1

Performance assessment of photovoltaic modules based on daily

energy generation estimation

Jing-Yi Wang: Beihang University, Beijing, 100191, China (email: wangjingyi1990@buaa.edu.cn);

Zheng Qian (corresponding author): Beihang University, Beijing, 100191, China (email:

qianzheng@buaa.edu.cn)

Hamidreza Zareipour: Department of Electrical and Computer Engineering, University of Calgary, Calgary,

AB T2N1N4, Canada (email: hzareipo@ucalgary.ca)

David Wood: Department of Mechanical and Manufacturing Engineering, University of Calgary, Calgary,

AB T2N1N4, Canada (email: dhwood@ucalgary.ca)

Declarations of interest: none.

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Performance assessment of photovoltaic modules based on daily

energy generation estimation

Abstract: Performance assessment can improve photovoltaic (PV) plant economics by identifying the need

for timely corrective actions. Performance assessment of PV plants is usually based on the comparison

between measured and modeled outputs of PV plants, i.e., an alarm occurs for abnormal operation when a

significant difference is detected. For such methods, it is critical to estimate the potential electricity

generation of PV plants in normal operation. However, unpredictable conditions affecting solar modules

pose challenges to develop reliable PV models. In this paper, a practical approach to improve estimating

daily energy generation of PV plants for performance assessment is proposed, which includes two main

components: (i) a data preprocessing method; (ii) sub-models in different weather conditions. The proposed

data preprocessing method detects outliers by comparing normalized outputs of adjacent inverters

instantaneously. It is robust against erroneous measurements in normal operation. Sub-models in different

weather conditions are developed using a Principal Component Analysis and Support Vector Machine

method for better representation of PV plant outputs. Results show that the proposed method can detect

deviations between the estimated and the measured daily energy generation of 10%. Moreover, false alarms,

i.e., an abnormal point identified while the system is operating normally, are significantly reduced.

Keywords: PV plant, performance assessment, daily energy generation estimation, data preprocessing,

weather conditions

Nomenclature:

|dG/dt| Absolute value of the derivative of irradiance

a,b Coefficients for the calculation of the expected power using Gβ

Apv Area of PV modules (m2)

Eest Daily energy estimation of PV modules (kWh)

Emeas Daily measurements of PV energy generation (kWh)

F1, …, F6 Indices defined in Appendix

f1, f2, f3 Constants to find ηc (%)

F7 7th index defined in equation (4)

G0 Extraterrestrial solar irradiance (W/m2)

G0,i Extraterrestrial solar irradiance of ith sample in a day (W/m2)

G0N,i Normalized value of extraterrestrial solar irradiance of ith sample in a day (W/m2)

Gd The deviation between G0 and Gs (W/m2)

Gs Solar irradiance on horizontal plane (W/m2)

Gs,i Solar irradiance on horizontal plane of ith sample in a day (W/m2)

GsN,i Normalized value of solar irradiance on horizontal plane of ith sample in a day (W/m2)

GSTC Solar irradiance at STC, 1000 W/m2

Gβ Measured solar irradiance on module surface (W/m2)

Gβ,i The measured solar irradiance on module surface (Gβ) of the ith sample (W/m2)

i The number of sample

j The number of day

Kfluc Fluctuation coefficient

kT Temperature coefficient (/C)

N Number of samples in a day

3

Nd Number of days in test dataset

Padj_STC The output power adjusted to a constant PV cell temperature of 25C (W)

PDC_est Estimation of output power of PV modules (W)

PDC_meas PV output power measured by inverters (W)

Pinv Observed output power of one inverter normalized over all inverters at a time point (W)

PSTC Output power of PV modules tested at STC (W)

Q1 First quartile

Q3 Third quartile

resE Relative residual between Emeas and Eest (%)

Ta Ambient temperature (C)

Tc PV cell temperature (C)

TSTC Operating PV cell temperature at STC, 25C

Vf Wind speed (m/s)

w Mounting coefficient

αMP Temperature factor of maximum power (%/C)

ηc PV efficiency (%)

ηref Reference PV efficiency at STC (%)

Abbreviation

CPR Corrected Performance Ratio

FA Firefly Algorithm

GA Genetic algorithms

GWO Grey Wolf Optimizer

IQR Interquartile range (Q3–Q1)

MPP Maximum Power Point

MPPT Maximum Power Point Tracking

PCA Principal Component Analysis

PR Performance Ratio

PSO Particle Swarm Optimization

PV Photovoltaic

RBF Radial Basis Function

RE Relative Error

RMSD Root Mean Square Deviation

RMSE Root Mean Square Error

SCA Sine Cosine Algorithm

SSA Salp Swarm Optimization

STC Standard Test Conditions, solar irradiance=1000W/m2, cell temperature=25C, Air Mass 1.5

SVM Support Vector Machine

1. Introduction

While the cost of photovoltaic (PV) energy is dropping, improving energy production over the life of a

plant is still a key factor in building the business case for a PV plant. Different factors such as shadowing,

soiling, aging of modules, and faults in the components can lead to a significant loss of energy production if

not detected and corrected in time [1].To this end, performance assessment can potentially improve PV plant

economics by identifying the need for timely corrective actions, and thus, preventing or reducing economic

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loss [1]. Generally, an indicator is selected to represent the performance of a solar plant. Performance

assessment systems are designed to compare the measured indicator with its “potential” value, and produce

an alarm if there are meaningful differences. Potential values of the selected indicator are generally defined

as the expected values of this indicator in normal operation. Normal operation is when the system is

physically intact, there is no soiling, and no unaccounted interruptions interfere with energy production.

