<p>In this Part 1 article of this series of articles, a new methodology to refine the Co-Content function <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="44291_2024_36_Article_IEq2.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="83" /> </InlineMediaObject> <EquationSource Format="TEX">\(\left(CC\left(V,I\right)\right)\)</EquationSource> <EquationSource Format="MATHML"><math> <mfenced close=")" open="("> <mi>C</mi> <mi>C</mi> <mfenced close=")" open="("> <mi>V</mi> <mo>,</mo> <mi>I</mi> </mfenced> </mfenced> </math></EquationSource> </InlineEquation> is proposed, consisting on fitting the current minus the short-circuit current <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="44291_2024_36_Article_IEq3.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="62" /> </InlineMediaObject> <EquationSource Format="TEX">\((I-{I}_{sc})\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mi>I</mi> <mo>-</mo> <msub> <mi>I</mi> <mrow> <mi mathvariant="italic">sc</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation><b>,</b> to an <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="44291_2024_36_Article_IEq4.gif" Format="GIF" Height="15" Rendition="HTML" Resolution="72" Type="Linedraw" Width="46" /> </InlineMediaObject> <EquationSource Format="TEX">\(N-1\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>N</mi> <mo>-</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation> order polynomial, where <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="44291_2024_36_Article_IEq5.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="86" /> </InlineMediaObject> <EquationSource Format="TEX">\({N}_{points}=N\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>N</mi> <mrow> <mi mathvariant="italic">points</mi> </mrow> </msub> <mo>=</mo> <mi>N</mi> </mrow> </math></EquationSource> </InlineEquation>, is the number of measured current–voltage <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="44291_2024_36_Article_IEq6.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="36" /> </InlineMediaObject> <EquationSource Format="TEX">\(\left(IV\right)\)</EquationSource> <EquationSource Format="MATHML"><math> <mfenced close=")" open="("> <mi>I</mi> <mi>V</mi> </mfenced> </math></EquationSource> </InlineEquation> points, and integrating it to calculate <InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="44291_2024_36_Article_IEq7.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="70" /> </InlineMediaObject> <EquationSource Format="TEX">\(CC\left(V,I\right)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>C</mi> <mi>C</mi> <mfenced close=")" open="("> <mi>V</mi> <mo>,</mo> <mi>I</mi> </mfenced> </mrow> </math></EquationSource> </InlineEquation>. The shunt resistance <InlineEquation ID="IEq8"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="44291_2024_36_Article_IEq8.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="40" /> </InlineMediaObject> <EquationSource Format="TEX">\(\left({R}_{sh}\right)\)</EquationSource> <EquationSource Format="MATHML"><math> <mfenced close=")" open="("> <msub> <mi>R</mi> <mrow> <mi mathvariant="italic">sh</mi> </mrow> </msub> </mfenced> </math></EquationSource> </InlineEquation>, the series resistance <InlineEquation ID="IEq9"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="44291_2024_36_Article_IEq9.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="33" /> </InlineMediaObject> <EquationSource Format="TEX">\(\left({R}_{s}\right)\)</EquationSource> <EquationSource Format="MATHML"><math> <mfenced close=")" open="("> <msub> <mi>R</mi> <mi>s</mi> </msub> </mfenced> </math></EquationSource> </InlineEquation>, the ideality factor <InlineEquation ID="IEq10"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="44291_2024_36_Article_IEq10.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="24" /> </InlineMediaObject> <EquationSource Format="TEX">\(\left(n\right)\)</EquationSource> <EquationSource Format="MATHML"><math> <mfenced close=")" open="("> <mi>n</mi> </mfenced> </math></EquationSource> </InlineEquation>, the light current <InlineEquation ID="IEq11"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="44291_2024_36_Article_IEq11.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="36" /> </InlineMediaObject> <EquationSource Format="TEX">\(\left({I}_{lig}\right)\)</EquationSource> <EquationSource Format="MATHML"><math> <mfenced close=")" open="("> <msub> <mi>I</mi> <mrow> <mi mathvariant="italic">lig</mi> </mrow> </msub> </mfenced> </math></EquationSource> </InlineEquation>, and the saturation current <InlineEquation ID="IEq12"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="44291_2024_36_Article_IEq12.