From 5525af2a1bced5efed371d91a8e01716300987b5 Mon Sep 17 00:00:00 2001 From: Mark Mikofski Date: Tue, 16 Jul 2024 00:24:33 -0700 Subject: [PATCH 1/8] Create Mismatch_from_cross-string_shade.md Initial draft --- Mismatch_from_cross-string_shade.md | 8 ++++++++ 1 file changed, 8 insertions(+) create mode 100644 Mismatch_from_cross-string_shade.md diff --git a/Mismatch_from_cross-string_shade.md b/Mismatch_from_cross-string_shade.md new file mode 100644 index 0000000..361720b --- /dev/null +++ b/Mismatch_from_cross-string_shade.md @@ -0,0 +1,8 @@ +Affects from shade are compiled, but can be summarized in two orthogonal categories: + +1. shade across all modules parallel to rows +2. shade across few modules perpendicular to rows + +## uniform shade +row-to-row shade on horizontal ground is an example of shade across all modules that +is parallel to rows. i covered this in another blog post From 8c03cd68d23641e972a851d44cbd658d3487f63c Mon Sep 17 00:00:00 2001 From: Mark Mikofski Date: Thu, 22 Aug 2024 00:35:56 -0700 Subject: [PATCH 2/8] add content to cross-string mismatch --- Mismatch_from_cross-string_shade.md | 8 ---- content/Mismatch_from_cross-string_shade.md | 46 +++++++++++++++++++++ 2 files changed, 46 insertions(+), 8 deletions(-) delete mode 100644 Mismatch_from_cross-string_shade.md create mode 100644 content/Mismatch_from_cross-string_shade.md diff --git a/Mismatch_from_cross-string_shade.md b/Mismatch_from_cross-string_shade.md deleted file mode 100644 index 361720b..0000000 --- a/Mismatch_from_cross-string_shade.md +++ /dev/null @@ -1,8 +0,0 @@ -Affects from shade are compiled, but can be summarized in two orthogonal categories: - -1. shade across all modules parallel to rows -2. shade across few modules perpendicular to rows - -## uniform shade -row-to-row shade on horizontal ground is an example of shade across all modules that -is parallel to rows. i covered this in another blog post diff --git a/content/Mismatch_from_cross-string_shade.md b/content/Mismatch_from_cross-string_shade.md new file mode 100644 index 0000000..3ed7477 --- /dev/null +++ b/content/Mismatch_from_cross-string_shade.md @@ -0,0 +1,46 @@ +Effects from shade are complicated, but can be summarized in two orthogonal categories: + +1. shade across all modules parallel to strings +2. shade across few modules perpendicular to strings + +## shade parallel to rows +Row-to-row shade in fixed-tilt systems, typically in winter, is an example of +shade across all modules that is parallel to strings. When I originally wrote +about [PV electrical mismatch]]({filename}PV-electrical-mismatch.md), I analyzed +this type of shade using [PVMismatch](https://sunpower.github.io/PVMismatch/) +to simulate shade across the bottom row of a single string of 10 modules in a +10 string system. The conclusion of that post was that the string performed as +well as the most shaded cell, so even though only the bottom cells were shaded, +the modules in the string lost most of their power. I shaded the bottom cells +80% to simulate only diffuse light, and the string lost roughly 80% of output. +The other 9 strings operated at full capacity, so the system only lost 8%. The +[NIST ground mount array](https://www.nist.gov/el/energy-and-environment-division-73200/heat-transfer-alternative-energy-systems/photovoltaic-1) +is an example of a system that will have row-to-row shade in winter that will +cause most of the strings to lose almost all of their output even when only +the bottom row of cells is shaded. + +![NIST Google](./images/nist-ground-array.png) + +This topic is also covered in + +## shade not parallel to rows +However, that post also contained a picture of a rooftop with non-uniform shade +that was not consistent across each module of the string. The shade cast from +the roofline cut diagonally across the modules in the string, which was wrapped +in two rows to fit. + +![non uniform shade on a roof](./images/20150923_170418.jpg) + +I didn't analyze the shade from this system in that post, so it raises the +question whether the rule of thumb I recommended would still apply? + +## shade perpendicular to rows +To simplify the question, the rest of this post analyzes a PV system with a +shade obstacle