Systems with micro-inverters or charge controllers allow assessment of an individual module

performance by observing the measurements of micro-inverters [2]; for PV systems with central or string

inverters, performance assessment is more difficult but equally important. Various methodologies for PV

systems with central or string inverters have been reported [3-6]. Some approaches use statistical analysis of

measurements as the performance indicator, (e.g., conversion efficiency, i.e., the ratio between output power

and incident solar irradiance [3]). Characteristics of a plant’s current-voltage (I-V) curves are also used [4].

Some others are based on comparison of normalized measurements from adjacent PV strings that are

measured simultaneously to detect a fault in one of the strings [5,6].

Another group of performance assessment algorithms is based on the comparison between measured

and modeled energy generation to identify anomalies [1,5,7-12]. For these methods, an accurate estimation

of the potential electricity production in normal operation is of great importance [11-14]. However, a large

number of unpredictable conditions, which affect the PV performance, pose a serious challenge to

developing a reliable model for estimation of PV output, and call for further research. Thus, different

techniques such as physical models [5,7,8], regression models [1,11,13,15], artificial intelligence techniques

[16] or commercial simulation packages [17] all use estimation models for performance assessment of PV

plants. Model development can be based on information about the physical characteristics supplied on a

manufacturer datasheet and necessary meteorological measurements along with adjustable parameters

[5,7,8], remote monitoring obtained from meteorological and satellite observation data [9] or historical

measurements [1,11,13,15]. Fault classification and detection can rely on the rate of power or energy losses,

i.e., the relative deviation of the measured production from the expected value [1,5,7-12].

In this paper, a methodology is proposed to improve the estimation of the potential energy production

of a solar PV plant for performance assessment. A PV plant consists of PV modules, junction boxes, inverters,

sensors, communication devices, etc. In this paper, we focus only on the performance assessment of the PV

modules. A PV performance assessment system typically consists of: (i) measuring PV power and

meteorological variables, (ii) modeling PV output and estimating the potential energy production, and (iii)

performance assessment of PV modules. In this paper, we focus on the second part.

The proposed methodology is focused on two particular issues: making the data preprocessing method

robust against measured abnormal values in normal operation such as erroneous measurements; and

capturing the characteristics of PV behavior in different operating conditions. In the next section, we provide

further elaboration on why data preprocessing and considering weather conditions are important.

Accordingly, the main contributions of this paper are: (i) a data preprocessing method is designed to be

robust against erroneous measurements in normal operation; (ii) sub-models of PV plants are developed for

different weather conditions to improve the estimation of energy generation.

The rest of this paper is organized as follows. In Section 2, the background of existing approaches to

improve potential output estimation for performance assessment is introduced. Section 3 describes the

modeled PV plant and analyzes the specific aspects of the measurements used in developing the proposed

method. Section 4 presents the proposed data preprocessing method and the sub-models development in

different weather conditions, as well as the other stages of the performance assessment of PV systems. In

Section 5, numerical results and discussions are presented. Finally, Section 6 concludes the paper.

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2. Background

This section presents a literature review of existing approaches improving potential output estimation

of PV plants for performance assessment. There are two commonly used approaches: data preprocessing and

sub-models development.

2.1 Data preprocessing

Data preprocessing is widely applied to exclude outliers from data sets used for developing the model

of PV modules for performance assessment, since that measurements from a PV plant are not always suitable

for model development. For example, the incident irradiance values on certain PV modules would be reduced

significantly by partial shading on PV arrays; and therefore measurements with low irradiance values are

eliminated in the presence of shading to avoid its influence [18]. Observations corresponding to zero power

output have also been removed because they are useless for model development [10]. In addition, some

common erroneous measurements have also been dealt with. For instance, the maximum power point

tracking (MPPT) capabilities of inverters are frequently poor to track the maximum power point (MPP) when

the irradiance changes rapidly [1,19]. Besides, in relatively large PV plants, the local pyranometers are

aligned and maintained regularly. Nevertheless, for residential or commercial PV systems, the pyranometers

may not be aligned with the PV modules; and the analyst may be unaware of the inaccurate measurements

[19].

In [11], the outliers were identified through visual inspection. In [1], the MPPs of PV modules are

roughly estimated using parameters in the datasheet. Then measurements differing by more than ±10% of

the estimation results are eliminated. In [19], the linear relationship between the MPP current and the solar

irradiance is used to check if the pyranometer is aligned with PV modules. The measurements with a non-

linear current-irradiance characteristic are eliminated. In [12,19,20] only the data for clear days were used

for modeling to avoid more uncertainty in changing weather conditions. However, these methods are not

robust against those erroneous measurements in normal operation, and may lead to a possible false alarm.

2.2 Sub-models development

In order to better estimate PV potential output, sub-models of PV plants for performance assessment

are developed in literature. The PV efficiency changes with the irradiance and cell temperature. In [11],

multiple models corresponding to different sunlight levels perform more accurately than a global model. In

[15], regression of different models over irradiance intervals is applied as a piecewise approach to capture

the variability owing to irradiance levels. In [21], the regression model is developed at different irradiance

values ranging from 0 to 1200 W/m2 in 10 W/m2 steps; as well as ambient temperature values from -10 to

+40C in 5C steps .

For PV power forecasting, weather is widely taken into consideration [22, 24]. Solar radiation is stable

on sunny days, but can fluctuate significantly on cloudy and rainy days [ 23 ]. PV power responds

instantaneously to the changes of solar irradiance, accordingly, is intermittent and undispatchable when solar

irradiance is varying [23]. As a result, the forecasting accuracy may be significantly different for sunny days

versus cloudy days [24].

In terms of PV performance assessment, weather is also important. The accuracy of PV models strongly

depends on the accuracy of measurements used for model development. If the pyranometer is misaligned

with the PV array, the errors of solar irradiance measurements are different on a sunny day and a partly

cloudy day. This is analyzed in Section 3.2. Besides, MPPs can be better identified on a sunny day. On a

partly cloudy day, MPPT may not be able to track MPPs accurately due to the fast changing irradiance [1],

which directly influences the accuracy of the developed model.

