A Novel Spectral Correction Method for Predicting the Annual Solar Photovoltaic Performance Ratio Using Short-Term Measurements

Abstract

A novel spectral-corrected Performance Ratio calculation method that aligns the short-term Performance Ratio calculation to the annual calculated Performance Ratio is presented in this work. The spectral-corrected Performance Ratio allows short-term measurements to reasonably estimate the annual Performance Ratio, which decreases the need for long-term measures and data storage and assists with routine maintenance checkups. The piece-wise empirical model incorporates two spectral variables, a geographical location-based variable, the air mass, a PV-technology-based variable, and a newly defined spectral correction factor that results in a universal application. The spectral corrections show significant improvements, resulting in errors across different air mass and clearness index ranges, as well as temporal resolutions. The results indicate that a spectral correction methodology is possible and a viable solution to estimate the annual Performance Ratio. The results further indicate that by correcting the spectrum, short-term measurements can be used to predict the annual Performance Ratio with superior performance compared to the well-known normal and weather-corrected PR calculation methods. This approach is the first documented effort to address the spectrum’s influence on the utility-scale Performance Ratio calculation from hourly measurements. The empirical formula suggested for the Performance Ratio calculation can be of extreme value for the real-time monitoring of PV systems and enhancing PV power forecasting accuracy when the spectrum is considered instead of its usual omission. The model can be universally applicable, as it incorporates location and technology, marking a groundbreaking start to comprehending and incorporating the spectral influence in utility-scale PV systems. The novel calculation has widespread application in the PV industry, performance modelling, monitoring, and forecasting.

1. Introduction

The Performance Ratio (PR) is a crucial metric for various stakeholders as the PR can capture plant performance [1,2]. PR is a unitless quantity that indicates the quality of the PV plant by assessing the produced energy with respect to the expected energy [2] and is a critical performance assessment metric for a PV system [3].

Stakeholders can use different temporal resolutions of performance metrics, such as hourly, monthly or annual PR estimations. The research indicates that the PR fluctuates throughout the year, which is hypothesised to be because of the spectrum. Thus, an hourly or monthly PR estimation does not indicate the annual PR performance. Ignoring this gives a false representation of the PV plant’s performance to stakeholders.

The calculation of PR adheres to the standard testing conditions (STCs) [4]. STCs are the industry-wide standard conditions to determine the PV module’s performance. The STC are a cell temperature of 25 °C ($T_{stc}$), an irradiance of 1000 $\mathrm{W}/{\mathrm{m}}^2$ ($G_{stc}$), and a reference spectrum at 1.5 air mass (AM1.5) [5]. These values correspond to the irradiance and spectrum of a clear sky upon a sun-facing 37°-tilted surface (${\theta }_T$) with a 48.19° solar zenith angle (${\theta }_Z$). STC approximates solar noon near the spring and autumn equinoxes in the United States of America.

The PV modules generally do not respond to the full AM1.5 spectrum [5]. Atmospheric conditions cause variations in spectral irradiance due to the scattering and absorption phenomenon, and the three main weather variables that have an effect are AM, AOD, and PW. All three of these variables change over a day and throughout the seasons [5,6], and for clear-sky conditions, the AM, AOD, and PW contribute more to the changes in the spectrum. Ignoring the changes in the spectrum’s distributions leads to unrealistic yield predictions [7].

Most of the Sun’s incident radiation at the top of Earth’s atmosphere falls within the 290 to 3000 nm wavelength range. The spectral distribution of this solar radiation is influenced by local environmental factors and the Sun’s position [8]. The Bird model, presented in [9], simulates solar spectral irradiance based on various spectral inputs. Different PV technologies exhibit varying absorption characteristics across the solar spectrum due to differences in their material composition, bandgap energies, and structural design. For instance, crystalline silicon modules typically absorb light efficiently in the visible spectrum but have reduced effectiveness at longer wavelengths, while thin-film technologies, such as cadmium telluride (CdTe) and copper indium gallium selenide (CIGS), can absorb a broader range of wavelengths. These variations in absorption lead to distinct relative spectral responses among different PV modules, affecting their overall efficiency and performance under diverse environmental conditions [7].

The bandgap energy of a PV cell decreases at higher operating temperature conditions, influencing the cell to absorb longer-wavelength photons. However, increasing the wind speed improves efficiency, as the cooling effect of the wind is more pronounced and blows away dust particles, which cleans the PV modules by removing dust deposits [10,11]. Farr and Stein assessed the spatial variations in temperature for a 16-module PV array, where the authors noted that the wind’s directionality caused trends in the spatial temperature variation [12]; however, the authors could not fully explain the observed trends. Established corrections, such as the temperature and irradiance or weather-corrected PR (WC-PR), do not consider spectral influences. The industry has yet to adopt spectra-corrected PR (SC-PR) [13].

Until now, there have been no adjustments to account for these spectral influences on the PR. There is also the problem that some form of spectral measurements is needed to correct for the spectrum. Ground-based solar spectrum measurements are scarce and costly and require high maintenance [7]. Satellite-based data map the annual average spectral influences over large geographical areas; however, these simulation techniques require further experimental validation [7].

The current international standard defines four temperature and four broadband irradiance power ratings but only one spectrum [14], despite the many studies that indicate that the spectrum does influence the output of PV technology [7,15,16,17,18,19,20]. Many studies have also noted seasonal variations in PR [21,22,23,24].

Even a small percentage error in yield estimations can have a significant impact when carried over multiple years [25]. Spectral data are limited and scarce [25], so the viable solution is spectral correction functions using various variables. Due to the extrapolation error, where the instantaneous PR is not equal to the annual PR in most scenarios, long-term data concerning PV power output and meteorological data have to be stored. Therefore, there remains a gap in the literature that illustrates how spectral corrections can be applied to the PR calculation and address these problems.

The manuscript is structured as follows: the introduction in Section 1 provides a comprehensive background to the challenges associated with performance estimations, the spectrum’s influence, and long-term PV system performance predictions. The development of a novel spectral correction method for calculating the Performance Ratio of PV systems (Section 2) follows the introduction. The methodology section, Section 3 provides details on the data used, comparison metrics, the model development process, and the selection and description of the final model. The results section (Section 4) presents the findings and validation of the spectrally corrected Performance Ratio. The manuscript concludes in Section 5 with a summary of the key insights and implications of the study.

2. Developing a Spectral Corrected Performance Ratio Calculation

NREL developed a weather-corrected PR formula, which is more specific to a geographical location [3,26]. The methodology accounts for the effects of ambient temperature ($T_a$), wind speed ($WS$) and irradiance on the performance of the PV module.

The normal PR calculation is as follows:

$$PR={\displaystyle \frac{{\sum }_{i}E{N}_{ac,i}}{{\sum }_i\left(P_{STC}·\frac{G_{POA,i}}{G_{STC}}\right)}}.$$

(1)

In Equation (1), $E{N}_{ac,i}$ is the measured PV energy generation, $P_{STC}$ is the sum of the nameplate ratings for all modules installed during the acceptance test, and $G_{POA,i}$ is the measured POA irradiance averaged over the time step i [4,26].

The weather-corrected PR calculation is as follows:

$$P{R}_{WC}={\displaystyle \frac{{\sum }_{i}E{N}_{ac,i}}{{\sum }_i\left(P_{STC}\frac{G_{PO{A}_i}}{G_{STC}}\right)·\left(1-\frac{{\delta }_{temp}}{100}\left(T_{cell,avg}-T_{cell,i}\right)\right)}}.$$

(2)

$T_{cell}$ is the cell temperature calculated from measured meteorological data, and ${\delta }_{temp}$ is the temperature coefficient for power (%/°C, negative in sign).

$$T_{cell,calc}=T_m+{\displaystyle \frac{G_{POA}}{G_{STC}}}\times \Delta T.$$

(3)

ΔT denotes the conduction temperature drop, and $T_m$ is the back-surface temperature of the module:

$$T_m=G_{POA}\times e^{a+b\times WS}+T_a.$$

(4)

In Equation (4), $e^{a+b\times WS}$ is the conduction/convection heat transfer coefficient, and the coefficients a, b and ΔT are coefficients for a glass/cell/polymer sheet on an open rack mounting structure (see

Table 1. Empirical convective heat transfer coefficients [27].

