Part 2 - Performance Engineering for Solar
This post is the second in a series related to performance engineering, and builds on the information previously presented. If you have not read the previous post, Part 1 may be found here.
When solar modules are manufactured, they are tested and labeled with a nominal power rating. For example, 445W. To determine this rating, each solar module is tested using a special device that simulates sunlight at a specific level and measures the temperature while the electrical charge characteristics of the module are measured and recorded. All of this information and the nominal power rating is based on an agreed upon reference point called Standard Test Conditions (STC).
STC is specified as an irradiance of 1,000W/m2 and a temperature of 25°C (e.g. 77°F for reference).
So, for our sample 445W solar module, it should nominally produce a power output of 445W when irradiance is 1,000W/m2 at a solar module cell temperature of 25°C.
So, what if it were a really cold day or a really hot day? How would that effect our estimate of the solar module’s performance?
To figure this out, we need a little more information about our solar module to perform what is referred to as ‘temperature correction’. Every solar module has a product datasheet that contains technical information about its electrical characteristics, including three (3) temperature coefficients (a multiplier value).
For our 445W module example I am using the datasheet from a First Solar Series 6 thin film solar module. Full disclosure, I used to work for First Solar up until mid 2012. This information however is applicable with any solar module make or model.
The full datasheet may be access at the following link … but here is the marked up portion I’m going to talk about for this conversation.
Of the three available coefficients, because we are talking about power, we’ll use the ‘Temperature Coefficient of Pmax’.
The -0.32%/°C nomenclature looks a little scary/confusing at first, but is pretty simple to use in practice. This coefficient represents the percentage of power loss (because it’s negative) for ever degree the solar module cell temperature is different (often referred to as ΔT, pronounced delta-tee) from STC.
Let’s work through a few samples:
First, a pretty hot day … let’s say our solar module cell temperature is 35°C (REF: 95°F). That means our solar module cell is 10°C hotter than the STC of 25°C. The math is (CellTemp – STCTemp = ΔT), or (35 – 25 = 10°C).
Next, we’re going to multiply our ΔT (of 10°C) by our temperature coefficient (-0.32%/°C). The math is (ΔT * Tk(Pmax)= %), or (10 * -0.32 = -3.2%).
This is telling us that we should expect our 445W (or Pmax) solar module to produce 3.2% less than the nominal rating. The math is (Pmax * (1 + %) ) = TemperatureCorrectedRating, or (445W * (1 + -3.2%) ) = 430.76W.
Second, a cooler day … let’s say our solar module cell temperature is 10°C (REF: 50°F). That means our solar module cell is 15°C colder than the STC of 25°C. The math is (CellTemp – STCTemp = ΔT), or (10 – 25 = -15°C).
Next, we’re going to multiply our ΔT (of -15°C) by our temperature coefficient (-0.32%/°C). The math is (ΔT * Tk(Pmax) = %), or (-15 * -0.32 = 4.8%).
This is telling us that we should expect our 445W solar module to produce 4.8% more than the nominal rating. The math is (Pmax * (1 + %) ) = TemperatureCorrectedRating, or (445W * (1 + 4.8%) ) = 466.36W.
The consolidated equation for this process looks like this:
Pmax * ( 1 – ( (CellTemp – STCTemp) * Tk(Pmax) ) ) = TemperatureCorrectedRating
In summary, the hotter the solar module the less efficient it will be, the cooler the solar module the more efficient it will be.
NOTE: There are upper and lower limits to how efficient a solar module will be before it limits itself or fails to operate. That discussion is beyond the scope of this post however, and represents abnormal conditions that are beyond the normal operating conditions we would expect normally.
While all that new information sinks in, we’re going to change gears for a minute and come back to temperature correction in a little bit.
Power vs. Energy
Until now, everything we’ve discussed is concerning the solar module’s power output (e.g. Watts, or W) and the fuel level of sunlight (e.g. irradiance, or W/m2). Both power and irradiance are instantaneous measurements, and change over time.
While our basic check of irradiance level to power output percentage is a good quick gut check, it is not suitable for performance evaluation over any length of time.
Energy (e.g. Wh, pronounced watt-hours) and Insolation (Wh/m2, pronounced watt-hours per meter squared) are the time-based expressions for power and irradiance, respectively.
NOTE: We’ll discuss calculating energy and insolation in a future post.
A very simple way to evaluate solar system performance over time is to compare how many energy units were generated for each fuel unit during a similar period of time. The math is (DailyEnergy / DailyInsolation = DailyPerformance). Where DailyPerformance represents the number of energy-units our system is generating for every insolation-unit of fuel.
