View source for Maximum power point tracking

{{Short description|Solar cell power extraction method}}
[[File:UP-curve of partially shaded solar generator.png|thumb|Power/Voltage-curve of a partially shaded PV system, with marked local and global MPP]]
'''Maximum power point tracking''' ('''MPPT'''),<ref>{{Cite journal |last1=Seyedmahmoudian |first1=M. |last2=Horan |first2=B. |last3=Soon |first3=T. Kok |last4=Rahmani |first4=R. |last5=Than Oo |first5=A. Muang |last6=Mekhilef |first6=S. |last7=Stojcevski |first7=A. |date=2016-10-01 |title=State of the art artificial intelligence-based MPPT techniques for mitigating partial shading effects on PV systems – A review |journal=Renewable and Sustainable Energy Reviews |volume=64 |pages=435–455 |doi=10.1016/j.rser.2016.06.053}}</ref><ref>{{Cite journal |last1=Seyedmahmoudian |first1=Mehdi |last2=Horan |first2=Ben |last3=Rahmani |first3=Rasoul |last4=Maung Than Oo |first4=Aman |last5=Stojcevski |first5=Alex |date=2016-03-02 |title=Efficient Photovoltaic System Maximum Power Point Tracking Using a New Technique |journal=Energies |volume=9 |issue=3 |pages=147 |doi=10.3390/en9030147|doi-access=free |hdl=10536/DRO/DU:30083526 |hdl-access=free }}</ref> or sometimes just '''power point tracking''' ('''PPT'''),<ref>{{Cite web|url=https://www.solar-electric.com/learning-center/mppt-solar-charge-controllers.html/|title=What is Maximum Power Point Tracking (MPPT)|website=Northern Arizona Wind & Sun}}</ref><ref>{{cite book |pages=1–17 |doi=10.1109/EnergyTech.2012.6304652 |chapter=A survey of maximum PPT techniques of PV systems |year=2012 |last1=Ali |first1=Ali Nasr Allah |last2=Saied |first2=Mohamed H. |last3=Mostafa |first3=M |last4=Abdel-Moneim |first4=T. M. |title=2012 IEEE Energytech |isbn=978-1-4673-1835-8 |s2cid=10207856 }}</ref> is a technique used with variable power sources to maximize energy extraction as conditions vary. The technique is most commonly used with [[photovoltaic]] (PV) solar systems, but can also be used with [[Wind turbine|wind turbines]], [[Power over fiber|optical power transmission]] and [[thermophotovoltaic]]s.

PV solar systems have varying relationships to inverter systems, external grids, battery banks, and other electrical loads.<ref>{{Cite journal |title=Simulation and Hardware Implementation of New Maximum Power Point Tracking Technique for Partially Shaded PV System Using Hybrid DEPSO Method |journal=IEEE Transactions on Sustainable Energy |date=2015-07-01 |issn=1949-3029 |pages=850–862 |volume=6 |issue=3 |doi=10.1109/TSTE.2015.2413359 |bibcode=2015ITSE....6..850S |first1=M. |last1=Seyedmahmoudian |first2=R. |last2=Rahmani |first3=S. |last3=Mekhilef |first4=A. |last4=Maung Than Oo |first5=A. |last5=Stojcevski |first6=Tey Kok |last6=Soon |first7=A. S. |last7=Ghandhari|s2cid=34245477 }}</ref> The central problem addressed by MPPT is that the [[Maximum power transfer theorem|efficiency of power transfer]] from the solar cell depends on the amount of available sunlight, shading, solar panel temperature and the [[Electrical load|load]]'s electrical characteristics. As these conditions vary, the load characteristic ([[Impedance_matching#Power_transfer|impedance]]) that gives the highest power transfer changes. The system is optimized when the load characteristic changes to keep power transfer at highest efficiency. This optimal load characteristic is called the ''maximum power point'' (MPP). MPPT is the process of adjusting the load characteristic as the conditions change. Circuits can be designed to present optimal loads to the photovoltaic cells and then convert the voltage, current, or frequency to suit other devices or systems.