Module Type	Mount	a	b	ΔT
Glass/cell/glass	Open rack	−3.47	0.0594	3
Glass/cell/glass	Close-roof mount	−2.98	−0.0471	1
Glass/cell/polymer sheet	Open rack	−3.56	−0.0750	3
Glass/cell/polymer sheet	Close-roof mount	−2.81	−00455	0
Polymer/thin-film/steel	Open rack	−3.58	−0.1130	3

) [27].

Finally, $T_{cell,avg}$ is the average calculated cell temperature weighted by irradiance of one year of weather data and is calculated using:

$$T_{cell,avg}={\displaystyle \frac{{\sum }_i\left[G_{POA,i}\times T_{cell,i}\right]}{{\sum }_i{G}_{POA,i}}}.$$

(5)

$G_{POA,i}$ and $T_{cell,i}$ are the POA irradiance and the calculated cell operating temperature for each hour.

$PW$ accounts for the presence of water vapour in the atmosphere and is calculated using [28,29,30]:

$$PW=0.1H_v·{\rho }_v.$$

(6)

$H_v$ is the apparent water vapour scale height (km) and ${\rho }_v$ is the surface water vapour density (${\mathrm{cm}}^2·{\mathrm{g}}^{-3}$).

$${\rho }_v=216.7·RH·{\displaystyle \frac{e_s}{T}}.$$

(7)

$e_s$ is the saturation water vapour pressure, and T is the temperature in Kelvin.

$H_v$ is determined by

$$H_v=0.4976+1.5265\left({\displaystyle \frac{T}{T_0}}\right)+\exp \left(13.6897\left({\displaystyle \frac{T}{T_0}}\right)-14.9188{\left({\displaystyle \frac{T}{T_0}}\right)}^3\right),$$

(8)

where $T_0=273.15$ Kelvin.

$e_s$ is estimated using

$$e_s=\sum _{i=0}^7{b}_i·T^i,$$

(9)

with the coefficients values defined as $b_0=6.1104546$, $b_1=0.4442351$, $b_2=1.402099\times {10}^{-2}$, $b_3=2.6454708\times {10}^{-4}$, $b_4=3.0357098\times {10}^{-6}$, $b_5=2.0972268\times {10}^{-8}$, $b_6=6.0487594\times {10}^{-11}$ and $b_7=-1.469687\times {10}^{-13}$ [28,29,30].

The general expression of AM is:

$$AM={\displaystyle \frac{1}{\cos {\theta }_Z}}.$$

(10)

The approximation in Equation (10) is reasonably accurate for ${\theta }_Z$ of up to 75°, which led Kasten and Young to propose a new equation to model the AM using [31]

$$AM={\left[\cos {\theta }_Z+0.5057·{\left(96.080-{\theta }_Z\right)}^{-1.634}\right]}^{-1}.$$

(11)

The absolute AM ($A{M}_a$) is the pressurised normalisation of AM, expressed as

$$A{M}_a=\mathrm{AM}\left({\displaystyle \frac{P}{P_o}}\right),$$

(12)

where P refers to the atmospheric pressure at the test site, and $P_o$ is the atmospheric pressure at sea level.

As the SC-PR aims to be a universal application, the $A{M}_a$ is used as a spectral variable instead of the $AM$ to include the altitude.

The definition of spectral mismatch ($MM$) is

$$MM={\displaystyle \frac{{\int }_{{\lambda }_1}^{{\lambda }_2}{G}_{ts\lambda }\left(\lambda \right)·SR\left(\lambda \right)d\lambda ·{\int }_{{\lambda }_3}^{{\lambda }_4}{G}_{ref}\left(\lambda \right)d\lambda }{{\int }_{{\lambda }_1}^{{\lambda }_2}{G}_{ref}\left(\lambda \right)·S{R}_{ref}\left(\lambda \right)d\lambda ·{\int }_{{\lambda }_3}^{{\lambda }_4}{G}_{ts\lambda }\left(\lambda \right)d\lambda }}.$$

(13)

${\lambda }_1$ and ${\lambda }_2$ are the lower and upper wavelength limits of spectral absorption of the PV module. ${\lambda }_3$ and ${\lambda }_4$ are the wavelength limits of the spectrum [7]. $G_{ref}$ is the reference spectral irradiance at STC, and $SR$ is the spectral response of the PV module. $G_{ts}$ is the measured solar spectral irradiance [32].

The formula multiplies the irradiance with the SR and the spectral irradiance and integrates over the wavelengths. When the SR at a specific wavelength, $\lambda$ is higher, and a larger amount of the spectral irradiance at wavelength $\lambda$ can generate a current in the PV module, compared to a lower SR. $MM$ indicates whether the PV performance will be more or less favourable to irradiance closer to reference conditions (STC).

$MM$ accounts for the spectrum’s influence on PV performance. The interpretation of $MM$ is that the PV module undergoes a spectral loss when $MM$ is less than unity and undergoes a spectral gain if $MM$ is greater than unity [17].

Figure 1 shows the default simulated SR of a poly-Si PV module used throughout the paper. The SR of a PV module indicates the spectral sensitivity and is a function of wavelength. The module can only absorb irradiance between certain wavelengths. In Figure 1, the SR is higher for wavelengths between 0.75 and 1 μm, which means that irradiance at these wavelengths can generate a higher current by the solar cells than the irradiance at lower wavelengths. It can also have the effect that larger variations in 0.75–1 μm in the spectral irradiance will have a more pronounced impact than lower wavelengths with a lower SR. The SR, therefore, encapsulates the PV technology’s sensitivity to the solar spectral irradiance.

$O_3$ can be successfully substituted with Equation (14) in spectral MM estimations, as studied in [33]:

$$O_3=235+\left(100+30\sin \left[0.9865\left(n+152.625\right)\right]+20\sin \left(2\left(Long-75\right)\right)\right)·\left({\sin }^2\left(1.5L\right)\right),$$

(14)

where $AO{D}_{500}$ can be substituted using a constant value of 0.123, also studied in [33].

The SC-PR must correct the WC-PR to account for spectral influences to estimate the annual PR:

$$P{R}_{SC,i}=P{R}_{S{C}_{annual}}=P{R}_{annual}=P{R}_{W{C}_{annual}}.$$

(15)

Thus, the equation should hold that

$$P{R}_{SC,i}={\displaystyle \frac{P{R}_{WC,i}}{C_i}}=P{R}_{annual}$$

(16)

where $C_{spec,i}$ is the spectral correction function:

$$C_{spec,i}={\displaystyle \frac{P{R}_{WC,i}}{P{R}_{annual}}}.$$

(17)

For the WC-PR, the following corrections for temperature ($C_{temp,i}$) and irradiance ($C_{irr,i}$) still hold:

$$P{R}_{WC,i}={\displaystyle \frac{E{N}_{ac,i}}{P_{STC}·C_{irr,i}\times C_{temp,i}}},$$

(18)

$$C_{temp,i}=\left[1-{\displaystyle \frac{\delta }{100}}\left(T_{c,avg}-T_{c,i}\right)\right],$$

(19)

$$C_{irr,i}={\displaystyle \frac{G_{PO{A}_i}}{G_{STC}}}.$$

(20)

Therefore, SC-PR becomes

$$P{R}_{SC,i}={\displaystyle \frac{E{N}_{ac,i}}{P_{STC}·C_{irr,i}\times C_{temp,i}\times C_{spec,i}}}.$$

(21)

The goal is to fit $C_{spec,i}$ to ensure $P{R}_{SC,i}=P{R}_{annual}$.