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3 Data analysis

In this section, we introduce the PV plant and provide a discussion on its historical data used for

numerical simulations. Our method is proposed based on the discussions.

3.1 PV plant

The methodology is developed and tested using historical measurements from a 9.8 MW roof-mounted

PV plant located in Southeast China (longitude 120°33’37”E, latitude 30°35’53”N, altitude 46m). It has a

subtropical monsoon climate with two seasons: a rainy summer and a dry winter.

Figure 1 shows the layout of the PV plant consisting of 38,786 TSM-PC05A polycrystalline silicon

modules installed on the roofs of eight buildings marked in the figure. Every 22 modules are connected in

series to form a string and every 16 strings are in parallel and connected to one junction box. The plant

contains 17 inverters. The numbers of junction boxes connected to the inverters are different.

Fig. 1 The overall layout of the PV plant

The characteristics of the solar module are in Table 1. The electrical parameters are found in the

datasheet provided by the manufacturer. The modules have been in operation since completion and

commissioning before Jan, 2014. In this case study, we use DC output power of inverters and local

meteorological data collected from June 24, 2015 to June 23, 2016. Therefore, the modules have been light-

exposed for more than one year before the start of the dataset, and the performance of modules is relatively

stable. The power generation of the PV modules degrades by 0.8% every year. We did not consider the

impacts of the degradation because the duration of the data is relatively short over which the degradation is

negligible.

For longer term use, the impact of performance degradation needs to be considered. The performance

degradation of PV modules is commonly caused by two types of reasons. The first type is long-term use

which is unavoidable and affect all the PV modules in the PV plant. The presented method does not aim to

detect this type of performance degradation. Consequently, data with this type of degradation will not be

considered as outliers. One potential solution to deal with this issue is to fit the model annually to update the

parameters of the model. The second type is damage on certain PV modules, such as cracks. In the data

preprocessing stage, which is presented later in Section 4.2, the comparison among the measurements of DC

output power of the different inverters is able to identify the salient performance degradation due to damage

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on certain modules.

Table 1 Characteristics of PV module

Manufacturer Trina Solar

Type Polysilicon

Model TSM-PC05A

Peak power 255 W

Reference PV efficiency (ηref) 15.6%

Temperature coefficient of power (αMP) -0.41%/C

The PV modules are installed close to the flat roofs which are all tilted at the same angle as the latitude

(30°35’53”) and facing due South (azimuth = 180°). There is no mutual shading or shading by adjacent

objects.

The power of PV modules measured by inverters and the meteorological measurements (one

pyranometer and one anemometer) are both averaged and sampled every minute. The measurements and

equipment are shown in Table 2.

Table 2 Data and Measurement Methods

Measurements Equipment

Power of PV modules MPPT of inverters (Sungrow SG630MX)

Solar irradiance Delta ohm LP PYRA 02 Pyranometer

Ambient temperature Pt100 (class B) temperature sensors (-50C to +70C)

Wind speed Thies Clima small wind sensor (0-40m/s) at 2m above of the PV array

3.2 Discussion of practical issues with measurements

The implementation of the PV model development poses several challenges that we discuss in this

section. One such problem is erroneous measurements. Among all the measurements used for PV model

development, the estimation accuracy is strongly dependent on irradiance and PV array output power

measured by inverters. For those PV plants without regular checks, erroneous measurements of irradiance

may be due to a misaligned or faulty pyranometer. Errors may likewise occur on the measurements of PV

array’s MPPs, since that the MPPT may not track the real MPPs precisely in fast changing irradiance

conditions [1,19]. This section analyzes the error sources of the used PV plant. Based on the analysis, our

methods are proposed.

1) Erroneous measurements of irradiance

The measured solar irradiance and output power of one inverter on a sunny day and a partly cloudy day

are presented in Figure 2. Observe the peaks of the measured solar irradiance and output power occur at

different time points on the sunny day while synchronous on the partly cloudy day.

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a. Mar 1, 2016, sunny day, the peaks are indicated by vertical lines

b. Nov 3, 2015, partly cloudy day

Fig. 2. Measured output power and solar irradiance

There are several reasons to explain the difference between the two figures. The PV power peaks at

12:18, and the measured solar irradiance peaks at 12:36 in Figure 2-a. The real local solar noon was 12:18

on March 1, 2016. This indicates a possible error in the measurement of the solar irradiance. This error could

be caused by an error in the azimuth angle of the pyranometer relative to the PV array. Note that solar

modules receive more direct solar radiation in clear days and more diffuse solar radiation in cloudy days.

Direct solar radiation is easily affected by the changes of the azimuth angle, while diffuse solar radiation is

barely affected by it. Therefore, the time difference between the peaks of the measured solar irradiance and

power is more obvious on a sunny day.

On the other hand, PV cell temperature strongly affects the PV performance, which should also be

analyzed. PV cell temperature (Tc) can be estimated using:

0.32

( )8.91 2

c a

f

T T w GV

(1)

Where Ta is ambient temperature, Gis irradiance incident on PV modules, Vf is wind speed, and w is

mounting coefficient and given as 1, 1.2, 1.8 and 2.4 for free standing, flat roof, sloped roof, and façade

integrated installation types, respectively [25]. We adjust the measured DC power of inverters (PDC_meas) to

a constant cell temperature of 25°C (TSTC), which is the Standard Test Conditions (STC), using the

temperature coefficient of maximum power (αMP). The adjusted power (Padj_STC) is calculated using:

_

_1 ( )

DC meas

adj STC

MP C STC

PP

T T

(2)

Figure 3-a depicts the measurements of G, Vf and Ta for March 1, 2016 which was a sunny day. Figure

3-b shows the adjusted power at STC versus the measurements of power. The power at STC peaks at 12:19,

which coincides closely with the local solar noon. This indicates that the PV modules are installed facing

south.