[[Solar cell]]s' non-linear relationship between temperature and total resistance can be analyzed based on the [[Current–voltage characteristic|Current-voltage (I-V) curve]] and the power-voltage (P-V) curves.<ref name="Seyedmahmoudian 402–414">{{Cite journal |title = A Comparative Study on Procedure and State of the Art of Conventional Maximum Power Point Tracking Techniques for Photovoltaic System |journal=International Journal of Computer and Electrical Engineering |pages=402–414 |volume=6 |issue=5 |doi=10.17706/ijcee.2014.v6.859 |first1=Mohammadmehdi |last1=Seyedmahmoudian |first2=Arash |last2=Mohamadi |first3=Swarna |last3=Kumary|year=2014 |doi-access=free }}</ref><ref>{{Cite journal |title=Analytical Modeling of Partially Shaded Photovoltaic Systems |journal=Energies |date=2013-01-04 |pages=128–144 |volume=6 |issue=1 |doi=10.3390/en6010128 |first1=Mohammadmehdi |last1=Seyedmahmoudian |first2=Saad |last2=Mekhilef |first3=Rasoul |last3=Rahmani |first4=Rubiyah |last4=Yusof |first5=Ehsan Taslimi |last5=Renani|doi-access=free |hdl=10536/DRO/DU:30080850 |hdl-access=free }}</ref> MPPT samples cell output and applies the proper resistance (load) to obtain maximum power.<ref>{{cite journal |last=Surawdhaniwar |first=Sonali |author-first2=Ritesh |author-last2=Diwan |title=Study of Maximum Power Point Tracking Using Perturb and Observe Method |journal=International Journal of Advanced Research in Computer Engineering & Technology |date=July 2012 |volume=1 |issue=5 |pages=106–110}}</ref> MPPT devices are typically integrated into an [[electric power converter]] system that provides voltage or current conversion, filtering, and regulation for driving various loads, including power grids, batteries, or motors. [[Solar inverter]]s convert DC power to AC power and may incorporate MPPT.

The power at the MPP (P<sub>mpp</sub>) is the product of the MPP voltage (V<sub>mpp</sub>) and MPP current (I<sub>mpp</sub>).

In general, the P-V curve of a partially shaded solar array can have multiple peaks, and some algorithms can get stuck in a local maximum rather than the global maximum of the curve.<ref>{{cite journal | doi=10.1016/j.sftr.2020.100020 | title=Classification and Evaluation Review of Maximum Power Point Tracking Methods | year=2020 | last1=Baba | first1=Ali Omar | last2=Liu | first2=Guangyu | last3=Chen | first3=Xiaohui | journal=Sustainable Futures | volume=2 | page=100020 | s2cid=219879843 | doi-access=free }}</ref>

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== Background ==
{{main|I-V curve}}
[[Image:Solar-Cell-IV-curve-with-MPP.png|thumb|Photovoltaic solar cell I-V curves where a line intersects the knee of the curves where the maximum power transfer point is located.]]

[[Photovoltaic cell]]s have a complex relationship between their operating environment and the [[Electric power|power]] they produce. The [[fill factor (solar cell)|fill factor]] (''FF'') characterizes the cell's non-linear electrical behavior. Fill factor is defined as the ratio of the maximum power from the cell to the product of [[Open-circuit voltage|open circuit voltage]] {{math|''V''<sub>oc</sub>}} and [[Short circuit|short-circuit current]] {{math|''I''<sub>sc</sub>}}. Tabulated data is often used to estimate the maximum power that a cell can provide with an optimal load under given conditions: 

:<math>P=FF \, V_{oc} I_{sc}</math>. 

For most purposes, {{math|''FF''}}, {{math|''V''<sub>oc</sub>}}, and {{math|''I''<sub>sc</sub>}} are enough information to give a useful approximate view of the cell's electrical behavior under typical conditions.

For any given set of conditions, cells have a single operating point where the values of the [[Electric current|current]] (''I'') and [[voltage]] (''V'') of the cell allow maximum [[Electric power|power]] output.<ref>{{Cite journal |title=Maximum power point tracking of partial shaded photovoltaic array using an evolutionary algorithm: A particle swarm optimization technique |journal=Journal of Renewable and Sustainable Energy |date=2014-03-01 |issn=1941-7012 |page=023102 |volume=6 |issue=2 |doi=10.1063/1.4868025 |first1=Mohammadmehdi |last1=Seyedmahmoudian |first2=Saad |last2=Mekhilef |first3=Rasoul |last3=Rahmani |first4=Rubiyah |last4=Yusof |first5=Ali Asghar |last5=Shojaei|hdl=1959.3/440382 |hdl-access=free }}</ref> These values correspond to a particular load [[electrical resistance|resistance]], which is equal to {{math|''V'' / ''I''}} as specified by [[Ohm's law]]. The power P is given by {{math|1=''P=V I''}}.

