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A simple stochastic method for modelling the uncertainty of photovoltaic power production based on measured data
Accepted Manuscript A simple stochastic method for modelling the uncertainty of photovoltaic power production based on measured data M. Barukčić, Ž. Hederić, M. Hadžiselimović, S. Seme PII:
S0360-5442(18)31904-2
DOI:
10.1016/j.energy.2018.09.134
Reference:
EGY 13828
To appear in:
Energy
Received Date: 17 December 2017 Revised Date:
14 September 2018
Accepted Date: 19 September 2018
Please cite this article as: Barukčić M, Hederić Ž, Hadžiselimović M, Seme S, A simple stochastic method for modelling the uncertainty of photovoltaic power production based on measured data, Energy (2018), doi: https://doi.org/10.1016/j.energy.2018.09.134. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
ACCEPTED MANUSCRIPT A Simple Stochastic Method for Modelling the Uncertainty of Photovoltaic
Power Production Based on Measured Data
M.Barukčić*a, Ž. Hederićb, M. Hadžiselimovićc, S. Semed J. J. Strossmayer University of Osijek, Faculty of Electrical Engineering, Computer Science
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a,b
and Information Technology Osijek, Kneza Trpimira 2B, 31 000 Osijek, Croatia c,d
University of Maribor, Faculty of Energy Technology, Hočevarjev trg 1, 8270 Krško, Slovenia
a
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* Corresponding author
[email protected], (Telephone: +385 31 224 685, Fax: +385 31 224 605) [email protected], c [email protected], d [email protected]
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b
Abstract: This paper describes statistical quantification tools for predicting photovoltaic (PV) production considering uncertainty in PV production at same irradiation levels and PV panel
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temperatures. When analysing measured data, it is observed that there are different PV power production levels for the same irradiation levels and panel temperatures. These PV power spread out can be caused by different causes, such as dust deposition over the panel, non-ideal working
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of maximal power point tracking devices, device efficiencies’ dependence on power, different temperatures over the PV panel, and others. Due to the stochastic character of these occurrences,
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they can be challenging when considered in the deterministic mathematical models usually used for PV power prediction. The probabilistic method for PV power production is proposed based on the probability density function with respect to the solar irradiation and the panel temperature. The simulation results are compared among the different models based on the probability density function with respect to the solar irradiation and panel temperature. The best overlapping between measured and calculated PV power production gives the proposed stochastic models dependent only on irradiance. The proposed stochastic model gives a PV energy prediction on a yearly basis with an error of less than 1%.
ACCEPTED MANUSCRIPT Key words: curve fitting, photovoltaic power production, probability density function, irradiance ranges, uncertainty
Introduction
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1
Many factors impact the production of photovoltaic (PV) panels; solar irradiance and PV panel
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temperature have the greatest influence. As is well-known, these two impact factors are modeled with a deterministic mathematical model. However, due to the stochastic character of the energy
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source (solar irradiance), the probabilistic approach is often employed in this case. In [1], the PV production is predicted based on the solar irradiance estimated by using the probability density function of the clearness index. The probability density function is modelled based on the Bayesian approach here. The Bayesian approach to prediction PV energy production is used in
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[2] with aim of finding daily pattern profiles of the PV production. The time series method for PV power forecasting depends on the day of year and the time of day is presented in [3]. In [4], the authors use an Artificial Neural Network (ANN) with ambient and panel temperatures and
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irradiance as input data to obtain PV power as output from ANN. ANN is also used in [5–9] for PV power estimation. In [10], a combination of different time series and ANN models tuned by
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using Genetic Algorithm (GA) is used for PV power prediction. The Genetic Programming (GP) to evolve the fuzzy inference system for PV power production is presented in [11]. The historical data processed by data mining technique are used in [12] to estimate PV power production. In [13] and [14], support vector machines (SVM) using the numerically forecasting weather data is used for PV energy and power prediction. The methods for short-term prediction of the PV power generation are presented in [15]. The random generated PV power with respect to time of the day is presented in [16]. This literature was also one of the motivation for the authors to conduct the research presented here. Most of the literature mentioned used a deterministic model
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of PV power production considering the stochastic character of the intensity of the energy source. Such an example of PV production prediction using the deterministic calculations can be found in the very recent literature [17,18] also. In [19], the software tools using deterministic calculations are also used for PV energy prediction. The inverse model for prediction of the PV
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production based on measuring the weather conditions is given in [20]. The analytic expression for PV power production is used in [21], in which models for estimation of PV panel temperature are researched. Typically, the stochastic model of irradiance and temperature is
