducers of the LACorporation, 55-P model (LEMwith Geneva, ® Company, transducers of the differential LA 55-P model (LEM were usedand to measure PV TA057 voltage probe (25 MHz, ±1400 V,Geneva, ±2%, Pico Switzerland) Technology, Tyler, TX, USA), Switzerland)LEM were used to measure PV andA, grid currents, GmbH, whileFribourg, PV and grid voltages were PR30 current to 20 kHz, ±20 were ±1%,measured LEM Europeusing Switzerland). and grid currents, while PVprobe and(dc grid voltages two voltage transducers of the ® Company, Geneva, ® (LEM measured using two voltage transducers of the LV 25-P model Additionally, two current transducers of the LA 55-P model (LEM Company, LV 25-P model (LEM® Company, Geneva, Switzerland). The switching frequencyGeneva, of the multilevel Switzerland) were used to measureofPVthe andmultilevel grid currents, while PV and grid voltages is were Switzerland). The switching frequency grid-connected inverter set to 2.5 kHz. grid-connected inverter is set 2.5 kHz. For digital implementation of current and voltage ® Company, measured using twotovoltage transducers of the LV 25-P model (LEM Geneva, controllers, For digital implementation of current and voltage controllers, asinverter well as the phase-locked loop Switzerland). The switching frequency of the grid-connected set to 2.5signal kHz. processor as well as the phase-locked loop (PLL) and themultilevel RCC-MPPT algorithms, ais digital (PLL) and the algorithms, a digital signal processor (DSP ForRCC-MPPT digital implementation of current and voltage controllers, as well as TMS320F28377D) the phase-locked loop was used to (DSP TMS320F28377D) was used to generate the PWM signals for the H-bridge and the LDN boards andsignals the RCC-MPPT algorithms, a digital processor (DSP TMS320F28377D) was used to inverter. An generate the(PLL) PWM for the H-bridge and signal the LDN boards of the multilevel LDN of the multilevel LDN inverter. interface board links DSP with power boards. generate the PWM signalsAn for optical the H-bridge and the LDN boards of the multilevel LDN inverter. An Results are optical interface board board links links DSPDSP with power boards. boards. Resultsshown are by shown by oscilloscope optical interface with power Results are oscilloscope shown by oscilloscope screenshots, elaborating and emphasizing the signals of interest. The main screenshots,screenshots, elaborating and emphasizing the signals of interest. The main parameters of the elaborating and emphasizing the signals of interest. The main parameters of the parameters of the experimental setup are given in Table 1. experimental setup are given in Table 1. experimental setup are given in Table 1. 0.62
0.66
0.64
0.68
0.7
Time (s)
*
1
MPP
*
2
0.6
0.62
0.64
0.66
0.68
0.7
Time (s)
Real PV source
Real PV source Linear-equivalent PV source
Linear-equivalent
PV source Figure 15. Circuit scheme of the experimental setup including the whole PV generation system.
Figure 15. Circuit scheme of the experimental setup including the whole PV generation system.
Figure 15. Circuit scheme of the experimental setup including the whole PV generation system.