3. Methodology

In continuation of the previous discussions of the importance of accurate long-term performance estimations derived from short-term measurements and the significant impact of spectral variations on PV system performance, this section outlines the methodological framework employed in this study. The methodology encompasses a detailed examination of the data sources, the criteria used for comparative analysis, and the systematic development of the spectral correction model for the Performance Ratio. The selection process for the final model was meticulously conducted, taking into account factors such as complexity, robustness, and temporal transferability, to ensure its broad applicability. Additionally, this section provides a comprehensive summary of the spectral correction Performance Ratio (SC-PR) model, setting the stage for the subsequent presentation of validation results in Section 4 and the study’s key findings and conclusions in Section 5.

3.1. Data

Previous research on the data analytics of the unique available dataset has been discussed regarding the spectral influences on performance [34] and the wind and temperature effects of long-term degradation within a utility-scale PV plant [35], situated in a semi-arid region in Northern Cape, South Africa. Figure 2 describes the four-weather-station layout within the PV plant. The correlation analysis of the POA irradiance from the weather stations linked the different inverters, shown with different colour groupings, and the interested reader is referred to [35]. The unique dataset has 84 inverters, each with their power output and associated weather station in a resolution of 5 min intervals. The quality control procedure excludes all data points where the POA is less than 20 $\mathrm{W}/{\mathrm{m}}^2$ and a calculated PR and WC-PR (Equations (1) and (2)) greater than 100%. Data from inverters that showed faulty data in the analysis of [36] were subsequently excluded from the dataset. The data used to formulate the spectral corrections are 5-min data of generated PV power, POA irradiance, $T_a$, $RH$, $AM$, $A{M}_a$, constant $AO{D}_{500}$, and estimated $O_3$.

3.2. Comparison Metrics

The baseline models are normal PR and WC-PR. The error is the difference between the models and the annual PR. Therefore, if the instantaneous PR calculated using Equation (1) is 75% but the annual PR is calculated as 80%, there is a 5% difference between what the normal PR predicted and the annual PR. Note that 5% as a fraction of 80% results in a 6.25% error, expressed as a percentage of the annual PR.

The comparison metrics are the RMSE, MAE and MBE, all expressed as a percentage of the annual PR.

3.3. Spectral Variable Selection for Correction Model

The spectral influences consist of two components: the input spectral influence (spectral irradiance) and the technology’s spectral influence (the PV module’s SR).

Figure 3 assesses the influences of different spectral variables on the short circuit current. The figure calculates the PW using Equation (6) ranges with varying $AM$, $RH$ and $T_a$. Figure 3 shows that the $AM$ significantly impacts the $I_{SC}$ and, in turn, the power output. The spectral correction methodology excludes PW, $O_3$ and $AO{D}_{500}$ due to their minimal impact on spectral irradiance.

The SR (Figure 1) must be considered part of the technology’s influence on performance. $A{M}_a$ is used to consider the input spectral influence, as it significantly influences the input spectral irradiance, as shown in Figure 3, compared to the other spectral variables.

The hypothesis is that this makes the contributions more universally applicable because $A{M}_a$ is determined based on the location, and the PV technology determines the SR.

The PV plant’s region is semi-arid, and the clear-sky Bird and Riordan model [9] is used to model the spectral irradiance, as most weathering profiles are sunny and clear-sky days for PV plants. Therefore, the novel spectral correction variable accounts for the change in spectral irradiance.

From Equation (13), $G_{POA}$ and $G_{STC}$:

$$G_{POA}={\int }_{{\lambda }_3}^{{\lambda }_4}E\left(\lambda \right),$$

(22)

$$G_{STC}={\int }_{{\lambda }_3}^{{\lambda }_4}{E}_{stc}\left(\lambda \right),$$

(23)

can thus simplify the spectral mismatch to the following [37]:

$$MM={\displaystyle \frac{{\int }_{{\lambda }_1}^{{\lambda }_2}E\left(\lambda \right)·SR\left(\lambda \right)d\lambda }{{\int }_{{\lambda }_1}^{{\lambda }_2}{E}_{ref}\left(\lambda \right)·S{R}_{ref}\left(\lambda \right)d\lambda }}·{\displaystyle \frac{G_{STC}}{G_{POA}}}.$$

(24)

The newly defined spectral correction factor $SC$

$$SC={\displaystyle \frac{{\int }_{{\lambda }_1}^{{\lambda }_2}E\left(\lambda \right)·SR\left(\lambda \right)d\lambda }{{\int }_{{\lambda }_1}^{{\lambda }_2}{E}_{ref}\left(\lambda \right)·S{R}_{ref}\left(\lambda \right)d\lambda }},$$

(25)

results in

$$MM=SC·{\displaystyle \frac{G_{STC}}{G_{POA}}}.$$

(26)

A similar analysis of $SC$, as discussed in [33] for spectral MM formulation, assesses the sensitivity of $AO{D}_{500}$ and $O_3$ on $SC$ simulations. Figure 4 shows the distribution of how the $SC$ compares for the measured versus estimated $SC$ and the RMSE of estimating $SC$. Figure 4a results indicate that a constant $AO{D}_{500}$ and estimated $O_3$ using Equation (14) correspond well with the measured $SC$. Figure 4b shows the RMSE of estimating $SC$ with the estimated $O_3$ and constant $AO{D}_{500}$. The results indicate a reasonable error when estimating $SC$ using estimated constant $AO{D}_{500}$ and estimated $O_3$ instead of measured variables.

The results indicate that the $SC$ is less sensitive to the $O_3$ and $AO{D}_{500}$ used for modelling. Therefore, the $SC$ is a viable spectral correction variable.

Table 2. Correlation between spectral variables and the goal correction value.

	${\mathit{AM}}_{\mathit{a}}$	PW	${\mathit{O}}_3$	$\mathbf{SC}$	${\mathit{C}}_{\mathit{spec},\mathit{i}}$
ine $A{M}_a$	1.0
PW	0.549	1.0
$O_3$	−0.13	−0.328	1.0
$SC$	−0.729	−0.397	−0.008	1.0
$C_{spec,i}$	−0.733	−0.396	0.031	0.562	1.0
ine

shows the correlation matrix between the goal correction value $C_{spec,i}$ and spectral variables. $A{M}_a$ is inversely correlated, and $SC$ has a moderately positive correlation with $C_{spec,i}$.

The following variables are chosen as possible correction variables as they indicate the greatest influence on the relationship to the fitted correction value: $A{M}_a$ represents the spectral irradiance and location, and $SC$ represents the technology.

$$C_{spec,i}=f\left(A{M}_a,SC\right).$$

(27)

Figure 5 and Figure 6 show the relationship between the correction factor $C_{spec,i}$ (from Equation (21)) and the $A{M}_a$ and SC.

Figure 5a and Figure 6a show binned $A{M}_a$ and $SC$, the mean correction factor in these bins and the total generated PV power. In Figure 5a, higher $A{M}_a$ levels have less generated PV power, but the correction factor drops to less than 1. Figure 6a indicates a more varied distribution of the generated PV power for different bins, and the $C_{spec,i}$ is less varied, though it decreases as $SC$ increases.

Figure 5b and Figure 6b show the $C_{spec,i}$, $A{M}_a$ and $SC$ relationships. As expected, there are considerably more data points at lower $A{M}_a$ levels with a negative correlation. Figure 6b shows a positively correlated relationship, capped close to 1 as $SC$ increases. Figure 5b shows the amount of generated power for different AM bins and the mean correction factor (a value greater than 1 represents an overestimated PR, and a value less than 1 represents an underestimated PR).

The relationship shown in Figure 5b shows an inverse relationship between the ideal correction and AM, also supported by the correlation matrix in Table 2. $SC$ has a moderate positive correlation with the desired correction factor (from Table 2).

3.4. Model Development

In the previous section, it was determined that $SC$ and $A{M}_a$ were the two variables that had the greatest influence on the fitted correction value. The next step is determining a function that encapsulates the spectral changes in the SC-PR formula in Equation (21).

The model is fitted using statistical methods such as regression. Regression techniques can take several forms, including linear, multi-linear, and non-linear approaches.