like a wind turbine tower or a telephone pole that casts shade +perpendicular to the rows. + +>TL;DR: When shade cuts perpendicular to strings, cells go into reverse bias, +bypass diodes activate in the shaded submodules, and the other modules operate +at higher voltage to match the voltage of unshaded parallel strings. + +01234567890123456789012345678901234567890123456789012345678901234567890123456789 \ No newline at end of file From 0995614e51f099493f08ee2d751086b40480776d Mon Sep 17 00:00:00 2001 From: Mark Mikofski Date: Fri, 23 Aug 2024 01:27:04 -0700 Subject: [PATCH 3/8] add front matter and references to cross-string mismatch --- content/Mismatch_from_cross-string_shade.md | 17 ++++++++++++++++- 1 file changed, 16 insertions(+), 1 deletion(-) diff --git a/content/Mismatch_from_cross-string_shade.md b/content/Mismatch_from_cross-string_shade.md index 3ed7477..ea1fa8b 100644 --- a/content/Mismatch_from_cross-string_shade.md +++ b/content/Mismatch_from_cross-string_shade.md @@ -1,8 +1,20 @@ +Title: PV Electric Mismatch in Silicon-Cell PV - Part 2 +Date: 2024-08-23 01:10 +Category: Solar +Tags: solar, modeling, code +Authors: Mark Mikofski +Summary: What happens when shade cuts across PV strings? + +# The many shades of PV electrical mismatch Effects from shade are complicated, but can be summarized in two orthogonal categories: 1. shade across all modules parallel to strings 2. shade across few modules perpendicular to strings +These categories were defined in the Fast Shade Model [1, 2] developed by Dr. +Bennet Meyers after simulating hundreds of different shade patterns and grouping +them by their electrical mismatch. + ## shade parallel to rows Row-to-row shade in fixed-tilt systems, typically in winter, is an example of shade across all modules that is parallel to strings. When I originally wrote @@ -43,4 +55,7 @@ perpendicular to the rows. bypass diodes activate in the shaded submodules, and the other modules operate at higher voltage to match the voltage of unshaded parallel strings. -01234567890123456789012345678901234567890123456789012345678901234567890123456789 \ No newline at end of file +## References + +1. Meyers, B., Mikofski, M. A., & Anderson, M. (2016). A Fast Parameterized Model for Predicting PV System Performance under Partial Shade Conditions. In IEEE (Ed.), 2016 IEEE 43rd Photovoltaic Specialists Conference (PVSC) (pp. 3173–3178). IEEE. https://doi.org/10.1109/PVSC.2016.7750251 +2. Meyers, B., & Mikofski, M. A. (2017). Accurate Modeling of Partially Shaded PV Arrays. In IEEE (Ed.), 2017 IEEE 44th Photovoltaic Specialist Conference (PVSC) (Vols. 2017-June, pp. 3354–3359). IEEE. https://doi.org/10.1109/PVSC.2017.8521559 From c446bc57d7350362338606554e7d03fae09c7371 Mon Sep 17 00:00:00 2001 From: Mark Mikofski Date: Fri, 23 Aug 2024 01:52:42 -0700 Subject: [PATCH 4/8] link to references in cross-string mismatch --- content/Mismatch_from_cross-string_shade.md | 10 +++++----- 1 file changed, 5 insertions(+), 5 deletions(-) diff --git a/content/Mismatch_from_cross-string_shade.md b/content/Mismatch_from_cross-string_shade.md index ea1fa8b..2af3ce5 100644 --- a/content/Mismatch_from_cross-string_shade.md +++ b/content/Mismatch_from_cross-string_shade.md @@ -11,9 +11,9 @@ Effects from shade are complicated, but can be summarized in two orthogonal cate 1. shade across all modules parallel to strings 2. shade across few modules perpendicular to strings -These categories were defined in the Fast Shade Model [1, 2] developed by Dr. -Bennet Meyers after simulating hundreds of different shade patterns and grouping -them by their electrical mismatch. +These categories were defined in the Fast Shade Model [[1, 2](#references)] +developed by Dr. Bennet Meyers after simulating hundreds of different shade +patterns and grouping them by their electrical mismatch. ## shade parallel to rows Row-to-row shade in fixed-tilt systems, typically in winter, is an example of @@ -57,5 +57,5 @@ at higher voltage to match the voltage of unshaded parallel strings. ## References -1. Meyers, B., Mikofski, M. A., & Anderson, M. (2016). A Fast Parameterized Model for Predicting PV System Performance under Partial Shade Conditions. In IEEE (Ed.), 2016 IEEE 43rd