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a. Measured meteorological parameters

b. Measured power versus adjusted power at STC

Fig. 3 Mar 1, 2016, Meteorological measurements and output power

In order to further validate the azimuthal misaligned pyranometer, we used TRNSYS software [26] to

simulate irradiance on two surfaces with different azimuthal angles. The results showed that an azimuthal

angle difference of 25° between the two surfaces would lead to a displacement of 18 min between the peaks

of the two simulated curves of irradiance. Accordingly, we concluded that the pyranometer was misaligned

with an azimuthal angle error of about 25° to the PV array.

2) Erroneous measurements of output power measured by inverters

The accuracy of the power of MPPs measured by inverters may be low due to poor MPPT when the

irradiance changes quickly [1, 19]. As shown in Figure 4, the solar irradiance has a quasi-linear dependence

with the adjusted power of the PV array at STC on a partly cloudy day. However, there are some points

which lie relatively dispersed, especially those with fast changing irradiance. It is because the MPPT is not

able to track the MPPs accurately on a partly cloudy day due to the fast changing irradiance [1]. We use the

absolute value of the derivative of irradiance to quantify the fluctuation of irradiance, which is denoted by

|dG/dt|. The value of |dG/dt| of the ith sample, i.e., |dG/dt|i, is calculated by equation (3).

, , 1

0 1

1

i

i i

idG dt

G G i (3)

where i denotes the ith sample; and Gβ,i denotes the measured solar irradiance on module surface (Gβ) of the

ith sample. Note that a majority of data with high values of |dG/dt| cause non-linear power-irradiance

characteristics in Figure 4.

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Fig. 4 Nov 3, 2015, example of output power at STC versus solar irradiance on a partly cloudy day

In summary, the erroneous measurements come from two sources: (i) the azimuthal misalignment of

the pyranometer relative to the PV array, and (ii) poor MPPT for fast changing irradiance. As stated in

Section 2.1, the common data preprocessing methods may detect the erroneous measurements in normal

operation as outliers, and thus lead to false alarms. Therefore, there is a need to develop a proper data

preprocessing method which is not only able to exclude outliers but also robust against the erroneous

measurements in normal operation, which is the focus of this paper.

3) Data analysis using Corrected Performance Ratio (CPR)

We need to investigate whether there are anomalies among the collected raw data primarily for further

research. The most important metric for the performance assessment of PV plants is Performance Ratio (PR)

defined in IEC 61724 [27]. PR is defined as follows [18]:

_

1

1

( )

N

DC meas

i

N

STC STC

i

P

PR

P G G

(4)

where PSTC denotes the output power of PV modules at STC; GSTC denote the STC irradiance, i.e., 1000W/m2;

and N denotes the number of samples within a day.

Since the definition of PR does not incorporate any factors except irradiance，a more precise metric

daily CPR considering the effects of PV cell temperature [20] is used instead, which is defined as follows:

_

1

1

( )(1 ( ))

N

DC meas

i

N

STC STC MP C STC

i

P

CPR

P G G T T

(5)

CPR values are compared to a predefined threshold to detect significant energy production loss. The

CPR values of the collected dataset are shown in Figure 5. The threshold of CPR is determined of 70% which

is related to its empirical base. There are three days detected as with significant degradation of energy

production: January 27, 2016; February 1, 2016 and February 2, 2016. Note that some CPR values exceed 1

in Figure 5, which seems to be unreasonable, but most of them are caused by the erroneous measurements

of irradiance whose values are lower than the real values.

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Fig. 5 Daily CPR of the PV plant

4 Methodology

In this section the proposed framework for estimation of daily PV energy production for performance

assessment is introduced.

4.1 General Framework

The flowchart of performance assessment is shown in Figure 6, in which the proposed PV power

production model is highlighted in red. Briefly, the proposed method to model and estimate potential PV

power production is divided into three steps:

Step 1) Data Collection and preprocessing: Data for 240 days are available from June 24, 2015 to June

23, 2016. The collected data are divided into a training dataset and a test dataset. Because there are merely

three days with significant degradation on CPR, these three abnormal days have been placed in the test

dataset in order to validate the effectiveness of the proposed method on detecting anomalies. The remaining

data are divided randomly in a uniformly distributed manner. The training dataset contains 167 days (about

70% of the total data), and the test dataset contains 73 days (about 30% of the total data). Then the proposed

data preprocessing approach is applied on the training dataset to eliminate outliers.

Step 2) Weather classification: In this stage, a weather classification model is constructed to classify

the weather conditions for each day. The features used to train the classification model are extracted from

solar irradiance measurements. Then Principal Component Analysis (PCA) is used to select the most

informative features. A machine learning algorithm, i.e., Support Vector Machine (SVM) is used to develop

a classifier.

Step 3) PV system modeling: In this stage, sub-models for different weather conditions are developed

using least square fits.

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Fig. 6 Structure of PV performance assessment

The performance assessment block is not the focus here, but it is used to demonstrate the proposed

method. This block is further discussed in Section 4.5.