A linear relationship can be expressed as

$$y=b_0+b_{1}x,$$

(28)

where y represents the dependent variable, x is the independent variable, $b_0$ is the intercept, and $b_1$ is the slope of the line. Regression analysis helps quantify the strength of the relationship between y and x [38].

To estimate $b_0$ and $b_1$, the least squares method is used, which minimises the sum of the squared differences between the observed and predicted values. This sum, known as the residual sum of squares (denoted $SSE$), is given by

$$SSE=\sum _{i=1}^n{\left(y_i-\widehat{y_i}\right)}^2,$$

(29)

where $\widehat{y_i}$ is the predicted value for each observation. Minimising the $SSE$ ensures that the regression line best fits the data points.

Polynomial regression extends this concept by modelling the dependent variable y as a polynomial of degree n with respect to x:

$$y=b_0+b_{1}x+b_2{x}^2+\cdots +b_r{x}^r.$$

(30)

This paper combines data science techniques and modelling, primarily using Python as the programming language [39], through the Anaconda interface [40], along with various supporting libraries [41,42,43,44,45]. The pandas library [42] is employed for efficient data manipulation and analysis. The pvlib library [41] provides essential functions and classes related to PV systems. Additionally, solar spectral irradiance is modelled using the pvlib spectrum module [43]. The SciPy library [44] is used for optimisation algorithms, particularly for fitting coefficients in the regression models outlined in this section. The SC-PR model was also calculated, and its performance metrics were evaluated.

The model development for SC-PR consists of considering many factors, such as the complexity and robustness of the empirical formula. An analysis of the model complexity compares different non-polynomials (or rational functions) of $A{M}_a$ and polynomials of $SC$, i.e., how complex the model has to be and how much the model improves the error. A model with less complexity and more robustness is a more ideal solution.

Order functions with positive and negative exponents are selected due to the non-linear relationships in Figure 5b and Figure 6b.

The spectral correction function takes the form of

$$f=f_{A{M}_a}·f_{SC},$$

(31)

where the AM function $f_{A{M}_a}$ can be in the inverse form due to its negative correlation with $C_i$ from Table 2:

$$f_{A{M}_a}=\sum _{n=0}^{n_{A{M}_a}}{a}_n{\displaystyle \frac{1}{{\left(A{M}_a\right)}^n}},$$

(32)

and the $SC$ function $f_{SC}$ can take the form of its moderately positive correlation with $C_i$ from Table 2:

$$f_{SC}=\sum _{n=0}^{n_{SC}}{b}_{n}S{C}^n.$$

(33)

The section further assesses a piece-wise model using $K_t$ versus having one continuous empirical formula. Thus,

$$f_{pw}=\left\{\begin{array}{cc}{f}_1\hfill & \mathrm{for}{K}_t<0.2,\hfill \\ f_2\hfill & \mathrm{for}0.2\le K_t<0.7,\hfill \\ f_3\hfill & \mathrm{for}{K}_t\ge 0.7.\hfill \end{array}\right.$$

(34)

The training–testing ratio of the data is a 60:40 split.

The model notation in the section from Equation (31) is denoted as

$${\mathit{f}}_t\left(n_{A{M}_a}=x,n_{SC}=y\right).$$

(35)

The subscript t refers to the model type, where continuous is denoted as $cont$-subscript and piece-wise as $pw$-subscript.

$n_{A{M}_a}$, as in Equations (32) and (35), refers to

$$\sum _{n=0}^{n_{A{M}_a}}{a}_n{\displaystyle \frac{1}{{\left(A{M}_a\right)}^n}}.$$

$n_{SC}$, as in Equations (33) and (35), refers to

$$\sum _{n=0}^{n_{SC}}{b}_{n}S{C}^n.$$

Thus, ${\mathit{f}}_{pw}\left(n_{A{M}_a}=2,n_{SC}=1\right)$ is the spectral correction function with the form

$${\mathit{f}}_{\mathit{piece}\text{-}\mathit{wise}}=\sum _{x=0}^2{a}_x{\displaystyle \frac{1}{{\left(A{M}_a\right)}^x}}·\sum _{y=0}^1{b}_{y}S{C}^y.$$

3.5. Model Selection

Figure 7 summarises the model selection methodology, leading to the two viable models discussed in this section. Two spectral variables representing the location and technology are used to developan SC-PR. The new model’s complexity, robustness and temporal transferability are important considerations in the model selection criteria.

3.5.1. Model Complexity, Temporal Sensitivity and Transferability

Figure 8 shows the MAE errors compared to the model’s complexity. The x-markers indicate continuous, and the dots indicate piece-wise models. The continuous models have higher errors than their piece-wise counterparts. Further, the error slowly improves as the model complexity increases, indicating there will be a trade-off between increasing performance by a minuscule amount for a much more complex model.

Figure 9 shows the MAE improvement of the different iterations combining Equations (32) and (33).

In Figure 9, the dashed lines indicate that the model is continuous, as described in Equation (34), and the solid lines indicate a piece-wise model. The piece-wise model outperforms the continuous model in all scenarios, being 5-min, hourly, and monthly aggregated. The results indicate that AM-only and $SC$-only functions show the worst performance. The conclusion is that the model needs to combine $A{M}_a$ and $SC$ instead of only $A{M}_a$ or $SC$.

Further, a more complex model can introduce uncertainties and additional computation, which may not be necessary to achieve the final result. A complicated model is not necessarily an ideal or practical model.

There has to be a trade-off between choosing a simpler model with a slightly higher error versus an extremely complex model with a slightly lower error. Therefore, the improvement of the $SC$ function from $n_{SC}=3$ to $n_{SC}=4$ and $n_{SC}=5$ is small enough that the conclusion is to limit the largest exponent of the $SC$ function (Equation (33)) to a maximum of 3. Similarly, with AM, the improvement from $n_{A{M}_a}=2$ to $n_{A{M}_a}=3$ (in Equation (32)) shows that both $n_{A{M}_a}=2$ and $n_{A{M}_a}=3$ can still be considered a viable option.

A further consideration is the temporal sensitivity and transferability. The model must transfer relatively well to hourly and monthly aggregated data, as shown in Figure 10 and Figure 11. The conclusions made from the 5-min datasets hold to these, indicating that $A{M}_a$- or $SC$-only models and $n_{SC}>3$ should be excluded as a possible spectral correction method for PR because significant improvement is not noticeable over a simpler model.

Table 3. Model complexity: 5-min results for AM and $K_t$ intervals.