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Energies 2017, 2037 The dc10,source
of 19 by aa resistive series The dc source has has been been implemented implemented by resistive voltage voltage supply supply with with variable variable 13 series resistance for for preliminary preliminarysteady-state steady-stateand andtransient transient tests, according to the scheme of Figure 15. resistance tests, according to the scheme of Figure 15. The The dc source has been implemented by a resistive voltage supply with variable series The corresponding parameters are given in Table corresponding parameters are given in Table 2. 2. resistance for preliminary steady-state and transient tests,of according to the scheme PV of Figure 15. The For realistic tests, a reduced-scale array two modules has For realistic tests, a reduced-scale array consisting consisting of two series-connected series-connected PV modules has corresponding parameters are given in Table 2. been adopted, introducing different irradiance conditions by covering/uncovering with a white sheet, been adopted, introducing different irradiance conditions by covering/uncovering with a white For realistic sunny tests, aand reduced-scale array consisting of two series-connected PVsurface modules has well cloudy conditions (the sun irradiance on the PV module ranges sheet,representing well representing sunny and cloudy conditions (the sun irradiance on the PV module surface been adopted, introducing different irradiance conditions by covering/uncovering with a white betweenbetween approximately 100% and 40%), as 40%), shownasinshown the right of Figure PV modules, on ranges approximately 100% and in side the right side 16. of Figure 16. PV placed modules, sheet, well representing sunny and cloudy conditions (the sun irradiance on the PV module surface the roofon of the supply the conversion system in the Lab at thein ground floor a connection placed thebuilding, roof of the building, supply the conversion system the Lab atusing the ground floor ranges(length between approximately 100% and 40%), asparameters shown in the right side of Figure 16. PVinmodules, cable of approximately 40 m). The main of the PV source are given Table 3, using a connection cable (length of approximately 40 m). The main parameters of the PV source are placed on the roof of the I-V building, supply the conversion system in the Lab at charging the ground floor whereas the corresponding and P-V characteristics, obtained from the Lab by the transient given in Table 3, whereas the corresponding I-V and P-V characteristics, obtained from the Lab by using a connection cable (length approximately 40 m). The main parameters of the PV source are of acharging capacitor (1 mF), are inof Figure 17. are given the transient of agiven capacitor (1 mF), in Figure 17. given in Table 3, whereas the corresponding I-V and P-V characteristics, obtained from the Lab by the charging transient of a capacitor (1 mF), are given in Figure 17.
DC supplies
DC supplies transducers
DSP F28377D board DSP F28377D board
interface board interface board
transducers voltage probes
LDN cells LDN H-bridge cells H-bridge
AC capacitor AC capacitor
voltage probes H-bridge capacitors LDN H-bridge capacitors LDN
1:10 transformer
1:10 transformer current probes AC grid current probes ACview grid of the experimental setup in the lab (left side) and arrangement of the PV Figure 16. Picture Figure 16. Picture view of the experimental setup in the lab (left side) and arrangement of the PV array (two modules) modules) on on the the roof, roof,free freeand andshadowed shadowedby by awhite whitesheet sheet∼(≅500 W/m22 and and 200 200 W/m W/m22, array 500 arrangement W/m Figure(two 16. Picture view of the experimental setup in thea lab (left side)(=and of the PV, right side). right array side). (two modules) on the roof, free and shadowed by a white sheet (≅500 W/m2 and 200 W/m2, right side).
Figure 17. P-V and I-V characteristics of the PV array including the cable (Table 2) with an irradiance of 500 W/m2 (top) and 200 W/m2 (white sheet, bottom) and Tcell ≅ 35 °C (reference is made to Figure 17. P-V and I-V characteristics of the PV array including the cable (Table 2) with an irradiance Figure 16). 17. P-V and I-V characteristics of the PV array including the cable (Table 2) with an irradiance of 2 (top) and 200 W/m2 (white sheet, bottom) and Tcell ≅ 35 °C (reference is made to of 500 W/m 500 W/m2 (top) and 200 W/m2 (white sheet, bottom) and Tcell ∼ = 35 ◦ C (reference is made to Figure 16). Figure 16).
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Table 1. Parameters of the conversion system. Label
Description
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CH CL Cf Rf , Label Lf fsw CH Vg CL f Cf
Parameters
dc-link H-bridge capacitance 2.2 mF dc-link LDN capacitance 3.3 mF Table 1. Parameters of the conversion system. ac filter capacitance 25 µF ac filter resistance and inductance 1.1 Ω, 8.8 mH Description Parameters switching frequency 2.5 kHz dc-link H-bridge capacitance 2.2 mF grid voltage (RMS) (1:10 transformer) 22/220 dc-link LDN capacitance 3.3 mF V grid (fundamental) frequency ac filter capacitance 2550 μFHz
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Rf, Lf ac filter resistance and inductance 1.1 Ω, 8.8 mH fsw frequency Table 2.switching Resistive dc source parameters. 2.5 kHz Vg grid voltage (RMS) (1:10 transformer) 22/220 V f Label grid (fundamental) frequency 50 Hz Description Parameters
VS R1 , R2
dc voltage supply dc source resistance (RS )
Table 2. Resistive dc source parameters.