Photovoltaic Specialists Conference (PVSC) (pp. 3173–3178). IEEE. https://doi.org/10.1109/PVSC.2016.7750251 -2. Meyers, B., & Mikofski, M. A. (2017). Accurate Modeling of Partially Shaded PV Arrays. In IEEE (Ed.), 2017 IEEE 44th Photovoltaic Specialist Conference (PVSC) (Vols. 2017-June, pp. 3354–3359). IEEE. https://doi.org/10.1109/PVSC.2017.8521559 +1. Meyers, B., Mikofski, M. A., & Anderson, M. (2016). A Fast Parameterized Model for Predicting PV System Performance under Partial Shade Conditions. In IEEE (Ed.), 2016 IEEE 43rd Photovoltaic Specialists Conference (PVSC) (pp. 3173–3178). IEEE. [https://doi.org/10.1109/PVSC.2016.7750251](https://doi.org/10.1109/PVSC.2016.7750251) +2. Meyers, B., & Mikofski, M. A. (2017). Accurate Modeling of Partially Shaded PV Arrays. In IEEE (Ed.), 2017 IEEE 44th Photovoltaic Specialist Conference (PVSC) (Vols. 2017-June, pp. 3354–3359). IEEE. [https://doi.org/10.1109/PVSC.2017.8521559](https://doi.org/10.1109/PVSC.2017.8521559) \ No newline at end of file From ed07ba19a32d8aaf8d2d771e815a61bbb10771cf Mon Sep 17 00:00:00 2001 From: Mark Mikofski Date: Fri, 23 Aug 2024 03:18:50 -0700 Subject: [PATCH 5/8] finish cross-string mismatch blog --- content/Mismatch_from_cross-string_shade.md | 68 ++++++++++++++---- .../pvmod0-20strings.png | Bin 0 -> 39113 bytes .../pvmod1-20strings.png | Bin 0 -> 41071 bytes .../pvstring-20strings.png | Bin 0 -> 41139 bytes .../pvsystem-20strings.png | Bin 0 -> 45938 bytes 5 files changed, 56 insertions(+), 12 deletions(-) create mode 100644 content/images/cross-string-mismatch/pvmod0-20strings.png create mode 100644 content/images/cross-string-mismatch/pvmod1-20strings.png create mode 100644 content/images/cross-string-mismatch/pvstring-20strings.png create mode 100644 content/images/cross-string-mismatch/pvsystem-20strings.png diff --git a/content/Mismatch_from_cross-string_shade.md b/content/Mismatch_from_cross-string_shade.md index 2af3ce5..17dbf0e 100644 --- a/content/Mismatch_from_cross-string_shade.md +++ b/content/Mismatch_from_cross-string_shade.md @@ -8,17 +8,17 @@ Summary: What happens when shade cuts across PV strings? # The many shades of PV electrical mismatch Effects from shade are complicated, but can be summarized in two orthogonal categories: -1. shade across all modules parallel to strings -2. shade across few modules perpendicular to strings +1. [shade parallel to strings](#shade-parallel-to-strings) +2. [shade perpendicular to strings](#shade-perpendicular-to-strings) These categories were defined in the Fast Shade Model [[1, 2](#references)] developed by Dr. Bennet Meyers after simulating hundreds of different shade patterns and grouping them by their electrical mismatch. -## shade parallel to rows -Row-to-row shade in fixed-tilt systems, typically in winter, is an example of -shade across all modules that is parallel to strings. When I originally wrote -about [PV electrical mismatch]]({filename}PV-electrical-mismatch.md), I analyzed +## shade parallel to strings +One example of shade across all modules that is parallel to strings, is row-to-row +shade in fixed-tilt systems, typically in winter. When I originally wrote +about [PV electrical mismatch]({filename}PV-electrical-mismatch.md), I analyzed this type of shade using [PVMismatch](https://sunpower.github.io/PVMismatch/) to simulate shade across the bottom row of a single string of 10 modules in a 10 string system. The conclusion of that post was that the string performed as @@ -33,9 +33,7 @@ the bottom row of cells is shaded. ![NIST Google](./images/nist-ground-array.png) -This topic is also covered in - -## shade not parallel to rows +## shade not parallel to strings However, that post also contained a picture of a rooftop with non-uniform shade that was not consistent across each module of the string. The shade cast from the roofline cut diagonally across the modules in the string, which was wrapped @@ -46,15 +44,61 @@ in two rows to fit. I didn't analyze the shade from this system in that post, so it raises the question whether the rule of thumb I recommended would still apply? -## shade perpendicular to rows +## shade perpendicular to strings To simplify the question, the rest of this post analyzes a PV system