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="38" /> </InlineMediaObject> <EquationSource Format="TEX">\(\left({I}_{sat}\right)\)</EquationSource> <EquationSource Format="MATHML"><math> <mfenced close=")" open="("> <msub> <mi>I</mi> <mrow> <mi mathvariant="italic">sat</mi> </mrow> </msub> </mfenced> </math></EquationSource> </InlineEquation>, are then deduced, in the case of a constant percentage noise or a percentage noise of the maximum current <InlineEquation ID="IEq13"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="44291_2024_36_Article_IEq13.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="45" /> </InlineMediaObject> <EquationSource Format="TEX">\(\left({I}_{max}\right)\)</EquationSource> <EquationSource Format="MATHML"><math> <mfenced close=")" open="("> <msub> <mi>I</mi> <mrow> <mi mathvariant="italic">max</mi> </mrow> </msub> </mfenced> </math></EquationSource> </InlineEquation>. In the former case, <InlineEquation ID="IEq14"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="44291_2024_36_Article_IEq14.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="21" /> </InlineMediaObject> <EquationSource Format="TEX">\({R}_{s}\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>R</mi> <mi>s</mi> </msub> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq15"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="44291_2024_36_Article_IEq15.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="111" /> </InlineMediaObject> <EquationSource Format="TEX">\({R}_{sh}, n, \text{and } {I}_{lig},\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>R</mi> <mrow> <mi mathvariant="italic">sh</mi> </mrow> </msub> <mo>,</mo> <mi>n</mi> <mo>,</mo> <mtext>and</mtext> <mspace width="0.333333em" /> <msub> <mi>I</mi> <mrow> <mi mathvariant="italic">lig</mi> </mrow> </msub> <mo>,</mo> </mrow> </math></EquationSource> </InlineEquation> can be deduced with less than 10% error, using only <InlineEquation ID="IEq16"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="44291_2024_36_Article_IEq16.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="40" /> </InlineMediaObject> <EquationSource Format="TEX">\({P}_{V}=\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>P</mi> <mi>V</mi> </msub> <mo>=</mo> </mrow> </math></EquationSource> </InlineEquation> 51 <InlineEquation ID="IEq17"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="44291_2024_36_Article_IEq17.gif" Format="GIF" Height="24" Rendition="HTML" Resolution="72" Type="Linedraw" Width="92" /> </InlineMediaObject> <EquationSource Format="TEX">\(\frac{number\, of \,points}{V}\)</EquationSource> <EquationSource Format="MATHML"><math> <mfrac> <mrow> <mi>n</mi> <mi>u</mi> <mi>m</mi> <mi>b</mi> <mi>e</mi> <mi>r</mi> <mspace width="0.166667em" /> <mi>o</mi> <mi>f</mi> <mspace width="0.166667em" /> <mi>p</mi> <mi>o</mi> <mi>i</mi> <mi>n</mi> <mi>t</mi> <mi>s</mi> </mrow> <mi>V</mi> </mfrac> </math></EquationSource> </InlineEquation>, even if the noise is as large as <InlineEquation ID="IEq18"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="44291_2024_36_Article_IEq18.gif" Format="GIF" Height="18" Rendition="HTML" Resolution="72" Type="Linedraw" Width="75" /> </InlineMediaObject> <EquationSource Format="TEX">\({p}_{n}=0.1\text{\%}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>p</mi> <mi>n</mi> </msub> <mo>=</mo> <mn>0.1</mn> <mtext>\%</mtext> </mrow> </math></EquationSource> </InlineEquation>, with a computation time around 80&#xa0;ms. <InlineEquation ID="IEq19"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="44291_2024_36_Article_IEq19.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="27" /> </InlineMediaObject> <EquationSource Format="TEX">\({I}_{sat}\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>I</mi> <mrow> <mi mathvariant="italic">sat</mi> </mrow> </msub> </math></EquationSource> </InlineEquation> needs <InlineEquation ID="IEq20"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="44291_2024_36_Article_IEq20.gif" Format="GIF" Height="18" Rendition="HTML" Resolution="72" Type="Linedraw" Width="82" /> </InlineMediaObject> <EquationSource Format="TEX">\({p}_{n}=0.05\text{\%}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>p</mi> <mi>n</mi> </msub> <mo>=</mo> <mn>0.05</mn> <mtext>\%</mtext> </mrow> </math></EquationSource> </InlineEquation> or less, and <InlineEquation ID="IEq21"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="44291_2024_36_Article_IEq21.