	AM Intervals (MAE [%])				AM Intervals (RMSE [%])				${\mathit{K}}_{\mathit{t}}$ Intervals (MAE [%])
	[0.5, 1.5)	[1.5, 6)	[6, 10)	[10, 30)	[0.5, 1.5)	[1.5, 6)	[6, 10)	[10, 30)	[0, 0.2)	[0.2, 0.7)	[0.7)
ine $f_{cont}\left(n_{A{M}_a}=2,n_{SC}=1\right)$	4.19	9.02	18.66	38.46	8.21	12.48	21.36	44.95	8.77	12.29	4.04
$f_{cont}\left(n_{A{M}_a}=2,n_{SC}=2\right)$	4.01	7.99	20.3	39.7	8.13	11.6	23.27	46.04	9.4	11.73	3.56
$f_{cont}\left(n_{A{M}_a}=2,n_{SC}=3\right)$	3.74	7.68	22.18	38.8	7.95	11.53	25.76	45.38	9.61	11.38	3.4
ine $f_{cont}\left(n_{A{M}_a}=3,n_{SC}=1\right)$	4.16	7.79	19.65	35.19	8.11	11.63	23.11	42.86	8.98	11.35	3.66
$f_{cont}\left(n_{A{M}_a}=3,n_{SC}=2\right)$	3.96	7.25	20.87	36.18	7.97	11.23	24.42	43.52	9.46	11.05	3.34
$f_{cont}\left(n_{A{M}_a}=3,n_{SC}=3\right)$	3.92	7.22	21.17	36.29	7.96	11.22	24.8	43.63	9.5	11.04	3.31
ine $f_{pw}\left(n_{A{M}_a}=2,n_{SC}=1\right)$	3.93	8.51	18.83	37.47	8.19	12.07	21.77	44.22	8.04	12.32	3.44
$f_{pw}\left(n_{A{M}_a}=2,n_{SC}=2\right)$	3.79	7.56	20.27	38.31	8.09	11.38	23.46	44.92	8.03	11.84	3.03
$f_{pw}\left(n_{A{M}_a}=2,n_{SC}=3\right)$	3.57	7.27	21.97	37.87	7.87	11.35	25.91	44.78	8.02	11.61	2.88
ine $f_{pw}\left(n_{A{M}_a}=3,n_{SC}=1\right)$	3.86	7.48	19.83	34.95	8.02	11.25	23.69	42.91	8.04	11.3	3.26
$f_{pw}\left(n_{A{M}_a}=3,n_{SC}=2\right)$	3.73	6.98	20.87	35.54	7.9	10.96	24.91	43.31	8.04	11.06	3.01
$f_{pw}\left(n_{A{M}_a}=3,n_{SC}=3\right)$	3.67	6.91	20.84	35.57	7.88	10.96	24.9	43.38	8.02	11.06	2.91
ine Normal	5.54	10.05	25.25	49.27	9.08	13.48	29.44	54.54	10.75	13.92	5.52
WC-PR	4.3	9.08	32.84	55.27	8.04	14.36	36.08	59.34	9.78	14.46	4.09

shows the 5-min results for different $A{M}_a$ and $K_t$ intervals. The normal and WC-PR show the worst performance in estimating the annual PR. The continuous models show poorer performance for lower $K_t$ levels.

Table 4. Model complexity: temporal resolution results.

	Hourly Aggregated			5-min			Monthly Aggregated
	RMSE	MAE	MBE	RMSE	MAE	MBE	RMSE	MAE	MBE
	[%]			[%]			[%]
ine $f_{cont}\left(n_{A{M}_a}=2,n_{SC}=1\right)$	10.09	6.37	−0.44	12.32	7.41	−0.06	1.11	0.89	0.36
$f_{cont}\left(n_{A{M}_a}=2,n_{SC}=2\right)$	9.99	5.86	−0.43	12.11	6.95	−0.07	0.79	0.63	0.25
$f_{cont}\left(n_{A{M}_a}=2,n_{SC}=3\right)$	10.22	5.68	−0.2	12.18	6.74	−0.02	0.88	0.71	0.26
ine $f_{cont}\left(n_{A{M}_a}=3,n_{SC}=1\right)$	9.86	5.83	−0.07	11.92	6.85	−0.04	0.92	0.69	0.19
$f_{cont}\left(n_{A{M}_a}=3,n_{SC}=2\right)$	9.82	5.53	−0.06	11.85	6.57	−0.03	1.13	0.94	0.21
$f_{cont}\left(n_{A{M}_a}=3,n_{SC}=3\right)$	9.87	5.51	−0.04	11.88	6.55	−0.03	1.05	0.86	0.21
ine $f_{pw}\left(n_{A{M}_a}=2,n_{SC}=1\right)$	9.73	5.95	−0.46	12.12	7.05	−0.06	0.93	0.74	0.14
$f_{pw}\left(n_{A{M}_a}=2,n_{SC}=2\right)$	9.61	5.46	−0.41	11.96	6.63	−0.06	0.8	0.63	0.01
$f_{pw}\left(n_{A{M}_a}=2,n_{SC}=3\right)$	9.91	5.32	−0.21	12.06	6.45	−0.01	0.75	0.6	−0.0
ine $f_{pw}\left(n_{A{M}_a}=3,n_{SC}=1\right)$	9.65	5.52	−0.09	11.78	6.56	−0.03	0.86	0.63	−0.04
$f_{pw}\left(n_{A{M}_a}=3,n_{SC}=2\right)$	9.62	5.28	−0.06	11.74	6.32	−0.02	1.02	0.84	−0.07
$f_{pw}\left(n_{A{M}_a}=3,n_{SC}=3\right)$	9.64	5.23	−0.06	11.74	6.26	−0.02	0.91	0.73	−0.06
ine Normal	13.45	8.6	−3.07	14.21	8.97	−1.53	2.46	2.24	0.06
WC-PR	15.35	8.65	−5.84	15.14	8.32	−4.0	0.91	0.7	−0.01

shows the sensitivity of transferability of different temporal resolutions. The normal and WC-PR have the worst performance. All the continuous models overestimate the monthly aggregated data. The piece-wise models perform marginally better than the continuous models for the different metrics.

The analysis showed that $n_{A{M}_a}=1$ was unsuitable. Additionally, $n_{SC}=4$ and $n_{SC}=5$ are unsuitable as they need to show significant improvement over the $n_{SC}=3$ function to be a considered option, confirming the conclusions made from Figure 8 and Figure 9. The piece-wise model shows better performance than the continuous model. However, $n_{A{M}_a}=2$ and $n_{A{M}_a}=3$ show promising results.

The results in Figure 8 and Figure 9 show that piece-wise models showed higher error improvements over WC-PR than continuous models, and the AM-only or $SC$-only models were not viable options. The results indicate that a combination of the two for a correction model would have higher accuracy in correcting PR for the spectrum.

3.5.2. Model Robustness

The ideal model should be robust while being as accurate as possible, which requires assessing how the coefficients vary throughout the PV plant. Due to the nature of the dataset, there are 84 inverters with their annual PR. Therefore, individual models are developed for each inverter and then compared to the entire PV plant to indicate if there is a significant change in coefficient variables. The ideal model would be one where very little variability is noticed and does not overfit on a certain variable, i.e., not large coefficients for $A{M}_a$ and small coefficients for $SC$ or vice versa, which can result in a universal application of the formula.

From the initial model complexity assessment, viable models were selected, and their robustness was further scrutinised. The analysis considered each model’s coefficient distribution to assess the robustness and evaluated the error improvement over WC-PR and temporal transferability across the entire plant. Some models showed excellent improvements, whereas others indicated overfitting or poor robustness, leading to the selection of two viable models, as summarised in

Table 5. $f=f_{A{M}_a}·f_{SC}$ summary after assessment.

		${\mathit{f}}_{\mathit{A}{\mathit{M}}_{\mathit{a}}}={\displaystyle \sum _{\mathit{n}=0}^{{\mathit{n}}_{{\mathit{AM}}_{\mathit{a}}}}}{\mathit{a}}_{\mathit{n}}\frac{1}{{\left(\mathit{A}{\mathit{M}}_{\mathit{a}}\right)}^{\mathit{n}}}$
		Piece-Wise				Continuous
		${\mathit{n}}_{{\mathit{AM}}_{\mathit{a}}}$ = 0	${\mathit{n}}_{{\mathit{AM}}_{\mathit{a}}}$ = 1	${\mathit{n}}_{{\mathit{AM}}_{\mathit{a}}}$ = 2	${\mathit{n}}_{{\mathit{AM}}_{\mathit{a}}}$ = 3	${\mathit{n}}_{{\mathit{AM}}_{\mathit{a}}}$ = 0	${\mathit{n}}_{{\mathit{AM}}_{\mathit{a}}}$ = 1	${\mathit{n}}_{{\mathit{AM}}_{\mathit{a}}}$ = 2	${\mathit{n}}_{{\mathit{AM}}_{\mathit{a}}}$ = 3
$f_{SC}={\displaystyle \sum _{n=0}^{n_{SC}}}{b}_{n}S{C}^n$	$n_{SC}=0$	✗	✗	✗	✗	✗	✗	✗	✗
	$n_{SC}=1$	✗	✗	✗	✗	✗	✗	✗	✗
	$n_{SC}=2$	✗	✗	✗	✗	✗	✗	✗	✗
	$n_{SC}=3$	✗	✗	✓	✓	✗	✗	✗	✗
	$n_{SC}=4$	✗	✗	✗	✗	✗	✗	✗	✗
	$n_{SC}=5$	✗	✗	✗	✗	✗	✗	✗	✗

, highlighted in green.