with a -shade obstacle like a wind turbine tower or a telephone pole that casts shade -perpendicular to the rows. +shade obstacle like a wind turbine tower, a telephone pole, or a chimney, that casts shade +perpendicular to the strings. My analysis is in this Jupyter notebook on Google Colaboratory: +[mismatch_vs_strings.ipynb](https://colab.research.google.com/drive/1b2Ll7G-4WBKPl57m-FPBhU8MLjLOTfIb) >TL;DR: When shade cuts perpendicular to strings, cells go into reverse bias, bypass diodes activate in the shaded submodules, and the other modules operate at higher voltage to match the voltage of unshaded parallel strings. +I simulated perpendicular shade on the first half of the first module in the +string, while the rest of the strings were unaffected. This could be like a +chimney. Then I increased the number of unaffected strings to see if it changed +the effect. The effect of a shadow perpendicular to the string caused bypass +diodes to trigger, but even after 20 strings, the IV curve of the system +appears unaffected and the total power loss is only 0.85% for the system +compared to unshaded. + +Here is the IV curve of the 20 string PV system with perpendicular shade on +1st module of 1st string from the Jupyter notebook. It looks unaffected! + +![20 string PV system with cross-string shade](./images/cross-string-mismatch/pvsystem-20strings.png) + +Now check out the IV curve of the string with the shaded module. It should +be generating about 3200[W], but even though it's lost about 500[W], it +still operates at 5[A], nearly the same current as the others strings. It +still has to operate at the same voltage as the other strings, 538.7[V] in +this example, so how does it do it with 2 bypass diodes activated? + +![20 string PV system with cross-string shade](./images/cross-string-mismatch/pvstring-20strings.png) + +A look at the module IV curves tells the rest of the story. The shaded module +still has to carry the 5[A] of the string, but 2 bypass diodes are triggered +so the voltage is down 75%. Note: these are SunPower/Maxeon 310[W] modules, +that have 96-cells in 8-columns with 3 bypass diodes in a 24-48-24 cell arrangement. + +![20 string PV system with cross-string shade](./images/cross-string-mismatch/pvmod0-20strings.png) + +However, the unshaded modules make up for the lost voltage in the shaded +module by operating just above the max power point. This is why the string +is operating at 5[A], to increase the voltage in the unshaded strings. +Very clever! Go team! Luckily for the shaded module, that current is +also very close to its max power point, which is only down 75% thanks +to the activated bypass diodes. Recall in the parallel shade scenario, +then entire string was down. + +![20 string PV system with cross-string shade](./images/cross-string-mismatch/pvmod1-20strings.png) + +## Conclusion +I wish I could say, "that's all there is to it." But as my first blog post +says, electrical mismatch in crystalline silicon is very counter-intuitive. +That's why I created PVMismatch to begin with. I was tired of guessing and +being wrong. So don't guess. Simulate with confidence, try PVMismatch, and +let me know what you learn! + ## References 1. Meyers, B., Mikofski, M. A., & Anderson, M. (2016). A Fast Parameterized Model for Predicting PV System Performance under Partial Shade Conditions. In IEEE (Ed.), 2016 IEEE 43rd Photovoltaic Specialists Conference (PVSC) (pp. 3173–3178). IEEE. 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zJ9pyjwuo>r;C`=4I)s}1ZH3@!LIcd5dkx7?R6 J%69YJ{cqP&-BbVo literal 0 HcmV?d00001 From 1401e2df8ed6368f79d658fe903932a18b579d9f Mon Sep 17 00:00:00 2001 From: Mark Mikofski Date: Fri, 23 Aug 2024 03:24:47 -0700 Subject: [PATCH 6/8] fix typo --- content/Mismatch_from_cross-string_shade.