A photovoltaic cell, for the majority of its useful curve, acts as a [[constant current source]].<ref>{{Cite web|url=http://geosci.uchicago.edu/~moyer/GEOS24705/2011/Notes/SolarPhysics.pdf|title=University of Chicago GEOS24705 Solar Photovoltaics EJM May 2011}}</ref> However, at a photovoltaic cell's MPP region, its curve has an approximately inverse exponential relationship between current and voltage. From basic circuit theory, the power delivered to a device is optimized (MPP) where the [[derivative]] (graphically, the slope) ''dI/dV'' of the I-V curve is equal and opposite the ''I/V'' ratio (where d''P/dV''=0)<ref>{{cite book |last=Sze |first=Simon M. |author-link=Simon M. Sze |date=1981 |edition=2nd |page=[https://archive.org/details/physicsofsemicon00szes/page/796 796] |title=Physics of Semiconductor Devices|publisher=Wiley |isbn=9780471056614 |url=https://archive.org/details/physicsofsemicon00szes |url-access=registration }}</ref> and corresponds to the "knee" of the curve.

A load with resistance ''R=V/I'' equal to the reciprocal of this value draws the maximum power from the device. This is sometimes called the 'characteristic resistance' of the cell. This is a dynamic quantity that changes depending on the level of illumination, as well as other factors such as temperature and cell condition. Lower or higher resistance reduces power output. Maximum power point trackers utilize control circuits or logic to identify this point.

[[File:Power-voltage (P -V) curve.png|thumb|Power-voltage (P-V) curve]]
If a full power-voltage (P-V) curve is available, then the maximum power point can be obtained using a [[bisection method]].

== Implementation ==
When directly connecting a load to cell, the operating point of the panel is rarely at peak power. The impedance seen by the panel determines its operating point. Setting the impedance correctly achieves peak power. Since panels are DC devices, [[DC-DC converter]]s  transform the impedance of one circuit (source) to the other circuit (load). Changing the duty ratio of the DC-DC converter changes the impedance (duty ratio) seen by the cell. The I-V curve of the panel can be considerably affected by atmospheric conditions such as irradiance and temperature.

MPPT algorithms frequently sample panel voltages and currents, then adjust the duty ratio accordingly.  [[Microcontrollers]] implement the algorithms. Modern implementations often utilize more sophisticated computers for analytics and load forecasting.

== Classification ==
Controllers can follow several strategies to optimize power output. MPPTs may switch among multiple algorithms as conditions dictate.<ref>Rahmani, R.; Seyedmahmoudian, M.;, Mekhilef, S.; Yusof, R.; 2013. Implementation of fuzzy logic maximum power point tracking controller for photovoltaic system. American Journal of Applied Sciences, 10: 209-218.</ref>

===Perturb and observe===
In this method the controller adjusts the voltage from the array by a small amount and measures power; if the power increases, further adjustments in that direction are tried until power no longer increases. This is called ''perturb and observe'' (P&O) and is most common, although this method can cause power output to oscillate.<ref name="zone"/><ref>{{cite web |url=http://sbrn.solarbuildings.ca/c/sbn/file_db/Doc_File_e/Advanced%20algorithm%20for%20MPPT.pdf |title=Advanced Algorithm for MPPT Control of Photovoltaic System |publisher=solarbuildings.ca |access-date=2013-12-19 |url-status=dead |archive-url=https://web.archive.org/web/20131219221448/http://sbrn.solarbuildings.ca/c/sbn/file_db/Doc_File_e/Advanced%20algorithm%20for%20MPPT.pdf |archive-date=2013-12-19}}</ref> It is also referred to as a ''[[hill climbing]]'' method, because it depends on the rise of the curve of power against voltage below the maximum power point, and the fall above that point.<ref name="CS">{{cite journal |doi=10.1002/pip.459 |title=Comparative Study of Maximum Power Point Tracking Algorithms|journal=Progress in Photovoltaics: Research and Applications|volume=11|pages=47–62|year=2003|last1=Hohm|first1=D. P.|last2=Ropp|first2=M. E.|s2cid=10668678|doi-access=free}}</ref> Perturb and observe is the most commonly used method due to its ease of implementation.<ref name="zone">{{cite web |url=http://zone.ni.com/devzone/cda/tut/p/id/8106 |title=Maximum Power Point Tracking |access-date=2011-06-18 |work=zone.ni.com |url-status=dead |archive-url=https://web.archive.org/web/20110416062158/http://zone.ni.com/devzone/cda/tut/p/id/8106 |archive-date=2011-04-16}}</ref> Perturb and observe method may result in top-level efficiency, provided that a proper predictive and adaptive hill climbing strategy is adopted.<ref>{{cite web |url=http://www.actapress.com/Abstract.aspx?paperId=23133 |title=Performances Improvement of Maximum Power Point Tracking Perturb and Observe Method |publisher=actapress.com |access-date=2011-06-18|date=2006-03-09 }}</ref><ref>{{cite journal |author-last1=Zhang |author-first1=Q. |author-first2=C. |author-last2=Hu |author-first3=L. |author-last3=Chen |author-first4=A. |author-last4=Amirahmadi |author-first5=N. |author-last5=Kutkut |author-first6=I. |author-last6=Batarseh |title=A Center Point Iteration MPPT Method With Application on the Frequency-Modulated LLC Microinverter |journal=IEEE Transactions on Power Electronics |date=2014 |volume=29 |issue=3 |pages=1262–1274 |doi=10.1109/tpel.2013.2262806|bibcode=2014ITPE...29.1262Z |s2cid=29377646 }}</ref>