100 V 40 Ω, 20 Ω
Label Description Parameters VS dc voltage supply 100 V Table 3. PV source parameters—Standard test conditions (STC). R1, R2 dc source resistance (RS) 40 Ω, 20 Ω
Label
Description
VOC ILabel SC VOC VMPP ISC IMPP V NSMPP RcI,MPP Lc
open circuit voltage Description short circuit current open circuit voltage voltage maximum photovoltaic short circuit current current maximum photovoltaic maximum photovoltaic voltage number of series-connected PV modules maximum photovoltaic current resistance and inductance of PV cable
Parameters
Table 3. PV source parameters—Standard test conditions (STC).
NS Rc, Lc
number of series-connected PV modules resistance and inductance of PV cable
43.4 V
Parameters 4.8 A 43.4 34 V V 4.8 A 4.4 A 34 V 2 4.4 Ω, A 40 µH 0.5 2 0.5 Ω, 40 μH
The first test is carried out in order to verify the input/output steady-state waveforms of the considered multilevel in the grid-connected configuration in Figure 15 with The first test isinverter, carried out in order to verify the input/output shown steady-state waveforms of resistive the dc supply (linear-equivalent PV source). In particular, Figure 18a shows grid andwith current considered multilevel inverter, in the grid-connected configuration shown in voltage Figure 15 resistive dc supply (linear-equivalent PV source). In particular, Figure 18a shows grid voltage and (top half screen) and PV voltage and current (bottom half screen), whereas Figure 18b shows the time (top voltage half screen) PV (top voltage current half and screen), whereas Figure zoomcurrent of inverter and and current halfand screen) and(bottom PV voltage current (bottom half18b screen). shows thethe time zoom ofvoltage inverterhas voltage and current (top half screen) and PVfive voltage andand current As expected, inverter a proper multilevel waveform over levels, inverter screen). sinusoidal As expected,despite the inverter voltage hasfrequency a proper multilevel waveform over fiveunity (grid)(bottom currenthalf is almost the switching being only 2.5 kHz, with levels, and inverter (grid) current is almost sinusoidal despite the switching frequency being only power factor. PV voltage and PV current have oscillations, including both first and second harmonic 2.5 kHz, with unity power factor. PV voltage and PV current have oscillations, including both first components (50 and 100 Hz, respectively), in phase opposition. and second harmonic components (50 and 100 Hz, respectively), in phase opposition.
(a)
(b)
Figure 18. Input/output steady-state waveforms of the converter with VS = 100 V, RS = 60 Ω, V = VMPP
Figure 18. Input/output steady-state waveforms of the converter with VS = 100 V, RS = 60 Ω, = 50 V: (a) grid voltage and current (top traces), PV voltage and current (bottom traces); (b) time V = VMPP = 50 V: (a) grid voltage and current (top traces), PV voltage and current (bottom traces); zoom: inverter voltage and current (top traces), PV voltage and current (bottom traces). (b) time zoom: inverter voltage and current (top traces), PV voltage and current (bottom traces).