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/content/Mismatch_from_cross-string_shade.md b/content/Mismatch_from_cross-string_shade.md index 17dbf0e..90e8971 100644 --- a/content/Mismatch_from_cross-string_shade.md +++ b/content/Mismatch_from_cross-string_shade.md @@ -88,7 +88,7 @@ is operating at 5[A], to increase the voltage in the unshaded strings. Very clever! Go team! Luckily for the shaded module, that current is also very close to its max power point, which is only down 75% thanks to the activated bypass diodes. Recall in the parallel shade scenario, -then entire string was down. +the entire string was down. ![20 string PV system with cross-string shade](./images/cross-string-mismatch/pvmod1-20strings.png) From aa2c744026f198327d0bce8f49117100525ea395 Mon Sep 17 00:00:00 2001 From: Mark Mikofski Date: Fri, 23 Aug 2024 03:42:36 -0700 Subject: [PATCH 7/8] add note about other variations --- content/Mismatch_from_cross-string_shade.md | 11 +++++++++-- 1 file changed, 9 insertions(+), 2 deletions(-) diff --git a/content/Mismatch_from_cross-string_shade.md b/content/Mismatch_from_cross-string_shade.md index 90e8971..092fdb6 100644 --- a/content/Mismatch_from_cross-string_shade.md +++ b/content/Mismatch_from_cross-string_shade.md @@ -48,7 +48,7 @@ question whether the rule of thumb I recommended would still apply? To simplify the question, the rest of this post analyzes a PV system with a shade obstacle like a wind turbine tower, a telephone pole, or a chimney, that casts shade perpendicular to the strings. My analysis is in this Jupyter notebook on Google Colaboratory: -[mismatch_vs_strings.ipynb](https://colab.research.google.com/drive/1b2Ll7G-4WBKPl57m-FPBhU8MLjLOTfIb) +[`mismatch_vs_strings.ipynb`](https://colab.research.google.com/drive/1b2Ll7G-4WBKPl57m-FPBhU8MLjLOTfIb) >TL;DR: When shade cuts perpendicular to strings, cells go into reverse bias, bypass diodes activate in the shaded submodules, and the other modules operate @@ -77,7 +77,7 @@ this example, so how does it do it with 2 bypass diodes activated? A look at the module IV curves tells the rest of the story. The shaded module still has to carry the 5[A] of the string, but 2 bypass diodes are triggered -so the voltage is down 75%. Note: these are SunPower/Maxeon 310[W] modules, +so the voltage is down 75%. Note: these are SunPower/Maxeon 320[W] modules, that have 96-cells in 8-columns with 3 bypass diodes in a 24-48-24 cell arrangement. ![20 string PV system with cross-string shade](./images/cross-string-mismatch/pvmod0-20strings.png) @@ -92,6 +92,13 @@ the entire string was down. ![20 string PV system with cross-string shade](./images/cross-string-mismatch/pvmod1-20strings.png) +Please check out [`mismatch_vs_strings.ipynb`](https://colab.research.google.com/drive/1b2Ll7G-4WBKPl57m-FPBhU8MLjLOTfIb) +because the 1-string example isn't limited by the voltage of parallel strings, so it's free to operate at the max power +point, and the losses are lower. Recall in the 20-string example, the shaded string lost about 15%, but in the 1-string +example, the loss was only about 8%. Another variation is allowing the shade to cross two or more strings. I covered +a scattershot of scenarios and found that parallel string voltage began to dominate sowmhere between 5 to 10 strings. +Of course, that only applies in this contrived example, but it was interesting nonetheless. + ## Conclusion I wish I could say, "that's all there is to it." But as my first blog post says, electrical mismatch in crystalline silicon is very counter-intuitive. From 913aba6cc0db7b45e48a25c93f24c3e91caf88a0 Mon Sep 17 00:00:00 2001 From: Mark Mikofski Date: Mon, 26 Aug 2024 01:24:28 -0700 Subject: [PATCH 8/8] Add link to nonuniform-rooftop-shade notebook - add link to Kurt's blog - fix some awkward language around adding more strings --- content/Mismatch_from_cross-string_shade.md | 35 ++++++++++++++------- 1 file changed, 24 insertions(+), 11 deletions(-) diff --git a/content/Mismatch_from_cross-string_shade.md b/content/Mismatch_from_cross-string_shade.md index 092fdb6..40ae44e 100644 --- a/content/Mismatch_from_cross-string_shade.md +++ b/content/Mismatch_from_cross-string_shade.md @@ -5,6 +5,10 @@ Tags: solar, modeling, code Authors: Mark Mikofski Summary: What happens when shade cuts across PV strings? +> Note: This post is part of a joint blog with my colleague Kurt Rhee: +[Primer on Electrical Mismatch](https://kurt-rhee.github.io/2024/04/15/a-primer-on-electrical-mismatch) + + # The many shades of PV electrical mismatch Effects from shade are complicated, but can be summarized in two orthogonal categories: @@ -46,7 +50,7 @@ question