===Incremental conductance===
In this method, the controller measures incremental current and voltage changes to predict the effect of a voltage change. This method requires more computation in the controller, but can track changing conditions more rapidly than P&O. Power output does not oscillate.<ref name="dspac">{{cite web |url=http://itee.uq.edu.au/~aupec/aupec06/htdocs/content/pdf/165.pdf |title=Evaluation of Micro Controller Based Maximum Power Point Tracking Methods Using dSPACE Platform |publisher=itee.uq.edu.au |access-date=2011-06-18 |url-status=dead |archive-url=https://web.archive.org/web/20110726191244/http://itee.uq.edu.au/~aupec/aupec06/htdocs/content/pdf/165.pdf |archive-date=2011-07-26}}</ref> It utilizes the incremental conductance (<math>dI/dV</math>) of the photovoltaic array to compute the sign of the change in power with respect to voltage (<math>dP/dV</math>).<ref name="pe">{{cite web |url=http://powerelectronics.com/power_semiconductors/power_microinverters_computercontrolled_improve_0409/ |title=MPPT algorithms |publisher=powerelectronics.com |access-date=2011-06-10|date=April 2009 }}</ref> The incremental conductance method computes MPP by comparison of the incremental conductance (<math>I_\Delta / V_\Delta</math>) to the array conductance (<math>I / V</math>). When these two are the same (<math>I / V = I_\Delta / V_\Delta</math>), the output voltage is the MPP voltage. The controller maintains this voltage until the irradiation changes and the process is repeated.

The incremental conductance method is based on the observation that at MPP, <math>dP/dV = 0</math>, and that <math>P = IV</math>. The current from the array can be expressed as a function of the voltage: 

<math>P = I(V)V</math>. 

Therefore, <math>dP/dV = VdI/dV + I(V)</math>. Setting this equal to zero yields: <math>dI/dV = -I(V)/V</math>. Therefore, MPP is achieved when the incremental conductance is equal to the negative of the instantaneous conductance. The power-voltage curve characteristic shows that: when the voltage is smaller than MPP, <math>dP/dV > 0 </math>, so <math> dI/dV > -I/V </math>; when the voltage is bigger than MPP, <math>dP/dV < 0 </math> or <math> dI/dV < -I/V </math>. Thus, a tracker can know where it is on the power-voltage curve by calculating the relation of the change of current/voltage and the current voltage themselves.

===Current sweep ===
The ''current sweep'' method uses a sweep waveform for the array current such that the I-V characteristic of the PV array is obtained and updated at fixed time intervals. MPP voltage can then be computed from the characteristic curve at the same intervals.<ref name="refmb1">{{cite journal |last=Esram |first=Trishan |author-first2=P. L. |author-last2=Chapman |title=Comparison of Photovoltaic Array Maximum Power Point Tracking Techniques |journal=IEEE Transactions on Energy Conversion |date=2007 |volume=22 |issue=2|pages=439–449 |doi=10.1109/TEC.2006.874230 |bibcode=2007ITEnC..22..439E |s2cid=31354655 }}</ref><ref name="refmb2">{{cite book |last=Bodur |first=Mehmet |author-first2=M. |author-last2=Ermis |title=Proceedings of MELECON '94. Mediterranean Electrotechnical Conference |chapter=Maximum power point tracking for low power photovoltaic solar panels |date=1994 |pages=758–761|doi=10.1109/MELCON.1994.380992 |isbn=0-7803-1772-6 |s2cid=60529406 }}</ref>