Using our example 100kW (or 100,000W) solar system, let’s consider the following 5 days of data:
| Day | Energy (Wh) | Insolation (wh/m2) | Energy/Insolation |
|---|---|---|---|
| 1 | 649,138 | 5,369 | 120.9 |
| 2 | 668,0978 | 5,627 | 118.7 |
| 3 | 491,280 | 5,394 | 91.1 |
| 4 | 475,464 | 3,986 | 119.3 |
| 5 | 347,268 | 2,844 | 122.1 |
For each day the energy output changes, and the insolation input changes too, but the DailyPerformance value has a very clear outlier. While four (4) our days have a nominal performance of ~120, day three (3) has a DailyPerformance of 91. We generated 29 energy-units less than we would have expected given the amount of solar-fuel (e.g. Insolation) we measured.
Something happened on day 3 of our data set.
Looking at our 5 days of data another way, by graphing power (the green area) and irradiance (the yellow-orange line) over the course of each production day, we can clearly see there was large outage in the early half of the day reducing the amount of energy generated.
This simple method can be used to compare productions hours, days, months, and even year-over-year operation to find outages (like in my example) or even longer term performance issues including soiling, module degradation, seasonal shading, and the effects of seasonal temperature.
What if we had more than one solar system though? What if we had ten (10) solar systems, and each were a different capacity (nameplate size)?
To use our daily performance concept across solar systems of different sizes, we need to normalize our calculated value based on the different capacities. This normalized value is called Performance Ratio (PR).
PR is calculated by dividing the measured energy generated by the energy expected based on the fuel.
Energy generated is our Energy (Wh) in the 5 days table above.
Energy expected based on fuel is the amount of Insolation (Wh/m2) multiplied by the DC capacity of the solar system.
Until now, when I have refereed to our sample solar system as having a capacity of 100kW, this is the AC capacity. AC capacity refers to the generating capacity of the solar system after all system losses and inverter limitations. The DC capacity is the total of all the solar module ratings before losses. For our sample system, we’ll assume the DC capacity is 135kWdc for this calculation.
To calculate PR, all we need to do is multiple the Energy/Insolation value by 135 and express the resulting value as a percentage.
| Day | Energy (Wh) | Insolation (wh/m2) | Energy/Insolation | PR |
|---|---|---|---|---|
| 1 | 649,138 | 5,369 | 120.9 | 89.6% |
| 2 | 668,0978 | 5,627 | 118.7 | 88.0% |
| 3 | 491,280 | 5,394 | 91.1 | 67.5% |
| 4 | 475,464 | 3,986 | 119.3 | 88.4% |
| 5 | 347,268 | 2,844 | 122.1 | 90.5% |
By using PR, we can now spot system level changes due to outages and long term factors, and compare these percentages across multiple solar systems for portfolio level analysis.
But what about the temperate correction?
In a real analysis, we would want to perform a temperature correction to the smallest interval of energy measured (say 1m, or 5m) to make sure that the actual temperature changes during the day were associated with the proper quantities of energy throughout the day. For this blog post however, I am going to use a daily average temperature that was calculated during the production hours of our solar system’s production.
| Day | Energy (Wh) | Insolation (wh/m2) | Energy/Insolation | PR | Avg. Temp (°C) | Corrected PR |
|---|---|---|---|---|---|---|
| 1 | 649,138 | 5,369 | 120.9 | 89.6% | -1.7 | 97.2% |
| 2 | 668,0978 | 5,627 | 118.7 | 88.0% | -0.6 | 95.2% |
| 3 | 491,280 | 5,394 | 91.1 | 67.5% | 2.4 | 72.3% |
| 4 | 475,464 | 3,986 | 119.3 | 88.4% | 6.9 | 93.5% |
| 5 | 347,268 | 2,844 | 122.1 | 90.5% | 4.3 | 96.5% |
In the table above I’ve added the daily average solar module cell temperatures (yes, these were very cold days) and applied our temperature correction to the 135kWdc nameplate capacity of our solar system. Due to the cold temperature, and our system being more efficient, those values were larger are larger than 135 and bring our actual and expected generation closer together. Our corrected PR values are now more similar to each other … although we can still see our fault on the third day.
These concepts: temperature correction, power vs energy, energy vs. fuel, normalization, and performance ratio establish the foundations for additional analysis techniques that will be discussed in future posts.