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="21" /> </InlineMediaObject> <EquationSource Format="TEX">\({P}_{V}\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>P</mi> <mi>V</mi> </msub> </math></EquationSource> </InlineEquation> equal or larger than 501 <InlineEquation ID="IEq22"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="44291_2024_36_Article_IEq22.gif" Format="GIF" Height="24" Rendition="HTML" Resolution="72" Type="Linedraw" Width="92" /> </InlineMediaObject> <EquationSource Format="TEX">\(\frac{number\, of\, points}{V}\)</EquationSource> <EquationSource Format="MATHML"><math> <mfrac> <mrow> <mi>n</mi> <mi>u</mi> <mi>m</mi> <mi>b</mi> <mi>e</mi> <mi>r</mi> <mspace width="0.166667em" /> <mi>o</mi> <mi>f</mi> <mspace width="0.166667em" /> <mi>p</mi> <mi>o</mi> <mi>i</mi> <mi>n</mi> <mi>t</mi> <mi>s</mi> </mrow> <mi>V</mi> </mfrac> </math></EquationSource> </InlineEquation>. For the latter case, <InlineEquation ID="IEq23"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="44291_2024_36_Article_IEq23.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="21" /> </InlineMediaObject> <EquationSource Format="TEX">\({R}_{s}\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>R</mi> <mi>s</mi> </msub> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq24"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="44291_2024_36_Article_IEq24.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="60" /> </InlineMediaObject> <EquationSource Format="TEX">\(\text{and } {I}_{lig},\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mtext>and</mtext> <mspace width="0.333333em" /> <msub> <mi>I</mi> <mrow> <mi mathvariant="italic">lig</mi> </mrow> </msub> <mo>,</mo> </mrow> </math></EquationSource> </InlineEquation> can be obtained with less than 10% error, using only <InlineEquation ID="IEq25"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="44291_2024_36_Article_IEq25.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="40" /> </InlineMediaObject> <EquationSource Format="TEX">\({P}_{V}=\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>P</mi> <mi>V</mi> </msub> <mo>=</mo> </mrow> </math></EquationSource> </InlineEquation> 251 <InlineEquation ID="IEq26"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="44291_2024_36_Article_IEq26.gif" Format="GIF" Height="24" Rendition="HTML" Resolution="72" Type="Linedraw" Width="92" /> </InlineMediaObject> <EquationSource Format="TEX">\(\frac{number\, of\, points}{V}\)</EquationSource> <EquationSource Format="MATHML"><math> <mfrac> <mrow> <mi>n</mi> <mi>u</mi> <mi>m</mi> <mi>b</mi> <mi>e</mi> <mi>r</mi> <mspace width="0.166667em" /> <mi>o</mi> <mi>f</mi> <mspace width="0.166667em" /> <mi>p</mi> <mi>o</mi> <mi>i</mi> <mi>n</mi> <mi>t</mi> <mi>s</mi> </mrow> <mi>V</mi> </mfrac> </math></EquationSource> </InlineEquation>, and <InlineEquation ID="IEq27"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="44291_2024_36_Article_IEq27.gif" Format="GIF" Height="18" Rendition="HTML" Resolution="72" Type="Linedraw" Width="75" /> </InlineMediaObject> <EquationSource Format="TEX">\({p}_{n}=0.1\text{\%}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>p</mi> <mi>n</mi> </msub> <mo>=</mo> <mn>0.1</mn> <mtext>\%</mtext> </mrow> </math></EquationSource> </InlineEquation>, or smaller, with total computation time around 49&#xa0;s. <InlineEquation ID="IEq28"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="44291_2024_36_Article_IEq28.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="110" /> </InlineMediaObject> <EquationSource Format="TEX">\({R}_{sh}, {I}_{sat}, \text{and } n\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>R</mi> <mrow> <mi mathvariant="italic">sh</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>I</mi> <mrow> <mi mathvariant="italic">sat</mi> </mrow> </msub> <mo>,</mo> <mtext>and</mtext> <mspace width="0.333333em" /> <mi>n</mi> </mrow> </math></EquationSource> </InlineEquation> needs that <InlineEquation ID="IEq29"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="44291_2024_36_Article_IEq29.gif" Format="GIF" Height="18" Rendition="HTML" Resolution="72" Type="Linedraw" Width="83" /> </InlineMediaObject> <EquationSource Format="TEX">\({p}_{n}\le 0.05\text{\%}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>p</mi> <mi>n</mi> </msub> <mo>≤</mo> <mn>0.05</mn> <mtext>\%</mtext> </mrow> </math></EquationSource> </InlineEquation>, and <InlineEquation ID="IEq30"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="44291_2024_36_Article_IEq30.