===Constant voltage===
''Constant voltage'' methods include one in which the output voltage is regulated to a constant value under all conditions and one in which the output voltage is regulated based on a constant ratio to the measured open circuit voltage (<math>V_{OC}</math>). The latter technique may also be labeled the "open voltage" method.<ref name="wseas">{{cite web |url=http://www.wseas.us/e-library/transactions/power/2008/27-545.pdf |title=Energy comparison of MPPT techniques for PV Systems |publisher=wseas |access-date=2011-06-18}}</ref> If the output voltage is held constant, there is no attempt to track MPP, so it is not strictly a MPPT technique, though it does function in cases when MPP tracking tends to fail, and thus it is sometimes used supplementally. In the open voltage method, power delivery is momentarily interrupted and the open-circuit voltage with zero current is measured. The controller then resumes operation with the voltage controlled at a fixed ratio, such as 0.76, of the open-circuit voltage <math>V_{OC}</math>.<ref name="ieeexplore.ieee.org">{{cite book |work=ieee.org|doi=10.1109/ICECE.2012.6471698 |title=2012 7th International Conference on Electrical and Computer Engineering |pages=908–911|year=2012 |last1=Ferdous |first1=S.M. |last2=Mohammad|first2=Mahir Asif|last3=Nasrullah|first3=Farhan|last4=Saleque|first4=Ahmed Mortuza|last5=Muttalib|first5=A.Z.M.Shahriar|isbn=978-1-4673-1436-7|s2cid=992906 }}</ref> This is usually a value that has been predetermined to be the MPP, either empirically or based on modelling, for expected operating conditions.<ref name="dspac"/><ref name="pe"/> The array's operating point is thus kept near MPP by regulating the array voltage and matching it to the fixed reference voltage <math>V_{ref}=kV_{OC}</math>. The value of <math>V_{ref}</math> may be chosen to give optimal performance relative to other factors as well as the MPP, but the central idea is that <math>V_{ref}</math> is determined as a ratio to <math>V_{OC}</math>. One of the inherent approximations in the method is that the ratio of MPP voltage to <math>V_{OC}</math> is only approximately constant, so it leaves room for further possible optimization.

=== Temperature method  ===
This method estimates the MPP voltage (<math>V_{mpp}</math>) by measuring the temperature of the solar module and comparing it against a reference.<ref>{{Cite journal|title=A MPPT approach based on temperature measurements applied in PV systems - IEEE Conference Publication|language=en-US|doi=10.1109/ICSET.2010.5684440|s2cid=8653562}}</ref> Since changes in irradiation levels have a negligible effect on the MPP voltage, its influences may be ignored - the voltage is assumed to vary linearly with temperature.

This algorithm calculates the following equation:

:<math>V_{mpp}(T)=V_{mpp}(T_{ref})+u_{V_{mpp}}(T-T_{ref})</math>

Where:

:<math>V_{mpp}</math> is the voltage at the maximum power point for a given temperature;

:<math>T_{ref}</math> is a reference temperature;

:<math>T</math> is the measured temperature;

:<math>u_{V_{mpp}}</math> is the temperature coefficient of <math>V_{mpp}</math> (available in the [[datasheet]]).

==== Advantages ====

* Simplicity: This algorithm solves one linear equation. Therefore, it requires little computation.
* Can be implemented as an analog or digital circuit.
* Since temperature varies slowly with time, oscillation and instability are non-factors.
* Low cost: [[List of temperature sensors|temperature sensors]] are usually cheap.
* Robust against [[Noise (electronics)|noise]].

==== Disadvantages ====

* Estimation error might not be negligible for low irradiation levels (e.g. below 200 W/m<sup>2</sup>).

===Comparison of methods===
Both P&O and incremental conductance are examples of "hill climbing" methods that can find the local maximum of the power curve for the array's operating condition, and so provide a true MPP.<ref name="Seyedmahmoudian 402–414"/><ref name="CS"/><ref name="dspac" />

P&O produces power output oscillations around the maximum power point even under steady state irradiance.