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researched with respect to weather conditions, the time of day, or the cloudiness index, but PV power generation is calculated by using a deterministic expression from stochastically
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determined irradiance and temperature. Some references research the stochastic in PV power itself as it is presented in [22–24]. The above given overview of the literature on the research prediction of the PV system production can be summarized as follows: prediction of the weather conditions (irradiance and temperature mainly) is researched in [1,3,4,15]; estimations of PV power using the weather conditions as input variables are presented in [5–10,13,14]; prediction
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of the PV production in the form of time series is presented in [10,11]. None of these sources researched the stochastic character of PV power for the same irradiance and temperature values.
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In other words, for same the irradiance and PV panel temperature, the output of PV system will always be equal, because the calculation of the PV power is deterministic. Distinct from the
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other studies, the research presented in this paper deals with PV production uncertainty in the case of the same irradiance and temperature input values, and the calculation of the PV power is stochastic. The research dealing with the uncertainty of the PV power production is presented in [12,16,23,24]. A stochastic model of PV production based on irradiance condition patterns (sunny, overcast, partly cloudy) is presented in [24]. In [12], [16], and [22], the uncertainty of the PV production is given depending on the period of the day (time). In [12] and [16], the parameters of the probability density function are time-dependent while linear combinations of the normal density functions (the Gaussian mixture method) with constant parameters are used
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in [23]. The stochastic model proposed here has irradiance and temperature dependent parameters of the probability density function, which is different from models presented in the cited literature. The impact of taking into account the PV production and other uncertainties on energy management in microgrids is presented in [25].
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During the analysis of the measured data of the PV plant, the authors observed that there are cases in which different PV powers were measured even though values of irradiance and PV panel temperature were the same (or very close each other). The causes of this can be due to dust
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deposition on PV panel surface [26], non-ideal work of MPPT, different spectral response of PV panel and PV cell used for the solar irradiance measurement, different temperatures over the PV
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panel, different air mass over time. Because of the stochastic character of these possible causes, they bring uncertainty to the estimation of PV production. This was the motivation to conduct this research, and it is presented in Fig. 1 and Fig. 5. As can be seen in Fig. 1, there are different levels of PV power generation even in case of the same the irradiance and temperature values. Furthermore, the shading effect can have a significant impact on PV production, as can be seen
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in [27]. Because this can be modeled by a deterministic model, as presented in [27] this impact is not considered here. The hypothesis is that the uncertainty of the PV power production can be
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quantified by using a probability density function that has parameters dependent on the solar irradiance. The proposed models can be used in probabilistic power flow in a distribution
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network containing PV systems to address uncertainty in PV system generation [28]. The statistical models of PV production can be of interest in case of the different optimization in networks with installed PV such is optimization considering the reactive power control presented in [29] or optimal control in hybrid microgrid [30]. Also, the importance of mathematical models that include uncertainty of PV power production can be found in [31–33]. The stochastic models proposed here are based on those presented in [22], and this is a continuation of the research presented in [22]. The probability density function parameters are determined based on historical measured PV plant data on a micro-location in Slovenia. In the present paper, the
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parameters of the probability density function are dependent on the irradiance and PV panel temperature, unlike the research presented in [22], in which only the dependence on the irradiance was considered. The contribution of the paper to the topic of PV power prediction is in proposing the new stochastic (probabilistic) model, which is distinct from much of the
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literature considering the deterministic calculation of PV power. The stochastic models of PV power estimation are presented in only small number of the literature, as can be seen from above overview. This paper also makes a contribution tp the topic of the stochastic models of PV
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production prediction. The innovation of the presented method is in proposing a stochastic model that has probability parameters depending on irradiance (and temperature), which is
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different than in the above-mentioned literature. The rest of the presented research is organized as follows: in Section 2, an overview of the proposed model is given; in Section 3. simulation results are presented and compared among the proposed models; in the last section a conclusion is given.