4.2 Data preprocessing

The model development requires a pre-filtering of the collected historical data. Observations

corresponding to low irradiance values are eliminated for which the measurement accuracy is significantly

reduced [11]. Accordingly, we define the lower limit of irradiance values as 400 W/m2, and eliminate data

with irradiance values less than 400 W/m2. The limit of irradiance is found by analyzing the regression

results of PV output power, which are based on the quasi-linear relationship between PV power and

irradiance at a range of irradiance values. We use different lower limits of irradiance values ranging from

100 to 800 W/m2 in 100 W/m2 steps. Different regression models of PV power are then developed using data

after filtering the observations with irradiance values lower than the different limits. Our findings showed

that the deviations of the estimated PV power values calculated using the developed regression models from

the real measured PV power decreased when the lower limit of irradiance increased. However, this

downward trend flattened when the lower limit of irradiance higher than 400 W/m2. Besides, the number of

data points used for model development in some days would be too small when using a lower limit of

irradiance higher than 400 W/m2 for data filtering. Accordingly, the lower limit of irradiance is roughly set

to 400 W/m2.

Subsequently, we compare the outputs of different inverters measured simultaneously to detect outliers

[5, 6], which aims to be robust against the erroneous measurements in normal operation. Note that the

erroneous measurements of irradiance have equal effect on the overall PV plant, not certain inverters.

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Therefore, the erroneous measurements of irradiance will not be detected by comparing the outputs of

different inverters. Accordingly, this method is robust against the erroneous measurements of irradiance. The

fast changing irradiance may have different effects on the MPPT techniques of different inverters. It may

lead to different degrees of error on the measurements of DC power of inverters. Then some of the erroneous

measurements of power may be detected as outliers by the proposed data preprocessing method. However,

it is hard to quantify and evaluate the different errors. This issue poses a challenge for the proposed data

preprocessing method to be robust against the erroneous measurements of power. We further discuss it in

Section 5.1 in details.

In this paper, we use boxplot rule for comparison among normalized DC power of all the different

inverters that are measured simultaneously. The boxplot rule classifies an observation as an outlier if

3 1.5invP Q IQR or1 1.5invP Q IQR [3]. Here, Pinv is the observed power of one inverter normalized over all

inverters at a time point; Q1 and Q3 are the first and third quantiles of all values of Pinv at this time point; and

IQR is the interquartile range (i.e., Q3-Q1).

It is noteworthy that the boxplot rule may erroneously classify normal data as outliers. For example,

given the area of the plant, some may be shaded by clouds but not others. In this case, the shaded modules

may show a significant degradation in output power with respect to other inverters such that the data of the

shaded modules are classified as outliers. To reduce this mistake, we consider that multiple consecutive

points outside the limits represent an outlier [11]. This strategy reduces the probability that brief low power

production periods are falsely considered as outliers. At the same time, it will impact the outlier detection

rate as well.

A proper selection of the number of the multiple consecutive points can reduce the negative influence

on the outlier detection rate of the strategy. An analysis of the shading periods caused by clouds in the

irradiance data reveals that the duration of shading periods hardly exceed five minutes [28]. Therefore, we

propose the strategy that five consecutive points outside the limits represent an outlier.

The irradiance and power measurements follow a strong linear relationship in normal operation at a

range of irradiance values [11]. In order to distinguish the erroneous measurements in normal operation from

abnormal measurements according to faults, we apply a rough and simple rule based on the Relative Error

(RE) between the output power adjusted to 25C and the expected power [19]:

_ ( )

( )

adj STCP aG bRE

aG b (6)

15% 15% RE (7)

where a and b are coefficients calculated by least square regression with the training dataset in normal

operation; the expected power is roughly estimated using the measured solar irradiance, which is denoted by

aGβ+b. As noted earlier, the erroneous measurements cause non-linear power-irradiance characteristics, so

does faulty data [11]. What differs from the erroneous measurements in normal operation to faulty data is

the degree of the deviation of the measured power from the expected power. Commonly, the faulty data will

lead to more significant RE than the erroneous measurements in normal operation do. Thus, we need to leave

some room for the erroneous measurements when detecting outliers. The data points inside the ±15% limits

are considered as normal operation; otherwise, they will be detected as anomalies and excluded from the

training dataset.

Figure 7 shows the collected data and the ±15% limits. The detected faulty data are mostly below the -

15% limit. Some faulty data represent the faults due to which the PV system produces significantly less

power than it should at the given irradiance level. Besides, faults may happen in the process of data recording

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and storage. In these circumstances, the records of measurements are commonly completed using

interpolation methods. Consequently, it may cause significant errors when there is a large time difference

between two discrete known points used for interpolation. For example, the faulty points fall in the green

ellipse are such interpolated points.

Fig. 7 Faults detection based on the linear irradiance-power characteristics.

Moreover, the outliers in the meteorological measurements are also excluded during the fitting process

of the regression model. At first, the PV model is fitted using the whole training dataset after excluding

outliers in the measurements of PV output power. Then the points whose confidence intervals of residuals

do not cover zero are considered as outliers, and are excluded from the training dataset. The process is

repeated until there is no outlier.

4.3 Classification based on weather identification

The goal of this stage is to classify measurements into different categories, and develop sub-models to

better estimate the energy production of a PV plant. Since the relation between solar irradiance and PV power

varies significantly with different weather conditions for the case described in Section 3.2, sub-models are

developed for different weather conditions. In order to assign each measurement correctly, it is necessary to

choose proper attributes of objects based on which the discrepancy between different categories of

measurements is maximized. Then we propose a PCA plus SVM method for classification of weather

conditions.

4.3.1 Feature extraction

In this stage, inputs to the classifier called feature vectors are extracted to represent each class. The

features are extracted from the incident solar irradiance on PV modules in this work to represent weather

conditions, referring to [23].

The distinctions between global solar radiation on the earth’s surface Gs and the extraterrestrial solar

radiation at the top of aerosphere G0 reflect the variation of solar irradiation and weather conditions [23].

Indexes defined by the extraterrestrial and surface solar irradiance are therefore used as feature vectors to

describe different weather conditions.