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The The following following tests tests are are carried carried out out in in order order to to verify verify the the dynamic dynamic performance performance of of the the different different The following tests are carried out in order to verify the dynamic performance of the different RCC-MPPT algorithms in the case of irradiance transients. In particular, two sets of experimental tests RCC-MPPT algorithms in the case of irradiance transients. In particular, two sets of experimental RCC-MPPT algorithms the case of presented irradiance particular, sets of experimental have been considered. Ininthe first tests intransients. Figures 19Inand 20and the linear-equivalent PV source tests have been considered. In the first tests presented in Figures 19 20two the linear-equivalent PV tests have been considered. In the first tests presented in Figures 19 and 20 the linear-equivalent PV has been selected in order to simulate extremely fast irradiance transients by switching on/off the series source has been selected in order to simulate extremely fast irradiance transients by switching source has series been selected in order to simulate extremely fastfour irradiance byalgorithms. switching resistances, performing dynamic comparative tests forcomparative all the types of transients RCC-MPPT on/off the resistances, performing dynamic tests for all the four types of on/off the series resistances, performing dynamic comparative tests for all the four types of The last tests presented in Figure 21 are performed considering the real PV source, introducing RCC-MPPT algorithms. The last tests presented in Figure 21 are performed considering the real PV RCC-MPPT algorithms. The last tests presented in Figure 21 are performed considering the real PV irradiance transients by shadowing/unshadowing the PV modules by a white sheet, well representing source, introducing irradiance transients by shadowing/unshadowing the PV modules by a white source, introducing irradiance transients by shadowing/unshadowing the effects of real clouds (Figures 16ofand sheet, well representing the effects real17). clouds (Figures 16 and 17). the PV modules by a white sheet, well representing the effects of real clouds (Figures 16 and 17).
(a) (b) (a) (b) Figure 19. Experimental results of RCC-MPPT algorithms in case of step transients of RS: Figure results of RCC-MPPT algorithms in case of RS : (a) low-pass Figure 19. 19.Experimental Experimental results of RCC-MPPT algorithms in step casetransients of step oftransients of RS: (a) low-pass filters by MAvg over T/2 (100 Hz) and (b) low-pass filters MAvg over T (50 Hz). grid Top filters by MAvg over T/2 (100 Hz) and (b) low-pass filters MAvg over T (50 Hz). Top traces: (a) low-pass filters and by MAvg over T/2 (100 Hz)PV and (b) low-pass filters MAvg over T (50 Hz). Top traces: grid voltage current. Bottom traces: voltage and current. voltage and current. Bottom traces: PV voltage and current. traces: grid voltage and current. Bottom traces: PV voltage and current.
(a) (b) (a) (b) Figure 20. Experimental results of modified RCC-MPPT algorithms in the case of step transients of Figure 20. Experimental results of modified RCC-MPPT algorithms the case of step stepfirst transients of RS: (a) using only the second harmonic component (100 algorithms Hz); and (b)in only harmonic Figure 20. Experimental results of modified RCC-MPPT inusing the case ofthe transients of RS: (a) using only the second harmonic component (100 Hz); and (b) using only the first harmonic component (50 Hz). Top traces:harmonic grid voltage and current. traces: PV voltage and current. R only the second component (100 Bottom Hz); and (b) using only the first harmonic S : (a) using component (50 (50 Hz). Hz). Top Top traces: traces: grid grid voltage voltage and and current. current. Bottom Bottom traces: traces: PV PV voltage voltage and and current. current. component
