Calculation of solar panel output

📐 The Foundational Solar Output Equation

A widely used formula to estimate the energy output of a photovoltaic (PV) system is the following [1]:$$\mathrm{Es}=\mathrm{Ein}<span\; class="tr\_"\; id="tr\_8"\; data-source=""\; data-orig="\times ">\times </span>\mathrm{<span\; class="tr\_"\; id="tr\_9"\; data-source=""\; data-orig="r">r</span>}<span\; class="tr\_"\; id="tr\_10"\; data-source=""\; data-orig="\times ">\times </span>H<span\; class="tr\_"\; id="tr\_11"\; data-source=""\; data-orig="\times ">\times </span>PR$$

Jedoch, to better integrate your specific variables, we can expand this into a more detailed form, commonly used for system sizing and implemented in recognized models like NREL’s PVWatts [4]:$$P_{p\mathrm{AM}}=H_{t\mathrm{ich}\mathrm{<span\; class="tr\_"\; id="tr\_15"\; data-source=""\; data-orig="l">l</span>}t}<span\; class="tr\_"\; id="tr\_16"\; data-source=""\; data-orig="\times ">\times </span>P_{stc}<span\; class="tr\_"\; id="tr\_17"\; data-source=""\; data-orig="\times ">\times </span>f_{t\mathrm{und}mp}<span\; class="tr\_"\; id="tr\_18"\; data-source=""\; data-orig="\times ">\times </span>f_{\mathrm{die}th\mathrm{und}\mathrm{<span\; class="tr\_"\; id="tr\_19"\; data-source=""\; data-orig="r">r</span>}}$$​

Let’s define each term in this expanded equation [4, 8]:

• PpvPpAM​ : The total energy output (in kWh) over a given period (z.B., daily, monthly, or annually) or the power output (in W) [4].
• PstcPstc​ : The total rated power of your solar array (in kWdc) under Standard Test Conditions (STC: irradiance of 1000 W/m², cell temperature of 25°C) [1, 4]. This is the “size” of your system.
• HtiltHtichlt​ : The daily, monthly, or annual solar irradiation (in kWh/m²) on the plane of your solar array (Plane of Array or POA). This is where latitude und panel angle are used to calculate the sunlight your specific setup receives [5, 7].
• ftempftundmp​ : The temperature derating factor (a decimal between 0 und 1). This accounts for the loss in efficiency as the solar panel’s cell temperature rises above 25°C [1, 2, 8].
• fotherfdiether​ : A combined factor for all other system losses (a decimal between 0 und 1). This includes soiling (dust), shading, wiring losses, inverter efficiency, and more [1, 4].

🔍 Breaking Down the Key Components

To make this equation work, you need to determine the specific values for$H_{t\mathrm{ich}\mathrm{<span\; class="tr\_"\; id="tr\_75"\; data-source=""\; data-orig="l">l</span>}t}$Htichlt​ and$f_{t\mathrm{und}mp}$ftundmp​.

1. Irradiation on a Tilted Surface ($H_{t\mathrm{ich}\mathrm{<span\; class="tr\_"\; id="tr\_79"\; data-source=""\; data-orig="l">l</span>}t}$Htichlt​)

This is the most complex part, as it combines your location (latitude) and panel angle. The annual optimal fixed tilt angle for a location is often approximated by its latitude [5]. Jedoch, for maximum accuracy, a more nuanced approach is needed.

• Fixed Tilt Angle: Die “golden rule” is to set the tilt angle equal to your latitude. Zum Beispiel, at a latitude of 35°N, panels are often installed with a 35° tilt [5].
• Calculating HtiltHtichlt​: Manually calculating the irradiation on a tilted plane is complex. It requires splitting horizontal solar radiation data into its direct and diffuse components and then transposing them to the tilted plane [7]. Aus diesem Grund, professionals use tools like the European Commission’s PVGIS (Photovoltaic Geographical Information System) [3] or NREL’s PVWatts in the United States [4]. By inputting your location (latitude/longitude), panel tilt, and orientation (azimuth), these tools provide an accurate value for $H_{t\mathrm{ich}\mathrm{<span\; class="tr\_"\; id="tr\_109"\; data-source=""\; data-orig="l">l</span>}t}$Htichlt​. More recent approaches even use machine learning to improve the accuracy of these estimates compared to traditional isotropic models [7].