The extracted features are defined in [23] in detail. The first feature, named clearness index, is the

proportion of the extraterrestrial solar radiation passing through the aerosphere to the earth’s surface. The

second one is the root mean square deviation (RMSD) of G0 and Gs, describing the shape difference between

them. The third one is the maximum value of the 3rd-order derivative of Gd ( 0d sG G G ), related to the

fluctuations of Gs. The ratio of maximum Gs to maximum G0 is selected as the fourth index. The fifth index

is the mean of the sum of squares of the differences of Gd from the mean of Gd. The sixth index describes

the variation tendency inconsistency of Gs and G0.

15

In addition to the six indexes described in [23] (see Appendix), the seventh index denoted byflucK is

the indicator proposed in this paper. It is defined by

1

7 ,

2

, , 1 , , 1

,

, , 1 , , 1

1 0

0 0

N

fluc i s i

i

s i s i s i s i

i s i

s i s i s i s i

F K ff G

where

if G G G Gff G

if G G G G

(8)

where Gs,i-1, Gs,i, Gs,i+1 denote Gs of the i-1th, ith and i+1th samples respectively. Note that the function

,i s iff G is the indicator describing the fluctuation tendency of Gs in the sampling interval [i-1, i+1]. If

, 0i s iff G , Gs does not fluctuates at the sampling point i; and if , 1i s iff G , it does. The fluctuation

tendency of Gs in a day can be measured by F7; the bigger the fluctuations of Gs in a day, the bigger the value

of F7 for this day.

4.3.2 Classification model

In this stage, we implement PCA plus SVM to develop a classifier for better estimation of energy

production considering weather conditions. The classification model is implemented in MATLAB.

1) Data dimension reduction

In this step, the PCA algorithm is used for variable selection to avoid overfitting of the classification

model [29]. PCA is a statistical procedure that uses an orthogonal transformation to convert a set of

observations of possibly correlated variables into a set of values of linearly uncorrelated variables called

principal components (PC). The importance of a variable in a PC model is indicated by the size of its residual

variance [30]. In this paper, the candidate PCs are ranked, and then the first 95% of the candidate PCs are

selected. The 95% limit comes from trial and error and should be carefully chosen based on the available

data.

2) Classification model development

In this step, SVM is used to develop a classifier to classify measurements into accurate groups. Note

that SVM requires inputs whose category membership is known [23]. The local weather can be divided into

two typical types: sunny and cloudy. The relationship between the output power of PV modules and the

measured solar irradiance in a sunny day is different from it in a cloudy day in the case in Section 3.2. An

example is given in Figure 8 to demonstrate this.

16

a. Mar 1, 2016, sunny day

b. Nov 3, 2015, partly cloudy day

Fig. 8 Examples of weather classification based on the linear irradiance-power characteristics.

Figure 8-a shows that the misalignment of the pyranometer could cause a non-linear power-irradiance

characteristic on a sunny day because the measurements of irradiance are different with the incident

irradiance on the PV modules. Besides, the deviations of the values of the measurements of irradiance from

the incident irradiance on the PV modules occur at the global irradiance range. As for partly cloudy days, it

is low accuracy in measuring the PV array maximum power due to poor MPPT of the inverter merely when

irradiance changes fast. Then nonlinearities between power and irradiance shows at the points with the fast

changing irradiance. Accordingly, the daily percentage of the data with non-linear power-irradiance

characteristics on a sunny day is higher than it on a partly cloudy day.

As noted above, the training dataset can be classified as a sunny day or a cloudy day based on the linear

power-irradiance characteristics. To this end, we use RE between the measurements of power and the

expected power, and set limits of ±5% for RE to classify weather conditions:

_ ( )

( )

adj STCP aG bRE

aG b (9)

5% 5% RE or RE (10)

where the coefficients a and b used for equation (9) are fitted by least square regression with daily

measurements without excluding low irradiance values.

RE is a metric to measure the linear relationship between the measurements of irradiance and output

power of PV modules. The greater the absolute value of RE is, the further the data away from the irradiance-

power straight line is. We consider that if the daily percentage of the data coincident with the rule in equation

(10) exceeds a threshold set to 70%, that analyzed day is considered as a sunny day and labeled as Class 1.

Or else it will be considered as cloudy and labeled as Class 2.

We use Figure 8 to explain the selection of the ±5% limits and the threshold of 70%. More than 70% of

the data points on a sunny day exceed the ±5% limits owing to inaccurate measurements of irradiance, while

less than 70% of the data points on one partly cloudy day outside the limits.

For a non-linear classification, SVM uses kernel functions mapping the inputs into high-dimensional

feature spaces to convert a not separable problem to a separable problem [23]. In the present work, a Radial

Basis Function (RBF) kernel is applied.

To optimize the result of classification by SVM, two parameters, i.e., limiting term and the kernel

parameter can be controlled. In the present work, these two parameters are selected using Genetic algorithms

17

(GA). The application of SVM plus GA are introduced in detail in [31].

4.4 Derivation of the output estimation equation

We apply linear regression models based on historical measurements for estimation of energy

production of a PV plant in this work for their lower degree of complexity and high accuracy [11]. Generally,

the PV efficiency (c ) [25] can be estimated as:

[1 ( )]Mc ref c STCP T T (11)

where ηref is the reference PV efficiency under STC listed in Table 1.

We modify (11) to allow c to be estimated using Gβ, Ta and Vf. Tc is estimated from equation (1).