Figure 19 shows grid voltage and current (top half screen) as well as PV voltage and current Figure 19screen) shows grid voltage and current (top half by screen) as well as PVthe voltage and current (bottom half19 thevoltage case ofand transients switching seriesand resistances Figure shows in grid current obtained (top half screen) as wellon/off as PV voltage current (bottom half screen) in the case of transients obtained by switching on/off the series resistances (Figure 15, 2), representing step irradiance transient. Diagrams in Figure 19a are obtained (bottom halfTable screen) in the case ofa transients obtained by switching on/off the series resistances (Figure 15, Table 2), representing a step irradiance transient. Diagrams in Figure 19a are obtained by the basic RCC scheme with low-pass filters based on MAvg over T/2 (100 Hz, corresponding to (Figure 15, Table 2), representing a step irradiance transient. Diagrams in Figure 19a are obtained by the basic RCC scheme with low-pass filters based on MAvg over T/2 (100 Hz, corresponding to Figure 11), RCC whereas diagrams in Figure 19b based are obtained theT/2 modified scheme with by the basic scheme with low-pass filters on MAvgby over (100 Hz,RCC corresponding to Figure 11), whereas diagrams in Figure 19bcorresponding are obtained to byFigure the modified RCC scheme with low-pass filters based on MAvg over T (50 Hz, 13). Figure 11), whereas diagrams in Figure 19b are obtained by the modified RCC scheme with low-pass low-pass filters based on MAvg over T (50 Hz, corresponding toisFigure both cases, as over expected, the steady-state MPP voltage equal13). to the half of the dc source filtersInbased on MAvg T (50 Hz, corresponding to Figure 13). In both cases, as expected, the steady-state MPP voltage is equal to half as of athe dc source is seen perturbation voltage (VS = 100V, VMPP = 50V). In this case, the (equivalent) irradiance stepthe voltage (VS = 100V, VMPP = 50V). In this case, the (equivalent) irradiance step is seen as a perturbation by the PV voltage controller. Correspondingly, the PV current shows a fast step transient. by the PV voltage controller. Correspondingly, the PV current shows a fast step transient.
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In both cases, as expected, the steady-state MPP voltage is equal to the half of the dc source voltage (VS = the 100V,basic VMPPRCC = 50V). In thisgiving case, the (equivalent) irradiance step seen asovershoots a perturbation Despite scheme acceptable results, it leads toislarger and by the PV voltage controller. Correspondingly, the PV current shows a fast step transient. higher settling times comparing to the modified RCC. Despite theshows basic RCC scheme giving acceptable it leads to larger overshoots and19, higher Figure 20 the same quantities with the results, same kind of transients as in Figure but settling times comparing to the modified RCC. referred to the modified RCC schemes employing only a specific harmonic component (reference is 20 shows same quantities with same20a kind transients in modified Figure 19,RCC but referred madeFigure to Figure 12). Inthe particular, diagrams inthe Figure areofobtained byas the scheme to the modified RCC schemes employing only a specific harmonic component (reference is made by to based on the 2nd harmonic component (100 Hz), whereas diagrams in Figure 20b are obtained Figure 12). In particular, diagrams 20aharmonic are obtained by the modified based on the modified RCC scheme based in onFigure the first component (50 Hz).RCC Bothscheme these modified the 2nd harmonic component (100 Hz), whereas diagrams in Figure 20b are obtained by the modified RCC-MPPT schemes give similar results, being similar the amplitude of voltage and current RCC scheme(Figures based on7the first8), harmonic component (50 Hz). Both modified harmonics and with smoothed overshoots andthese short settlingRCC-MPPT times for schemes the PV give similar results, being similar the amplitude of voltage and current harmonics (Figures 7 and 8), variables. with In smoothed overshoots and short settling times for the PV variables. particular, the modified RCC scheme employing the first harmonic component (50 Hz) In particular, modified scheme first harmonic component Hz) seems seems to be more the effective, alsoRCC offering the employing advantage the of using the highest harmonic(50 amplitude in to more modulation effective, also offering of using the highestschemes harmonic amplitude in the thebe typical index rangethe foradvantage grid-connected PV generation (i.e., m between 0.7 typical modulation index range grid-connected generationscheme schemes m between 0.7last andtests 0.8). and 0.8). For these reasons, onlyfor this last modifiedPV RCC-MPPT is (i.e., considered in the For these reasons, only21, thisconsidering last modified scheme is considered in theinlast tests presented in presented in Figure a RCC-MPPT real PV source (two PV modules), real