Thus, two possible models that could be viable asan SC-PR are

• ${\mathit{f}}_{pw}\left(n_{A{M}_a}=2,n_{SC}=3\right)$:

  $$f_{piece-wise}=\left(\sum _{x=0}^2{a}_x{\displaystyle \frac{1}{{\left(A{M}_a\right)}^x}}\right)·\left(\sum _{y=0}^3{b}_{y}S{C}^y\right),$$
• ${\mathit{f}}_{pw}\left(n_{A{M}_a}=3,n_{SC}=3\right)$:

  $$f_{piece-wise}=\left(\sum _{x=0}^3{a}_x{\displaystyle \frac{1}{{\left(A{M}_a\right)}^x}}\right)·\left(\sum _{y=0}^3{b}_{y}S{C}^y\right).$$

Figure 12 shows how coefficients change across the PV plant for each inverter for the function ${\mathit{f}}_{pw}\left(n_{A{M}_a}=2,n_{SC}=3\right)$. The coefficient distribution across the three bins is relatively stable.

Figure 13 shows the MAE improvement over WC-PR for each inverter block. Figure 13a,b indicates that the model significantly improves over the WC-PR for the entire plant. The monthly aggregated data in Figure 13c show marginal improvements in most inverters.

Figure 14 shows how coefficients change across the PV plant for each inverter for ${\mathit{f}}_{pw}\left(n_{A{M}_a}=3,n_{SC}=3\right)$.

The $A{M}_a$ coefficients in Figure 14a are close to zero, and the b coefficients vary significantly across the plant. Figure 14b,c shows a more stable coefficient distribution across the different inverters. The results in Table 3 and Table 4 indicate that the model does better than the piece-wise counterpart. In Table 4, the piece-wise function significantly outperforms the Normal and WC-PR for hourly and monthly aggregated data.

Figure 13 shows the distribution of the PV plant for improvements over the WC-PR of each inverter. The 5-min resolution (Figure 13a) shows ranges of improvement over WC-PR between 15 and 28%, with the hourly aggregated resolution (Figure 13b) ranging between 24 and 41% improvement and the monthly aggregated resolution (Figure 13c) between a 6.9% decrease to 30% increase (−6.9% to 30%).

Figure 15 shows the distribution of the PV plant for improvements over the WC-PR of each inverter. The 5-min resolution (Figure 15a) shows ranges of improvement over WC-PR between 18 and 31%, with the hourly aggregated resolution (Figure 15b) ranging between 26 and 43% improvement and the monthly aggregated resolution (Figure 15c) between a 16% decrease to a 9% increase (−16% to 9%).

A final comparison of the two models assesses the most suitable spectral correction function for PR estimations.

3.5.3. Final Model Selection

Table 6. Comparison of two viable SC-PR models vs. normal and WC-PR (5-min resolution).

	5-min
	RMSE	MAE	MBE
	[%]
ine $f_{pw}\left(n_{A{M}_a}=2,n_{SC}=3\right)$	12.06	6.45	−0.01
$f_{pw}\left(n_{A{M}_a}=3,n_{SC}=3\right)$	11.74	6.26	−0.02
ine Normal	14.21	8.97	−1.53
WC-PR	15.14	8.32	−4.0

summarises Table 4 and indicates that the two models significantly outperform the normal and WC-PR. There is a significant reduction in error in the RMSE and MAE. The models were originally fitted for 5-min resolution data to have as many data as possible; however, the hourly metrics are still an important consideration for the final model selection. There is a marginal difference between the two new models regarding performance in the 5-min resolution.

Table 7. Comparison of two viable SC-PR models vs. normal and WC-PR (5-min resolution).

	${\mathbf{AM}}_{\mathbf{a}}$ Intervals (MAE [%])				${\mathbf{AM}}_{\mathbf{a}}$ Intervals (RMSE [%])
	[0.5, 1.5)	[1.5, 6)	[6, 10)	[10, 30)	[0.5, 1.5)	[1.5, 6)	[6, 10)	[10, 30)
ine $f_{pw}\left(n_{A{M}_a}=2,n_{SC}=3\right)$	3.57	7.27	21.97	37.87	7.87	11.35	25.91	44.78
$f_{pw}\left(n_{A{M}_a}=3,n_{SC}=3\right)$	3.67	6.91	20.84	35.57	7.88	10.96	24.9	43.38
ine Normal	5.54	10.05	25.25	49.27	9.08	13.48	29.44	54.54
WC-PR	4.3	9.08	32.84	55.27	8.04	14.36	36.08	59.34

summarises from Table 3 and shows the RMSE and MAE metrics for the 5-min data for different AM intervals. The lower AM intervals (0.5 AM to 1.5 AM) generate the most PV power, as shown in Figure 5a, showing improvement in reducing the WC-PR and normal PR for both models.

After 6 AM, the amount of generated PV power compared to power generated for lower $A{M}_a$ levels becomes incomparable. It is important to focus on improving the periods where the most PV power is generated rather than focusing on higher AM, which is associated with lower PV power generation. Figure 5b shows that most data are below 10 AM. The mid-range $A{M}_a$ levels (1.5 AM to 6 AM) show that the $n_{A{M}_a}=3$ function performs marginally better than $n_{A{M}_a}=2$ with the $n_{SC}=3$ function. For higher $A{M}_a$ (greater than 6), there is a significant improvement in reducing the error over WC-PR.

Table 8. Comparison of two viable SC-PR models vs. normal and WC-PR (5-min resolution).

	${\mathbf{K}}_{\mathbf{t}}$ Intervals (MAE [%])
	[0, 0.2)	[0.2, 0.7)	[0.7)
ine $f_{pw}\left(n_{A{M}_a}=2,n_{SC}=3\right)$	8.02	11.61	2.88
$f_{pw}\left(n_{A{M}_a}=3,n_{SC}=3\right)$	8.02	11.06	2.91
ine Normal	10.75	13.92	5.52
WC-PR	9.78	14.46	4.09

summarises Table 3, showing the MAE for different $K_t$-intervals. The results indicate that the two models considerably improve over WC-PR in estimating the annual PR compared to WC-PR.

Table 9. Comparison of two viable SC-PR models vs. normal and WC-PR (hourly aggregated).

	Hourly Aggregated
	RMSE	MAE	MBE
	[%]
ine $f_{pw}\left(n_{A{M}_a}=2,n_{SC}=3\right)$	9.91	5.32	−0.21
$f_{pw}\left(n_{A{M}_a}=3,n_{SC}=3\right)$	9.64	5.23	−0.06
ine Normal	13.45	8.6	−3.07
WC-PR	15.35	8.65	−5.84

summarises Table 4 and shows the hourly metrics, an important criterion the model has adhered to viable spectral correction method. The results indicate considerable improvements over the WC-PR, reducing the MBE to almost zero. The normal and WC-PR all show slightly under estimations for annual PR from the instantaneous measurements.

Table 10. Comparison of two viable SC-PR models vs. normal and WC-PR (monthly aggregated).

	Monthly Aggregated
	RMSE	MAE	MBE
	[%]
ine $f_{pw}\left(n_{A{M}_a}=2,n_{SC}=3\right)$	0.75	0.6	−0.0
$f_{pw}\left(n_{A{M}_a}=3,n_{SC}=3\right)$	0.91	0.73	−0.06
ine Normal	2.46	2.24	0.06
WC-PR	0.91	0.7	−0.01

summarises Table 4 and shows the monthly aggregated data. The monthly model is outside this manuscript’s scope; however, a discussion showing its temporal transferability indicates potential future improvements.

The normal PR has the worst performance in estimating the annual PR. The $n_{A{M}_a}=3$ function is comparative with WC-PR, whereas the $n_{A{M}_a}=2$ function marginally reduces the RMSE and MAE better. The metrics in Table 10 are all mean-based. As the number of monthly aggregated data is considerably less than the hourly or 5-min data, the magnitude of the errors is influenced.