Thenc can be formulated as:

0.32

(1 ) ( )8.91 2

c ref MP STC ref MP a ref MP

f

T T w GV

(12)

Then simplify the symbolic constants in equation (12) by:

1 (1 )ref MP STCf T (13)

2 ref MPf (14)

3 0.32 ref MPf w (15)

Finally, equation (12) is simplified to:

3

1 28.91 2

c a

f

f Gf f T

V

(16)

The actual conversion efficiency of PV modules can be calculated by equation (17) where Apv denotes

the area of the PV array:

_

(%) 100DC meas

c

pv

P

A G

(17)

Accordingly, the coefficients f1, f2 and f3 of (16) are determined based on the calculated c and the

measurements ofaT , G

and fV by the least squares regression method. Then the estimation of the daily

energy production of the overall PV modules (Eest) in a PV plant on a new day is calculated based on the

estimated ηc by:

1

( )N

est c pv

i

E A G

(18)

4.5 Performance assessment of PV systems

In this stage, we apply the method which sets normal operation limits on the residual between the

estimated and measured daily energy production (Emeas) to detect abnormal operations. The residual (resE) is

formulated as:

18

_

1

1

( )

100%

N

c pv DC meas

est meas i

E N

estc pv

i

A G PE E

resE

A G

(19)

The normal operation limits are determined by analysis of the probability distribution of residuals. If

the calculated value of residual lies outside the normal operation limits, this point will be labelled as

abnormal.

5 Numerical Results

In order to prove the effectiveness of the proposed data preprocessing and the sub-models, we use them

for performance assessment of the PV modules described in Section 3. Moreover, several data preprocessing

and model development methods in literature are applied for comparison. We evaluate the applied methods

from two aspects: (i) the accuracy of the estimation of the potential energy output, and (ii) the results of

performance assessment of PV modules.

5.1 Results of estimation of the potential energy output

In this section, the proposed methods are used to estimate the potential energy output for the test days.

1) The results of the proposed data preprocessing method

We applied the proposed data preprocessing method to the training data set to exclude outliers. The

distribution of the percentages of the detected outliers of one inverter within one day is shown in Figure 9.

Note that in most cases, the percentages of the detected outliers in one day are less than 2%.

It is noteworthy that larger values of the daily percentages of the detected outliers occur a little more

frequently on cloudy days than on sunny days. As mentioned in section 4.2, the proposed data preprocessing

method may detect erroneous measurements of power of certain inverters as outliers. The erroneous

measurements of power occur mostly on cloudy days when irradiance changing fast. Therefore, the detected

outliers on cloudy days include some of the erroneous measurements of power, while the detected outliers

on sunny days do not. However, the difference between sunny days and cloudy days is not significant.

Consequently, the erroneous measurements of power on cloudy days have not been extensively detected as

outliers. From this point of view, the proposed data preprocessing method is quasi-robust against the

erroneous measurements of power.

Fig. 9 Distribution of the percentages of the detected outliers within one day

As an illustration, the daily result of the proposed data preprocessing method of one inverter is shown

in Figure 10. This unit’s data for February 3, 2016 is chosen to present the result of data preprocessing since

it has a large number of outliers, i.e., 10.89% of the data are detected as outliers.

19

Fig. 10 Identification of outliers in February 3, 2016

Figure 10 shows that the upper bound and lower bound fluctuate strongly from 9:00 to 10:30. It is

probably caused by large changes of incident irradiance on the PV arrays due to moving clouds. Then the

bounds change more slowly from 10:30 to 13:00 since irradiance tends to be smoother then. However, the

DC power of the inverter still fluctuates. Accordingly, the points outside the limits from about 10:30 to 13:00

are detected as outliers, which may be caused by system faults or interpolation errors. Meanwhile, the

abnormal measurements caused by the fast changing irradiance but normal operation from 9:00 to 10:00 are

kept. Thus, the outliers owing to possible abnormal operation are detected, and erroneous measurements in

normal operation are kept.

2) The comparison of estimation results with other methods

In order to demonstrate the effectiveness of the proposed method, we also apply other methods for

comparison. Another data preprocessing method, i.e., excluding erroneous measurements owing to poor

MPPT of the inverters when the irradiance changes rapidly from the training dataset [12,19,20], is used to

demonstrate the effectiveness of the proposed data preprocessing method. Two other methods, i.e., a global

regression model and separate models for different irradiance ranges, are used to demonstrate the

effectiveness of the proposed sub-models. For the separate models for different irradiance ranges, we assume

that irradiance above 250W/m2 is high irradiance. If the percentage of time with high irradiance during a day

exceeds a threshold that is empirically set to 40%, the day is used to train the first model. The remaining

days are used to train the second model.

There are several scenarios by combinations of certain data preprocessing methods and model

development methods as follows:

Scenario (i) EECG: Excluding erroneous measurements by poor MPPT of inverters [12,19,20,32]

combined with the global model [11,33,34];

Scenario (ii) DPCG: The proposed data preprocessing method combined with the global model

[11,33,34];

Scenario (iii) DPCI: The proposed data preprocessing method combined with the separate models for

different irradiance ranges [11];

Scenario (iii) DPCW: The proposed data preprocessing method combined with the proposed sub-

models for varying weather conditions.

These methods are measured based on the estimation accuracy of output energy production of the PV

plant in this Section. We apply Root Mean Square Error (RMSE) to measure the accuracy of the models,

which is defined as:

20

2

1100%

dN

E

j

d

res

RMSEN

(20)

where Nd is the number of the test days, resE is the residual between the estimated and the measured daily

energy production of day j.

The RMSE values calculated for the four scenarios, i.e. EECG, DPCG, DPCI and DPCW, are 18.57%,

14.03%, 13.78% and 13.25% respectively. Observe that the proposed DPCW method performs better than

others.

Note that the RMSE value of DPCG method is lower by 4.54% than for the EECG method although

they use the same global model. This is mainly because the model of EECG is developed merely on data in

clear days after excluding the erroneous measurements of power in fast changing irradiance conditions.

Consequently, this model inaccurately estimates the energy generation on cloudy days. Thus, the DPCG

method using the proposed data preprocessing leads to a more accurate estimation of energy production of

the PV array.