environmental Figure 21, considering a real PV source (two PV modules), in real environmental operating conditions, operating conditions, and with realistic irradiance transients. Increasing and decreasing sun and with realistic irradiance transients. Increasing and decreasing sun irradiance transients obtained by irradiance transients obtained by shadowing and unshadowing the PV modules by a white sheet shadowing and unshadowing the PV modules by and a white have beenofconsidered, corresponding have been considered, corresponding to the P-V I-V sheet characteristics Figure 17 (500 W/m2 and 2 and 200 W/m2 ). to the P-V and I-V characteristics of Figure 17 (500 W/m 200 W/m2). In current (top half of the screen), together withwith PV In particular, particular,Figure Figure21 21shows showsgrid gridvoltage voltageand and current (top half of the screen), together voltage and the estimation of dP/dV (bottom half of the screen). As expected, the estimation of dP/dV PV voltage and the estimation of dP/dV (bottom half of the screen). As expected, the estimation of fails during the initial of part the transient, but without introducing particular drawbacks in PV and dP/dV fails during the part initial of the transient, but without introducing particular drawbacks in grid variables. In fact, both for increasing and decreasing irradiance transients, grid current amplitude PV and grid variables. In fact, both for increasing and decreasing irradiance transients, grid current has a smoothed fast profile, without significant amplitude has aand smoothed and fast profile, without overshoots. significant overshoots.
(a)
(b)
Figure 21. 21. Decrease Decrease and and increase increase sun sunirradiance irradiancetransients transients(500 (500W/m W/m22–200 –200 W/m W/m22)) using the Figure using only only the 50-Hz harmonic component (Fourier). 50-Hz harmonic component (Fourier).
6. Efficiency Analysis in Comparison to the Basic H-Bridge Configuration 6. Efficiency Analysis in Comparison to the Basic H-Bridge Configuration As a final remark, the proposed multilevel conversion scheme, in Figure 1b, is compared with As a final remark, the proposed multilevel conversion scheme, in Figure 1b, is compared the basic H-bridge conversion scheme, in Figure 1a, from the point of view of the overall efficiency. with the basic H-bridge conversion scheme, in Figure 1a, from the point of view of the In particular, a comparative estimation of power losses is carried out introducing some simplifying overall efficiency. In particular, a comparative estimation of power losses is carried out introducing assumptions. some simplifying assumptions. Generally speaking, the multilevel converter itself has one additional leg (with two additional Generally speaking, the multilevel converter itself has one additional leg (with two additional power switches), i.e., three legs instead of two, with a consequent increase in conduction and power switches), i.e., three legs instead of two, with a consequent increase in conduction and switching switching losses being the main disadvantage of multilevel configurations. On the other hand, the losses being the main disadvantage of multilevel configurations. On the other hand, the reduced reduced harmonic distortion in multilevel output voltage (inter-level voltage excursion is half, passing from 3 to 5 levels) makes it possible to reduce the inductance (Lf) of the ac-link inductor to
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harmonic distortion in multilevel output voltage (inter-level voltage excursion is half, passing from 3 to 5 levels) makes it possible to reduce the inductance (Lf ) of the ac-link inductor to obtain the same grid current distortion, reducing inductor losses. This is one of the known advantages of multilevel configurations. With reference to the multilevel inverter losses, for the LDN leg in series with the two H-bridge legs, they share the same current, so each leg has the same conduction losses. The steady-state voltage the half in the LDN leg, as discussed in Section 2, the switching losses in the LDN leg are the half compared to the switching losses in the individual H-bridge legs. Supposing conduction and switching losses equally shared, as in most of the switching converter design, the additional LDN leg introduces 50% more of the conduction losses, and 25% more of the switching losses. So, comparing to the basic H-bridge inverter, the multilevel inverter has almost 