2. The Temperature Derating Factor ($f_{t\mathrm{und}mp}$ftundmp​)

Solar panels operate less efficiently as they get hot. This factor corrects for this effect [1, 2]. The formula, implemented in models like PVWatts, ist wie folgt: [4, 8]:$$f_{t\mathrm{und}mp}=1+[\mathrm{<span\; class="tr\_"\; id="tr\_118"\; data-source=""\; data-orig="\gamma ">\gamma </span>}<span\; class="tr\_"\; id="tr\_119"\; data-source=""\; data-orig="\times ">\times </span>(T_{c\mathrm{und}\mathrm{<span\; class="tr\_"\; id="tr\_120"\; data-source=""\; data-orig="l">l</span>}\mathrm{<span\; class="tr\_"\; id="tr\_121"\; data-source=""\; data-orig="l">l</span>}}<span\; class="tr\_"\; id="tr\_122"\; data-source=""\; data-orig="-">-</span>T_{stc})]$$

• γγ : The power temperature coefficient provided by the manufacturer. For crystalline silicon, it is typically expressed in %/° C and is negative [6, 10].
• TcellTcell​ : The estimated operating cell temperature (° C). More sophisticated models also account for wind speed and irradiance [1, 9].
• TstcTstc​ : The cell temperature at standard test conditions (STC), which is always 25° C [4].

Zum Beispiel, according to industry data, for a module with $\mathrm{<span\; class="tr\_"\; id="tr\_142"\; data-source=""\; data-orig="\gamma ">\gamma </span>}=<span\; class="tr\_"\; id="tr\_143"\; data-source=""\; data-orig="-">-</span>0.4\mathrm{\%}\mathrm{/}\mathrm{°}C$γ=−0.4%/°C, $T_{c\mathrm{und}\mathrm{<span\; class="tr\_"\; id="tr\_146"\; data-source=""\; data-orig="l">l</span>}\mathrm{<span\; class="tr\_"\; id="tr\_147"\; data-source=""\; data-orig="l">l</span>}}=65\mathrm{°}C$Tcell​=65°C, und $T_{stc}=25\mathrm{°}C$Tstc​=25°C, the power loss is significant [6]. The calculation is:$$f_{t\mathrm{und}mp}=1+[<span\; class="tr\_"\; id="tr\_154"\; data-source=""\; data-orig="-">-</span>0.004<span\; class="tr\_"\; id="tr\_155"\; data-source=""\; data-orig="\times ">\times </span>(65<span\; class="tr\_"\; id="tr\_156"\; data-source=""\; data-orig="-">-</span>25)]=1+(<span\; class="tr\_"\; id="tr\_157"\; data-source=""\; data-orig="-">-</span>0.16)=0.84$$

This means the panel is operating at only 84% of its rated power due to the high temperature.

Typical Temperature Coefficient ($\mathrm{<span\; class="tr\_"\; id="tr\_161"\; data-source=""\; data-orig="\gamma ">\gamma </span>}$γ) Values

The table below presents typical values for different panel technologies, based on research and industry data [2, 6, 10]:

Panel Technology	Typical Temperature Coefficient ($\mathrm{<span\; class="tr\_"\; id="tr\_168"\; data-source=""\; data-orig="\gamma ">\gamma </span>}$γ)	Aufzeichnungen
Monocrystalline Silicon (Older BSF)	-0.45% zu -0.50% /° C	Older technology with higher temperature losses [6].
Monocrystalline Silicon (Modern PERC)	-0.35% zu -0.40% /° C	Common technology with improved performance [6].
Monocrystalline Silicon (N-type TOPCon)	-0.29% zu -0.35% /° C	Advanced technology with a very good coefficient [6].
Monocrystalline Silicon (HJT – Heterojunction)	-0.25% zu -0.30% /° C	Premium technology with the best coefficient [6].
Polycrystalline Silicon	-0.40% zu -0.50% /° C	Older technology, generally higher coefficient [6].
Thin-Film (CdTe)	-0.24% zu -0.25% /° C	Very good performance in heat [6].
Field-Aged Modules	-0.5% /° C (for ηm)	Measurements on aged modules confirm these orders of magnitude [2].

3. Other Derating Factors ($f_{\mathrm{die}th\mathrm{und}\mathrm{<span\; class="tr\_"\; id="tr\_193"\; data-source=""\; data-orig="r">r</span>}}$fdiether​)

This is a catch-all for real-world inefficiencies. A typical value for a well-designed system might be around0.75 zu 0.85 [1]. You can calculate it by multiplying individual factors together [4].

💡 A Practical Example

Let’s combine these for a simplified annual estimate for a1 kWdc system using the PVWatts formula [4, 8].

