UPDATE - I update this question today(28.12.2022), sorry for being not specific earlier.
I have a bunch of solar panels, each one is connected to an inverter. For each solar panel I have two sensors, a wind and a west sensor. From every inverter, I collect the power output.
The readings are collected for every 5 minutes, and I have a dataset over 7 days.
I know that the power can be estimated using the equation: $$\rho_A=irr_s*A_e*(1+\eta_T*(T_m-25°C))$$ where,
$\rho_A$ the expected poweroutput
$irr_s$ the solare irradiation on the PV-Panel
$A_e$ the effective are of the PV-Panel (a Product of the effective effeciency of the Panel and the actual area),
$\eta_T$ the Temperaturecoefficient of the Panel and
$T_m$ the Modul temperature
Here is what I did so far: I only used the west sensor as I thought the wind sensor has little effect on the power output of a solar panel(common sense?). Then, I grouped the data by hour and then day, for example, for on the 22.12.2022 i got 24 readings, and so on.
I rearranged the equation, to get something like Ax + Bxy + C = 0, see here:
$$A_e * irr_s + A_e * \eta_T * (irr_s * T_m + irr_s * 25) - rho_A = 0$$
if I say, A = $irr_s$ and B = $irr_s * T_m + irr_s * 25$ and C = $-rho_A$,
Now I calculated the two unknowns, $A_e$ and $\eta_T$. I used matrixes to solve different pairs of equations using:
Let us assume the two pairs look like this:
$A_1x + B_1xy + C_1 = 0$ and
$A_2x + B_2xy + C_2 = 0$
M1 = $$\begin{matrix} A_1 & B_1 \\ A_2 & B_2 \\ \end{matrix} $$
and
M2 = $$\begin{matrix} C_1 \\ C_2 \\ \end{matrix} $$
ROOTS = $$\begin{matrix} R_1 \\ R_2 \\ \end{matrix} $$
Then, $ROOTS = M1^{-1}.M2$
I plotted them on a graph, see below: Click here
Do you think my approach is good? How else could one approach this problem? Thank you for your inputs in advance!
First. let me simplify the notation by replacing $\rho_A$ and $irr_s$ with their ratio $r = \dfrac{\rho_A}{irr_s}$ and $T_m$ with $T = T_m - 25\ ^\circ C$. And because carrying around a useless subscript is just a nuisance, call the other two variables just $A$ and $\eta$. Then your equations have the form
$$r_i = A(1 + \eta T_i)$$
Where $i = 0,1,2, ...$ represents the various measurements you've taken. For two measurements , let $K = \dfrac{r_i - r_j}{T_i - T_j}$ Then $$A = r_i - KT_i = r_j - KT_j\\\eta = \frac KA$$
I assume this is what you did. If so, then the most likely explanation of your graph is that the points on the line represent pairs of measurements where $T_i$ and $T_j$ were significantly different, while the outliers are pairs where $T_i$ and $T_j$ are close in value. The scatter you see is simply cancellation error. But that is a guess. There are other possible causes for the outliers.
In any case, the line is a strong indication that the outliers are the result of error, and $A$ and $\eta$ are not constants, but vary with each other, apparently with the relationship $$A = 30(1 -\eta)$$ Apparently the efficiency of the panels depends on the Temperature Coefficient, which itself is not constant, but depends on the other state variables. That dependence is not visible in this graph. You would need to plot $\eta$ against $T$ or $r$ to see that.