whether the rule of thumb I recommended would still apply? ## shade perpendicular to strings To simplify the question, the rest of this post analyzes a PV system with a -shade obstacle like a wind turbine tower, a telephone pole, or a chimney, that casts shade +shade obstacle like a wind turbine, a telephone pole, or a chimney, that casts shade perpendicular to the strings. My analysis is in this Jupyter notebook on Google Colaboratory: [`mismatch_vs_strings.ipynb`](https://colab.research.google.com/drive/1b2Ll7G-4WBKPl57m-FPBhU8MLjLOTfIb) @@ -55,21 +59,22 @@ bypass diodes activate in the shaded submodules, and the other modules operate at higher voltage to match the voltage of unshaded parallel strings. I simulated perpendicular shade on the first half of the first module in the -string, while the rest of the strings were unaffected. This could be like a -chimney. Then I increased the number of unaffected strings to see if it changed +string, while the rest of the strings were unshaded. For example, this shade could be caused by a +chimney. Then I increased the number of unshaded strings to see if it changed the effect. The effect of a shadow perpendicular to the string caused bypass -diodes to trigger, but even after 20 strings, the IV curve of the system -appears unaffected and the total power loss is only 0.85% for the system -compared to unshaded. +diodes to trigger. Adding more strings did not stop the bypass diodes from triggering, even after 20 parallel +strings. The IV curve of the system had a kink until 4 unshaded parallel strings were added, but after 9 unshaded parallel strings were added, the IV curve +appeared unaffected. After 19 unshaded parallel strings were added, the total power loss was only 0.85% for the system +compared to unshaded. However, the power loss in the shaded string was about 15%. -Here is the IV curve of the 20 string PV system with perpendicular shade on -1st module of 1st string from the Jupyter notebook. It looks unaffected! +Here is the IV curve of the 20 string PV system with perpendicular shade on the +1st module of the 1st string from the Jupyter notebook. It looks unaffected! ![20 string PV system with cross-string shade](./images/cross-string-mismatch/pvsystem-20strings.png) Now check out the IV curve of the string with the shaded module. It should be generating about 3200[W], but even though it's lost about 500[W], it -still operates at 5[A], nearly the same current as the others strings. It +still operates at 5[A], nearly the same current as the other strings. It still has to operate at the same voltage as the other strings, 538.7[V] in this example, so how does it do it with 2 bypass diodes activated? @@ -100,12 +105,20 @@ a scattershot of scenarios and found that parallel string voltage began to domin Of course, that only applies in this contrived example, but it was interesting nonetheless. ## Conclusion -I wish I could say, "that's all there is to it." But as my first blog post -says, electrical mismatch in crystalline silicon is very counter-intuitive. +I wish I could say, that's all there is to it, but as I said in my first blog post, +electrical mismatch in crystalline silicon is very counter-intuitive. That's why I created PVMismatch to begin with. I was tired of guessing and being wrong. So don't guess. Simulate with confidence, try PVMismatch, and let me know what you learn! +## Epilogue +So back to that rooftop with the diagonal shade line. It's a bit of both +categories right? How do you think it will perform? Will it lose nearly +all of its power or will bypass diodes active and save the day? Or maybe +something in between or completely different. Try to analyze it using +PVMismatch. If you need help I analyzed it in this Google Colab notebook: +[`nonuniform-rooftop-shade.ipynb`](https://colab.research.google.com/drive/1wOSF9aNvxUc2t1iduNKN1Dn-vBW_j92w) + ## References 1. Meyers, B., Mikofski, M. A., & Anderson, M. (2016). A Fast Parameterized Model for Predicting PV System Performance under Partial Shade Conditions. In IEEE (Ed.), 2016 IEEE 43rd Photovoltaic Specialists Conference (PVSC) (pp. 3173–3178). IEEE. [https://doi.org/10.1109/PVSC.2016.7750251](https://doi.org/10.1109/PVSC.2016.7750251)