Defining Uncertainty Parameters from Measured Data
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The proposed stochastic model is based on next hypothesis: the uncertainties in PV panel
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production can be represented by random numbers. The random number is generated by using normal (Gaussian) distribution that has variable (no constant) distribution mean and standard
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deviations. The stochastic character of the PV system power generation can be observed from the presentation of the measured powers given in Fig. 1. As mentioned before, even in the case of the same values of the irradiance and PV panel temperature, the PV power production can be different.
79 W
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T = 36.83°C T = 36.23°C
84.98 W
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T = 39.63°C
Fig. 1.: Measured power generation of PV system over a year and detail for an irradiance value
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(mapped data are calculated according to (7) and represent normalized data with respect to the reference temperature)
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As mentioned before, in [22], the stochastic models considering the distribution parameters dependent only on irradiance are presented. Because those models are compared with the proposed method, the models are briefly represented here and details can be found in [22]. The mean Pmean of the normal distribution dependent on the solar irradiance G is approximated by a polynomial of the H order in both approaches in [22]: H
Pmean ( G ) = ∑ ah ⋅ G h h=0
where ah are polynomial coefficients of the power mean.
(1)
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Dependence according to (1) is stated based on the experimentally measured PV generated power. In addition, to determine the standard deviation dependent on the irradiance, the measured data set of the PV power is divided in a number of irradiance ranges according to Fig.2 [22] with
Gavr ,i =
Gmin,i + Gmax,i 2
; i ∈ (1L N r )
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range average irradiance as: (2)
where Gavr,i is the average value of the irradiance for each irradiance range, Gmin,i and Gmax,i are
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range bounds of the i–th range, and Nr is number of the ranges.
In the first approach used in [22] (hereinafter denoted as Mod A), the standard deviation in each
σ A,i =
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irradiance range is calculated as: Ni 2 1 ⋅ ∑ ( Pmeas ,i , j − Pmi ) N i − 1 j =1
(3)
where, Pmeas,i , j is measured the j-th PV power within the i-th irradiance range, Pmi is average
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values (mean) of the measured PV powers of the i-th irradiance range and Ni is number of measured PV powers in i-th irradiance range.
The second approach (hereinafter denoted as Mod B) is based on the standard deviation of the
Ni 2 1 ⋅ ∑ ( PdiffB ,i , j − PdiffB ,i ) N i − 1 j =1
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σ resB ,i =
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differences (residuals) between measured and fitted PV powers (Fig.4.): ;PdiffB = Pmeas − Pmean ( G )
where Pmean in (4) is calculated according to (1).