The coefficient distribution across the entire PV plant indicates how robust the model is. Figure 12 shows the $n_{A{M}_a}=2$ with the $n_{SC}=3$ function, where the coefficients were quite stable for all the inverters. Figure 12a shows that the $A{M}_a$ has more influence in the lower $K_t$ ranges, whereas the $SC$ has a greater influence in the mid and higher $K_t$ ranges.

Figure 14 shows $n_{A{M}_a}=3$ with the $n_{SC}=3$ function, where the coefficients were quite stable for all the inverters, except in the lower $K_t$ range (Figure 14a), where there was also overfitting on the b coefficient.

Overall, the final verdict is to use the simpler model, ${\mathit{f}}_{pw}\left(n_{A{M}_a}=2,n_{SC}=3\right)$, over the more complex model.

3.6. Novel Spectra-Corrected Performance Ratio Estimation

Step 1: Data

• Subhourly data points (ideally 5–10 min intervals) of the data required that we determine the weather-corrected Performance Ratio. At least one full year is required for weather-corrected PR to determine the weighted cell temperature:

  (a)

  Generated PV Power;

  (b)

  GHI for $K_t$;

  (c)

  POA irradiance (or use GHI with the correct combination of decomposition and transposition model);

  (d)

  Wind speed, ambient temperature, and relative humidity.

• Spectral variables:

  (a)

  Estimate $O_3$, PW, and AOD₅₀₀;

  (b)

  Determine $AM$ and $A{M}_a$;

  (c)

  SR of PV technology;

  (d)

  Simulate the spectral correction factor $SC$ using $O_3$, AM, PW, and AOD₅₀₀.

• Quality control of the dataset.

  (a)

  All calculated PR must be less than 1 (WC-PR and Normal-PR);

  (b)

  Remove all faulty data/inverters/periods where non-spectrum-related issues occurred (such as breakdowns, curtailments);

  (c)

  Remove all POA less than 20 $\mathrm{W}/{\mathrm{m}}^2$.

Step 2: Spectra-Corrected Performance Ratio

• Calculate for each interval (5-min or 10-minutely ideal);
• Spectral correct each interval using Equation (36) below;
• Aggregating to hourly by summing the PV power and the spectral corrected over each hour to determine the hourly PR.

$$P{R}_{SC,i}={\displaystyle \frac{E{N}_{ac,i}}{P_{STC}·C_{irr,i}\times C_{temp,i}\times C_{spec,i}}},$$

(36)

where

$$C_{temp,i}=\left[1-{\displaystyle \frac{\delta }{100}}\left(T_{c,avg}-T_{c,i}\right)\right],$$

(37)

$$C_{irr,i}={\displaystyle \frac{G_{PO{A}_i}}{G_{STC}}},$$

(38)

$$C_{spec,i}=(a_0+{\displaystyle \frac{a_1}{A{M}_a}}+{\displaystyle \frac{a_2}{A{M}_a^2}})·(b_0+b_{1}SC+b_{2}S{C}^2+b_{3}S{C}^3),$$

(39)

where

Table 11. Coefficients for the spectral correction function.

	${\mathit{a}}_0+\frac{{\mathit{a}}_1}{\mathit{A}\mathit{M}}+\frac{{\mathit{a}}_2}{\mathit{A}{\mathit{M}}^2}$			${\mathit{b}}_0+{\mathit{b}}_1\mathit{S}\mathit{C}+{\mathit{b}}_2\mathit{S}{\mathit{C}}^2+{\mathit{b}}_3\mathit{S}{\mathit{C}}^3$
	a0	a1	a2	b0	b1	b2	b3
$K_t<0.2$	2.3088	2.4439	−1.4213	0.2957	0.0199	−0.0604	0.0357
$0.2\le K_t<0.7$	0.2651	0.4640	−0.3032	1.7805	2.6742	−3.5864	1.4292
$K_t\ge 0.7$	0.6235	0.1191	−0.0638	0.8686	2.1181	−2.1200	0.6562

shows the coefficients.

4. Results

The previous section discussed the methodology of selecting a model to correct the spectrum for PR estimates. Two spectral variables, $A{M}_a$ and $SC$, representing the location and technology, were selected to make the formula universally applicable. Section 3.6 discusses incorporating and applying the SC-PR formula.

In this section, the novel SC-PR formulation’s results are presented, and the three temporal resolutions, hourly, 5-min, and monthly, are discussed. However, the 5-min and hourly resolutions precede the monthly resolution, as this is the most common application. The spectrum changes rapidly, and the model better reflects higher-resolution data.

4.1. Annual Results

4.1.1. Year 1

Table 12. Year 1 temporal results of SC-PR.

	Hourly Aggregated			5-min Aggregated			Monthly Aggregated
	RMSE	MAE	MBE	RMSE	MAE	MBE	RMSE	MAE	MBE
	[%]			[%]			[%]
ine Normal	13.19	8.44	−2.78	12.71	8.3	−1.43	2.35	2.2	−0.08
WC-PR	15.17	8.64	−5.81	13.87	7.74	−4.19	1.22	0.96	−0.0
SC-PR	9.1	5.1	−0.66	9.94	5.6	−0.38	0.81	0.7	0.06

shows the normal, WC-PR, and SC-PR results of Year 1. The SC-PR estimates the annual PR from hourly and 5-min aggregated data with a lower error than normal and WC-PR. The MBE is also significantly reduced for higher-temporal-resolution data. The monthly aggregated data show slightly worse performance than WC-PR.

Figure 16 visualises the normal, WC-PR and SC-PR of Year 1 when $A{M}_a<10$ and POA $>50\mathrm{W}/{\mathrm{m}}^2$. The SC-PR shows less variability than normal and WC-PR. Figure 16a,b shows overestimations (SC-PR > 1) and underestimations, as do the normal and WC-PR.

Figure 17 shows the distribution of estimating PR compared to normal and WC-PR, which indicates that the SC-PR estimates more data points closer to the annual PR, compared to WC-PR and normal PR.

Figure 18 demonstrates how using SC-PR improves the estimate of annual PR from instant data by removing timestamps with low PV power generation ($A{M}_a<3$ and POA $>200\mathrm{W}/{\mathrm{m}}^2$).

4.1.2. Year 2

Table 13. Year 2 temporal results of SC-PR.

	Hourly Aggregated			5-min Aggregated			Monthly Aggregated
	RMSE	MAE	MBE	RMSE	MAE	MBE	RMSE	MAE	MBE
	[%]			[%]			[%]
ine Normal	13.02	8.35	−2.91	12.75	8.34	−1.67	2.46	2.21	0.26
WC-PR	14.91	8.54	−5.49	13.82	7.85	−3.96	1.11	1.02	0.14
SC-PR	9.35	5.03	−0.05	10.07	5.56	−0.14	0.89	0.75	0.13

shows the normal, WC-PR and SC-PR results of Year 2. The results are similar to Year 1. The SC-PR estimates the annual PR from hourly and 5-min aggregated data with a lower error than normal and WC-PR. The MBE is also significantly reduced for data with higher temporal resolution. The monthly aggregated data performs better than WC-PR and normal PR, in contrast to Year 1.

Figure 19 visualises the normal, WC-PR, and SC-PR of Year 2 for data points where $A{M}_a<10$ and POA $>50\mathrm{W}/{\mathrm{m}}^2$. The SC-PR shows less variability than normal and WC-PR. Figure 19a,b shows overestimations (SC-PR > 1) and underestimations, as do the normal and WC-PR.

Figure 20 shows the distribution of SC-PR compared to normal and WC-PR, which shows that both the WC-PR and normal PR for more ideal scenarios underestimate the PR.

In Figure 21, eliminating timestamps featuring low PV power generation ($A{M}_a<3$ and POA $>200\mathrm{W}/{\mathrm{m}}^2$) highlights the improved accuracy in estimating the annual PR using SC-PR with instantaneous data.