Moreover, the proposed method for sub-models development considering weather conditions improves

the estimation of daily energy production compared to other methods. The improvements of the proposed

method are presented by comparing the estimation results of the proposed DPCW method with DPCG

method and DPCI method, since that they use the same data preprocessing method. Although the

improvement of DPCW method is not significant from a RMSE viewpoint, the effectiveness of the proposed

sub-models development method is demonstrated in the application of performance assessment of the PV

modules in Section 5.2.

We have also applied several metaheuristic optimization methods, i.e., Genetic Algorithm (GA) [35],

Particle Swarm Optimization (PSO) [36], Firefly Algorithm (FA) [37], Grey Wolf Optimizer (GWO) [38],

Sine Cosine Algorithm (SCA) [39] and Salp Swarm Optimization (SSA) [40], to DPCW method for

parameter optimization of the models. Compared to the least square fits method, the metaheuristic techniques

did not have significant improvement on the accuracy of the developed models, and operated slower than

the least square fits method did. However, this may not always be the case depending on the dataset. We

suggest the appropriate method for model fitting to be customized for the data at hand.

5.2 Results of performance assessment

We apply the estimated potential energy outputs for performance assessment of PV modules in this

section. The residual between the measured and the estimated daily energy production is used as the indicator

representing the power losses. The days with residual exceeding the predefined threshold are defined as

abnormal. Based on our analysis, over 80% of the residuals fall within the (-10%, 10%) interval. Thus, we

arbitrarily use this interval as the boundaries of normal operation.

Figure 11 depicts the results of the residuals of daily energy production (resE) expressed as a percent

for the test days. As Figure 11 reveals, residuals of the three days with abnormal operation, i.e., January 27,

2016; February 1, 2016 and February 2, 2016, exceed the 10% limit. Besides, it is noteworthy that these days

are among those detected by either of EECG, DPCG, DPCI and DPCW methods. However, the methods

falsely identify a number of other normal operation days as abnormal - see other days whose residuals fall

outside the ±10% bands.

21

Fig. 11 Performance assessment results using EECG, DPCG, DPCI and DPCW methods based on residual of daily energy

production

Thus, we determine the rate of false alarms for the models. False alarms are defined as those where a

normal operation day is identified as an abnormal operation day. High false alarm rate would result in a great

waste of human and material resources. Therefore, reducing false alarms is critical. There are 70 normal

operation points in the test dataset; the false alarms detected by EECG, DPCG, DPCI and DPCW methods

are 33, 13, 11 and 7 respectively. The proposed DPCW method leads to fewer false alarms compared to the

other approaches.

Comparison among the four scenarios can show the effectiveness of the proposed data preprocessing

method. The DPCG method and the EEPG method apply different data preprocessing methods, while they

both use the global model. The results show that, by using the proposed data preprocessing method, the

DPCG method reduces more than 70% false alarms when compared to the EEPG method.

Moreover, the proposed sub-models development shows significant improvements in the final

performance assessment results when compared to other methods. Note that the false alarms detected by the

DPCW method reduces, respectively, by 6 and 4 false alarms when compared to the DPCG and DPCI

methods. Since that DPCG, DPCI and DPCW use the same proposed data preprocessing method, the

improvements are made by the proposed sub-models development considering weather conditions.

Note that once the misalignment error of the pyranometer is quantified, the erroneous measurements of

irradiance can be corrected prior to developing the estimation model to further improve the estimation

accuracy. To this end, future work will primarily divide global irradiance into direct beam and diffuse

irradiance; then correct the azimuthal error on the direct irradiance. The division of global irradiation into

direct beam and diffuse irradiation can depend upon variables such as the clearness index, solar elevation,

atmospheric precipitable water, etc. [41].

6 Conclusion

This paper develops a method to improve the estimation of daily energy generation of PV modules for

performance assessment. Our method has two main contributions, i.e., a data preprocessing method and a

sub-models development methodology considering weather conditions. The proposed data preprocessing

method is robust against the erroneous measurements in normal operation. It identifies outliers in the

measured DC power of certain inverters by comparing the DC power of all inverters that are measured

simultaneously when there is no shading. Besides, we propose separate sub-models for sunny days and

cloudy days. The data are classified by a Principal Component Analysis and Support Vector Machine

algorithm.

22

The proposed method is evaluated using data from a roof-mounted PV plant located in Southeast China.

The results show that the proposed methods are capable to improve the performance assessment of PV

modules, when compared to other methods in literature.

Appendix. Equations used for feature extraction in weather conditions

identification

The following formulas are the first six indexes defined in [23]:

24 1

, 1 ,0 1

1 24 1

0, 1 0,0 10

( ) ( )

( )( )

N

s s i s ii

N

i ii

G t dt G GF

G GG t dt

(1)

2

2 0 , ,

1

1( )

N

N i sN i

i

F G GN

(2)

where 0,

0 ,

0,1, ,

100max

i

N i

ii N

GG

G

,

,

,

,1, ,

100max

sN i

sN i

sN ii N

GG

G

, 1, ,i N

3

0, ,

3 31, ,

( )max { }

i s i

i N

d G GF

dt

(3)

,

1, ,

4

0,1, ,

max { }

max { }

s ii N

ii N

GF

G

(4)

2

5 0, , 0, ,

1 1

1 1( ) ( )

N N

i s i i s i

i i

F G G G GN N

(5)

6 0 , 0,

2

, ,N

s i s i i

i

F f G G f G G

(6)

where

, , 1 0, 0, 1

, 0,

, , 1 0, 0, 1

1 0,

0 0

s i s i i i

i s i i

s i s i i i

if G G G Gf G G

if G G G G

Acknowledgment

We would like to thank the financial support from the Natural Science Foundation of China (61573046).

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