Incremental conductance can determine the maximum power point without oscillating.<ref name="zone"/> It can perform MPPT under rapidly varying irradiation conditions with higher accuracy than P&O.<ref name="zone"/> However, this method can produce oscillations and can perform erratically under rapidly changing atmospheric conditions. The sampling frequency is decreased due to the higher complexity of the algorithm compared to P&O.<ref name="pe"/>

In the constant voltage ratio (or "open voltage") method, energy may be lost during the time the current is set to zero.<ref name="pe"/> The approximation of 76% as the <math>V_{MPP}/V_{OC}</math> ratio is not necessarily accurate.<ref name="pe"/> Although simple and low-cost to implement, the interruptions reduce array efficiency and do not ensure finding the actual MPP. However, efficiencies of some systems may reach above 95%.<ref name="ieeexplore.ieee.org"/>

== Placement ==
Traditional [[solar inverter]]s perform MPPT for the entire array. In such systems the same current, dictated by the inverter, flows through all modules in the string (series). Because different modules have different I-V curves and different MPPs (due to manufacturing tolerance, partial shading,<ref>Seyedmahmoudian, M.; Mekhilef, S.; Rahmani, R.; Yusof, R.; Renani, E.T. Analytical Modeling of Partially Shaded Photovoltaic Systems. Energies 2013, 6, 128-144.</ref> etc.) this architecture means some modules will be performing below their MPP, costing efficiency.<ref name="sciam">{{cite web |url=http://blogs.scientificamerican.com/solar-at-home/invert-your-thinking-squeezing-more-power-out-of-your-solar-panels/ |title=Invert your thinking: Squeezing more power out of your solar panels |publisher=blogs.scientificamerican.com |access-date=2015-05-05}}</ref>

Instead, MPPTs can be deployed for individual modules, allowing each to operate at peak efficiency despite uneven shading, soiling or electrical mismatch.

Data suggest having one inverter with one MPPT for a project that has identical number of east and west-facing modules presents no disadvantages when compared to having two inverters or one inverter with more than one MPPT.<ref>{{cite web |url=http://www.interpv.net/tech/tech_print.asp?idx=418&part_code=030140054 |title=InterPV.net - Global PhotoVoltaic Business Magazine |work=interpv.net}}</ref>

== Battery operation ==
At night, an off-[[Electrical grid|grid]] PV system may use batteries to supply loads. Although the fully charged battery pack voltage may be close to the PV panel's MPP voltage, this is unlikely to be true at sunrise when the battery is partially discharged. Charging may begin at a voltage considerably below the PV panel MPP voltage, and an MPPT can resolve this mismatch.

When the batteries are fully charged and PV production exceeds local loads, an MPPT can no longer operate the panel at its MPP as the excess power has no load to absorb it. The MPPT must then shift the PV panel operating point away from the peak power point until production matches demand. (An alternative approach commonly used in spacecraft is to divert surplus PV power into a resistive load, allowing the panel to operate continuously at its peak power point in order to keep the panel as cool as possible.<ref>{{Cite web|url=https://electronics.stackexchange.com/questions/475063/why-is-it-desired-to-divert-the-surplus-pv-power-into-a-resistive-load|title=solar cell - Why is it desired to divert the surplus PV power into a resistive load?|website=Electrical Engineering Stack Exchange}}</ref>)

In a grid-connected system, all delivered power from solar modules is sent to the grid. Therefore, the MPPT in a grid connected system always attempts to operate at MPP.

==References==
{{Reflist}}

==Further reading==
*{{cite journal |last1=Bialasiewicz |first1=J.T. |title=Renewable Energy Systems With Photovoltaic Power Generators: Operation and Modeling |journal=IEEE Transactions on Industrial Electronics |date=July 2008 |volume=55 |issue=7 |pages=2752–2758 |doi=10.1109/TIE.2008.920583|s2cid=20144161 }}
*{{cite journal |last1=Poponi |first1=Daniele |title=Analysis of diffusion paths for photovoltaic technology based on experience curves |journal=Solar Energy |date=April 2003 |volume=74 |issue=4 |pages=331–340 |doi=10.1016/S0038-092X(03)00151-8|bibcode=2003SoEn...74..331P }}
*{{cite book |editor1-last=Markvart |editor1-first=Tomas |title=Solar electricity |date=July 2000 |publisher=Wiley |isbn=978-0-471-98852-6 |pages=[https://archive.org/details/solarelectricity02edunse/page/298 298] |edition=2nd |url=https://archive.org/details/solarelectricity02edunse/page/298 }}

== External links ==
{{Portal|Renewable energy|Energy}}
{{Commonscatinline|Maximum power point tracker}}
* [http://www.wpi.edu/Pubs/E-project/Available/E-project-042408-181823/unrestricted/PPTReport.doc MPPT tracker by Daniel F. Butay] ([[Microchip PIC]] based)

{{Photovoltaics}}

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