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="40" /> </InlineMediaObject> <EquationSource Format="TEX">\({P}_{V}=\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>P</mi> <mi>V</mi> </msub> <mo>=</mo> </mrow> </math></EquationSource> </InlineEquation> 751 <InlineEquation ID="IEq31"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="44291_2024_36_Article_IEq31.gif" Format="GIF" Height="24" Rendition="HTML" Resolution="72" Type="Linedraw" Width="92" /> </InlineMediaObject> <EquationSource Format="TEX">\(\frac{number\, of \,points}{V}\)</EquationSource> <EquationSource Format="MATHML"><math> <mfrac> <mrow> <mi>n</mi> <mi>u</mi> <mi>m</mi> <mi>b</mi> <mi>e</mi> <mi>r</mi> <mspace width="0.166667em" /> <mi>o</mi> <mi>f</mi> <mspace width="0.166667em" /> <mi>p</mi> <mi>o</mi> <mi>i</mi> <mi>n</mi> <mi>t</mi> <mi>s</mi> </mrow> <mi>V</mi> </mfrac> </math></EquationSource> </InlineEquation> or larger. A computation time expression of the form <InlineEquation ID="IEq32"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="44291_2024_36_Article_IEq32.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="129" /> </InlineMediaObject> <EquationSource Format="TEX">\(time=E{{N}_{points}}^{m}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>t</mi> <mi>i</mi> <mi>m</mi> <mi>e</mi> <mo>=</mo> <mi>E</mi> <msup> <mrow> <msub> <mi>N</mi> <mrow> <mi mathvariant="italic">points</mi> </mrow> </msub> </mrow> <mi>m</mi> </msup> </mrow> </math></EquationSource> </InlineEquation>, is deduced. The methodology proposed in this article is appliable to unevenly/randomly distributed <i>IV</i> data points, and it is implemented in Part 2 in solar cells’ and photovoltaic modules’ experimental <InlineEquation ID="IEq33"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="44291_2024_36_Article_IEq33.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="23" /> </InlineMediaObject> <EquationSource Format="TEX">\(IV\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="italic">IV</mi> </mrow> </math></EquationSource> </InlineEquation> reported in the literature, to deduce their five solar cell parameters.</p>

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Refinement of the co-content function, through an integration of a polynomial fit of \(I-{I}_{sc}\). Part 1 theoretical analysis and proposal

  • Victor-Tapio Rangel-Kuoppa

摘要

In this Part 1 article of this series of articles, a new methodology to refine the Co-Content function \(\left(CC\left(V,I\right)\right)\) C C V , I is proposed, consisting on fitting the current minus the short-circuit current \((I-{I}_{sc})\) ( I - I sc ) , to an \(N-1\) N - 1 order polynomial, where \({N}_{points}=N\) N points = N , is the number of measured current–voltage \(\left(IV\right)\) I V points, and integrating it to calculate \(CC\left(V,I\right)\) C C V , I . The shunt resistance \(\left({R}_{sh}\right)\) R sh , the series resistance \(\left({R}_{s}\right)\) R s , the ideality factor \(\left(n\right)\) n , the light current \(\left({I}_{lig}\right)\) I lig , and the saturation current \(\left({I}_{sat}\right)\) I sat , are then deduced, in the case of a constant percentage noise or a percentage noise of the maximum current \(\left({I}_{max}\right)\) I max . In the former case, \({R}_{s}\) R s , \({R}_{sh}, n, \text{and } {I}_{lig},\) R sh , n , and I lig , can be deduced with less than 10% error, using only \({P}_{V}=\) P V = 51 \(\frac{number\, of \,points}{V}\) n u m b e r o f p o i n t s V , even if the noise is as large as \({p}_{n}=0.1\text{\%}\) p n = 0.1 \% , with a computation time around 80 ms. \({I}_{sat}\) I sat needs \({p}_{n}=0.05\text{\%}\) p n = 0.05 \% or less, and \({P}_{V}\) P V equal or larger than 501 \(\frac{number\, of\, points}{V}\) n u m b e r o f p o i n t s V . For the latter case, \({R}_{s}\) R s , \(\text{and } {I}_{lig},\) and I lig , can be obtained with less than 10% error, using only \({P}_{V}=\) P V = 251 \(\frac{number\, of\, points}{V}\) n u m b e r o f p o i n t s V , and \({p}_{n}=0.1\text{\%}\) p n = 0.1 \% , or smaller, with total computation time around 49 s. \({R}_{sh}, {I}_{sat}, \text{and } n\) R sh , I sat , and n needs that \({p}_{n}\le 0.05\text{\%}\) p n 0.05 \% , and \({P}_{V}=\) P V = 751 \(\frac{number\, of \,points}{V}\) n u m b e r o f p o i n t s V or larger. A computation time expression of the form \(time=E{{N}_{points}}^{m}\) t i m e = E N points m , is deduced. The methodology proposed in this article is appliable to unevenly/randomly distributed IV data points, and it is implemented in Part 2 in solar cells’ and photovoltaic modules’ experimental \(IV\) IV reported in the literature, to deduce their five solar cell parameters.