4.1.3. Year 3

Table 14. Year 3 temporal results of SC-PR.

	Hourly Aggregated			5-min Aggregated			Monthly Aggregated
	RMSE	MAE	MBE	RMSE	MAE	MBE	RMSE	MAE	MBE
	[%]			[%]			[%]
ine Normal	13.86	8.86	−3.07	14.0	8.95	−1.86	2.43	2.14	0.16
WC-PR	15.69	8.83	−5.86	14.98	8.2	−4.43	0.42	0.36	0.07
SC-PR	10.49	5.64	−0.01	11.58	6.21	−0.19	0.44	0.38	0.08

shows the normal, WC-PR, and SC-PR results of Year 3. The results are similar to Years 1 and 2. The SC-PR estimates the annual PR from hourly and 5-min aggregated data with a lower error than normal and WC-PR. The MBE is also significantly reduced for higher-temporal-resolution data. The monthly aggregated data perform similarly to WC-PR but considerably improve on normal PR.

Figure 22 visualises the normal, WC-PR, and SC-PR of Year 2 when $A{M}_a<10$ and POA $>50\mathrm{W}/{\mathrm{m}}^2$. The SC-PR shows less variability than normal and WC-PR. Figure 22a,b show overestimations (SC-PR > 1) and underestimations, as do the normal and WC-PR.

Figure 23 shows the distribution of SC-PR compared to normal and WC-PR with similar results to Year 1 and Year 2. The SC-PR estimates more of the data points closer to the annual PR compared to WC-PR and Normal PR.

Figure 24 demonstrates the removal of timestamps with lower PV power generation ($A{M}_a<3$ and POA $>200\mathrm{W}/{\mathrm{m}}^2$), highlighting how using SC-PR improves the estimation of annual PR based on instantaneous data.

4.1.4. Year 4

Table 15. Year 4 temporal results of SC-PR.

	Hourly Aggregated			5-min Aggregated			Monthly Aggregated
	RMSE	MAE	MBE	RMSE	MAE	MBE	RMSE	MAE	MBE
	[%]			[%]			[%]
ine Normal	13.49	8.66	−3.08	13.53	8.7	−1.93	2.61	2.41	0.25
WC-PR	15.39	8.59	−5.83	14.62	7.91	−4.42	0.69	0.54	0.11
SC-PR	10.46	5.49	0.16	11.12	5.91	−0.12	0.79	0.63	0.07

shows the normal, WC-PR, and SC-PR results of Year 4. The results are similar to Years 1–3. The SC-PR estimates the annual PR from hourly and 5-min aggregated data with a lower error than normal and WC-PR. The MBE is also significantly reduced for higher-temporal-resolution data. The monthly aggregated data show worse performance to WC-PR but considerably improve on normal PR.

Figure 25 visualises the normal, WC-PR, and SC-PR of Year 2 when $A{M}_a<10$ and POA $>50\mathrm{W}/{\mathrm{m}}^2$. The SC-PR shows less variability than Normal and WC-PR. Figure 25a,b show overestimations (SC-PR > 1) and underestimations, as do the normal and WC-PR.

Figure 26 shows the distribution of SC-PR compared to normal and WC-PR with similar results to Year 1 to Year 3. The SC-PR estimates more of the data points closer to the annual PR compared to WC-PR and normal PR for the hourly aggregated data.

Figure 27 removes lower PV power generation timestamps, $A{M}_a<3$ and POA $>200\mathrm{W}/{\mathrm{m}}^2$, which indicates the improvement the SC-PR makes in estimating the annual PR from instantaneous data.

4.2. Discussion

The newly defined spectral correction methodology for Performance Ratio successfully uses instantaneous measurements to estimate annual PR, with reasonable accuracy. The model shows significantly improved accuracy for higher AM values, where the spectral influence is non-linear. The model has a universal application due to its incorporation of the location and technology, but further testing and validation will be required to obtain additional sets of a and b coefficients. The method is a piece-wise empirical model including two spectral variables: $A{M}_a$, representing the location and irradiance, and a newly defined spectral correction factor, $SC$, representing the technology.

The new spectral corrections show promising results with reduced RMSE, MAE, and MBE for different $A{M}_a$ and $K_t$ bins and the overall dataset. To the author’s knowledge, there have been no attempts in the literature to correct the Performance Ratio with spectral corrections. The novel methodology addresses and successfully estimates PR from aggregated hourly values with reasonable accuracy to estimate a PV plant’s annual PR. The model does show slight overestimations and underestimations for higher $A{M}_a$ levels, but the results could also indicate faults with the plant.

The data used in the model have several limitations, such as the decomposition and transposition models: when using GHI with the decomposition and transposition models to approximate POA irradiance, these models may introduce biases, especially in non-standard or extreme weather conditions. Errors in irradiance, temperature, and wind speed sensors could propagate through the model. Miscalibrations or sensor degradation could skew the Performance Ratio, leading to false results. Removing faulty data and periods with curtailments or breakdowns may inadvertently remove valid periods of low production due to spectral or operational conditions. The aggregation of spectral-corrected data to hourly intervals could lead to the loss of finer details in PR performance variations, especially during rapid changes in irradiance or atmospheric conditions. Non-linear factors like inverter efficiency, partial shading, or temperature-induced degradation might not be fully captured in the weather-corrected PR, leading to deviations from actual performance.

Different PV technologies have varying spectral responses, and the correction factor might not generalise well across all types of panels, leading to inaccuracies in the calculated spectral-corrected PR. The lack of more available data for various technologies, geographical locations, and climates hinders the validation process. Errors of faulty measurement or plant equipment can be masked in the statistics closer to STC or ideal operational conditions. The SC-PR model compensates for the spectrum in the non-ideal spectral scenarios, and high $A{M}_a$ could exacerbate these errors, indicating faults in the plant worth addressing. These limitations could further refine and improve the spectral correction methodology and increase the model’s robustness.

The model shows an adequate temporal transference from 5-min to hourly data. The recommendation for the model is to use higher temporal resolution, as the spectrum is highly variable. The model corrects the PR for the spectrum, especially for higher $A{M}_a$ levels with lower generation of PV power. Therefore, the recommendation is to use periods of the day when the PV module will be in higher production and the spectral irradiance is greater.

5. Conclusions

The newly developed spectral correction methodology for estimating the PR demonstrates effective accuracy when extrapolating instantaneous measurements to the annual PR. Notably, the model exhibits a substantial enhancement in performance as the AM increases, particularly in scenarios where the solar spectrum deviates from STC. The new model boasts universal applicability due to its capacity for geographical location and technology-specific factors.

The methodology comprises a piece-wise empirical model that integrates two key spectral variables: $A{M}_a$, which reflects the location and irradiance, and a newly defined spectral correction factor denoted as $SC$, signifying the technology in use. These new spectral corrections yield promising outcomes, showcasing reduced RMSE, MAE, and MBE across various AM_a and $K_t$ intervals and different temporal resolutions.

Minor overestimations and underestimations are discernible for higher AM levels, indicating potential maintenance issues within the PV plant that require attention. Additionally, the model demonstrates satisfactory temporal transferability when transitioning from 5-min to hourly data to monthly data. Given these findings, it is advisable to consider the model’s application at higher temporal resolutions, leveraging the inherent adaptability of the solar spectrum. The model is particularly effective in correcting the PR when dealing with higher AM levels and reduced PV power generation compared to conventional PR estimations. It is advisable to operate the model during times of the day when the PV modules operate at peak production and spectral irradiance is more predictable.

Validation of the novel spectral correction methodology is essential for various technology types and climates. Its application must be updated in software tools as the new model can indicate the financial implications to stakeholders.

The proposed novel approach marks the first documented effort in the literature to rectify PR using spectral corrections for large utility-scale PV plants. The groundbreaking methodology effectively addresses and accomplishes estimating PR based on hourly aggregated values, yielding satisfactory accuracy in projecting a large PV plant’s annual performance.