37.5% of additional losses. With reference to the ac-link inductor, in case of multilevel inverter, it can be designed for the half of the inductance (Lf /2) compared to the basic H-bridge scheme (Lf ), since the voltage harmonic distortion is almost half, leading to almost the same current harmonic distortion. Supposing the use of the same amount of copper for the reactor winding, and neglecting the core losses (if any), the copper √ losses are reduced to 50%, with the winding resistance reduced to half (turns are 2 times less and the √ wire cross area can be 2 times more to have the same copper weight), and the ac-link reactor current the same. All in all, a real benefit in terms of efficiency can be experienced in the case of the multilevel inverter (H-bridge + LDN) compared to basic inverter (only H-bridge) especially in case of an overall conversion system design with lower inverter losses (that are increased 37.5%) compared to ac-link inductor losses (that are decreased by 50%). 7. Conclusions In this paper, a single-phase single-stage PV generation system with multilevel output voltage waveforms and an improved RCC-MPPT algorithm has been proposed and examined in detail. The multilevel inverter is implemented by an LDN cell, as a kind of retrofit to the basic H-bridge cell, increasing the output voltage levels from three to five. In addition to the second harmonic resulting from the basic H-bridge configuration, a multilevel configuration introduces additional voltage and current harmonics on the input (PV) side. In particular, a relevant first-order harmonic is noticeable, the third harmonic is slightly appreciable, whereas higher-order harmonics are completely negligible. Both PV voltage and current harmonic amplitudes are analytically calculated in the whole modulation range, offering the possibility of a precise and effective design of the dc-link capacitor to satisfy the ripple requirements. Due to the additional harmonics, the basic RCC-MPPT scheme becomes inadequate, leading to a misestimation of dP/dV. A modified RCC scheme extracting the amplitude of a specific harmonic from PV voltage and current waveforms has been proposed in order to overcome this drawback. Reference has been made to the harmonic with highest amplitude in order to maximize the resolution, leading to a correct and precise estimation of dP/dV. It has been verified that a correct estimation of dP/dV can be simply obtained by doubling the time window of low-pass filters in the RCC scheme (i.e., moving average, from T/2 to T). It has been proven that the resulting dP/dV is a weighted average of all the dP/dV estimations performed by the individual PV voltage and current harmonics, the weight being the individual voltage harmonic amplitude itself. Numerical and experimental tests have been carried out to prove the effectiveness of the whole PV generation scheme, including multilevel waveforms and improved RCC-MPPT algorithms. Both the linear-equivalent PV source and real PV sources have been implemented in the experimental setup, considering both steady-state and fast sun irradiance transient conditions in order to verify the dynamic performance of the different RCC-MPPT methods.
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Author Contributions: Manel Hammami contributed to manuscript composition and analytical developments, and she specifically provided the simulation and experimental results. Gabriele Grandi contributed to analytical developments, supervising the manuscript composition and the experimental tests. Conflicts of Interest: The authors declare no conflict of interest.
Appendix A The coefficients Ak and Bk in Equation (7) are calculated as: Zαm
Ak (m) =
m sin ϑ cos(kϑ) dϑ =
0
Bk (m) =
π/2 R αm
h i p 1 k sin(k αm ) + 4m2 − 1 cos(k αm ) − 2m , −2
2k2
(1 − m sin ϑ) cos(kϑ) dϑ = − 1k sin(k αm ) +
1
2k2 −2
h
k sin(k αm ) +
√
i 4m2 − 1 cos(k αm ) ,
(A1)
(A2)
with αm = arcsin(1/2m). By introducing the power balance (Equation (10)) and considering sinusoidal output current with unity power factor (Equation (11)), it gives: i
H
# " ∞ 1 = m + (2U0 − U2 ) sin ϑ − m cos 2ϑ − ∑ (Un+1 − Un−1 ) sin nϑ Iac 2 n =3
(A3)
From Equation (A3), the amplitudes of the low-frequency input H-bridge current harmonic components are:
I1H
I H = I2H = 21 m Iac = 21 (2U0 − U2 ) Iac n ≥ 3 odd integer. InH = 12 |Un+1 − Un−1 | Iac
(A4)
References 1. 2.
3. 4. 5. 6. 7. 8. 9.
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