1. Array Power (PstcPstc​): 1 kWdc
2. Tilted Irradiation (HtiltHtichlt​): Let’s assume you’ve used an online tool like PVGIS [3] for your specific latitude and chosen tilt. The tool outputs an annual HtiltHtichlt​ of 1700 kWh/m².
3. Temperature Factor (ftempftundmp​): Based on your local climate and panel specifications (z.B., $\mathrm{<span\; class="tr\_"\; id="tr\_220"\; data-source=""\; data-orig="\gamma ">\gamma </span>}=<span\; class="tr\_"\; id="tr\_221"\; data-source=""\; data-orig="-">-</span>0.4\mathrm{\%}\mathrm{/}\mathrm{°}C$γ=−0.4%/°C [6]), you calculate an average annual $f_{t\mathrm{und}mp}$ftundmp​ of 0.90.
4. Other Losses (fotherfdiether​): You estimate a combined factor of 0.80 for inverter losses, soiling, wiring, etc. [1, 4].

Your estimated annual energy output ($P_{p\mathrm{AM}}$PpAM​) would be [4]:$$P_{p\mathrm{AM}}=1\text{ <span class="tr\_" id="tr\_237" data-source="" data-orig="kWdc">kWdc</span>}<span\; class="tr\_"\; id="tr\_238"\; data-source=""\; data-orig="\times ">\times </span>1700\text{ <span class="tr\_" id="tr\_239" data-source="" data-orig="kWh/m²">kWh/m²</span>}<span\; class="tr\_"\; id="tr\_240"\; data-source=""\; data-orig="\times ">\times </span>0.90<span\; class="tr\_"\; id="tr\_241"\; data-source=""\; data-orig="\times ">\times </span>0.80=1224\text{ kWh}$$PpAM​=1 kWdc×1700 kWh/m²×0.90×0.80=1224 kWh

This means your 1 kWdc system is expected to generate about 1224 kWh of electricity per year under these conditions.

🧠 Recommendations for the Most Accurate Results

• Use Professional Tools: For the most reliable $H_{t\mathrm{ich}\mathrm{<span\; class="tr\_"\; id="tr\_249"\; data-source=""\; data-orig="l">l</span>}t}$Htichlt​ values, I strongly recommend using established tools like PVGIS [3] oder PVWatts [4]. They handle the complex geometry of sun position and radiation conversion for you [7].
• Consult the Datasheet: The most accurate value for the temperature coefficient ($\mathrm{<span\; class="tr\_"\; id="tr\_257"\; data-source=""\; data-orig="\gamma ">\gamma </span>}$γ) will always come from the manufacturer’s datasheet for the specific solar panel model you are using [6, 10]. Look for “Temperature Coefficient of Pmax” oder “Power Temperature Coefficient”.
• Gather Quality Input Data: The accuracy of your equation depends on your inputs. Use site-specific data for average temperatures and the exact technical details of your panels [1, 2, 9].

I hope this detailed analysis helps you develop a robust model for your solar energy calculations.

📚 Reference List

[1] MDPI (2022). Implicit Equation for Photovoltaic Module Temperature and Efficiency via Heat Transfer Computational Model.MDPI

[2] NIH (2023). Tabelle 3: Average temperature coefficients of the 3 field-aged PV modules.Heliyon

[3] Scilit (undated). PV-GIS: a web-based solar radiation database for the calculation of PV potential in Europe.Scilit

[4] NREL (2013). PVWatts Version 1 Technical Reference.National Renewable Energy Laboratory (NREL)

[5] Hugging Face (undated). Fiacre/PV-system-expert-500 · Datasets.Hugging Face

[6] Tongwei (2025). Mono Silicon Solar Panel Efficiency丨Temperature Coefficient, Low Light Performance, Attenuation Rate.Tongwei Co., Ltd.

[7] Energy Conversion and Management (2024). A universal tool for estimating monthly solar radiation on tilted surfaces from horizontal measurements: A machine learning approach.Energy Conversion and Management

[8] pvlib-python Documentation (undated). pvlib.pvsystem.pvwatts_dc.Read the Docs

[9] UNT Digital Library (1981). Analytical and experimental system studies of combined photovoltaic/thermal systems. Technical status report No. 12. University of North Texas

[10] IEEE (1997). Temperature coefficients for PV modules and arrays: measurement methods, difficulties, and results.Conference Record of the Twenty-Sixth IEEE Photovoltaic Specialists Conference