(4)
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i-th irradiance range
i+1 irradiance range
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j-th power in i-th range
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Gavr,i
Gavr,i+1
Fig.2.: Grouping measured data in irradiance ranges [22]
Based on the range standard deviations calculated according to (3) (in Mod A) or to (4) (in Mod
(
)
K
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B), the standard deviation dependent on the irradiance is estimated as [22]: σ G,Gavr ,1:N r , σ1:N r or σ res ,1:N r = ∑ bk ⋅ G k k =0
(5)
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where bk are polynomial coefficients of the power standard deviations. As is well-known, there are deterministic models of PV power generation dependent on the
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panel temperature and the solar irradiation. This model is incorporated into the stochastic models proposed here. According to this deterministic model, the mean Pcalcm of the normal distribution dependent on the solar irradiance G and panel temperature T is approximated by deterministic PV power dependent on the irradiance and panel temperature as: Pcalcm ( G,T ) = PSTC ⋅
G ⋅ 1 + γ ⋅ (T − TSTC ) GSTC
(6)
where are: PSTC, GSTC and TSTC referent panel power, irradiance, and panel temperature, respectively (here these values are referred to Standard Test Conditions (STC) because these
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data are easy available from the PV panel datasheet), γ power temperature coefficient (available from the datasheet). The first step in defining the standard deviation of this model (with irradiance and temperature dependent mean) is the mapping of measured PV power values (Pmeas) into new values (Pmapp)
Pmapp =
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as:
Pmeas 1 + γ ⋅ (T − TSTC )
(7)
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This mapping (this is the normalization of the power values with respect to a reference temperature) is done to eliminate influence of panel temperature on estimation of the stochastic
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model parameters because the influence can be deterministically modelled. The normalized values of the PV power generation with respect to the reference temperature are obtained by data mapping according to (7). As can be seen from Fig. 1, based on the presentation of the mapped power values, the temperature influence (PV power temperature dependence considered by using
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(7) as the common PV power temperature dependence) is not the only cause of the power dispersion at some irradiance value. Thera are some impacts that are not mathematically described through the common expression according to (6). The hypothesis here is that this
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dispersion of the PV power production can be described stochastically. After the mapping
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according to (7) is done, the standard deviation according to (3) and (4) are calculated as (using mapped PV powers from (7) and the mean fitted on the mapped data according to (1)), PMmapp(G): σ C ,i =
Ni 2 1 ⋅ ∑ ( Pmapp ,i , j − Pmmapp ,i ) N i − 1 j =1
(8)
where Pmmapp,i is average values (mean) of the mapped PV powers of the i-th irradiance range σ resD ,i =
Ni 2 1 ⋅ ∑ ( PdiffD ,i , j − PdiffD ,i ) N i − 1 j =1
;PdiffD ( G ) = PMmapp ( G ) − Pmapp ( G )
(9)
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The proposed model using (7) and (8) is denoted here as Mod C, and the model using (7) and (9) is denoted as Mod D. For both models, the irradiance dependent distribution standard deviation is estimated according to (5). One more stochastic model of PV power generation is proposed including the power distribution
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mean dependent on the irradiance and panel temperature. In this model, the differences between the measured and according to (6) estimated PV powers are calculated (Fig.5.): PdiffE = Pmeas − Pcalcm
(10)
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The difference mean Pdiffm (G) and standard deviation are fitted according (1) and (5), respectively, to obtain their dependences on irradiance. The standard deviations of the irradiance
σ diffE ,i =
Ni 2 1 ⋅ ∑ ( PdiffE ,i , j − PmdifEf ,i ) N i − 1 j =1
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ranges are determined by using the irradiance range deviations as:
(11)
Because the mean of these differences (Fig. 5.) is not zero in this model, the PV power is estimated as summation of the calculated value (6) and random generated value with the mean
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and standard deviations of the differences (13). This stochastic model is here denoted as Mod E (11).
In addition to these models, one simpler model is proposed, named Mod F, in which the
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irradiance-dependent normal distribution mean is the same as in Mod A and B. The standard
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deviation in this case is constant, and it is calculated according to (8), considering all differences (residuals) between the Pmean (1) and measured values (residuals for PV powers fitted according to (1) are given on Fig. 4.) as: σF =
N 2 1 ⋅ ∑ ( PdiffB ,k − PmdiffB ) N − 1 k =1
(12)
where N is number of measured data over the year. After the normal distribution parameters are defined for each proposed stochastic model, the PV system power is estimated as the normal distributed random values according to:
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Mod A : P ( G ) Mod B : P ( G )
N ( Pmean ( G ) , σ A ( G ) )
N ( Pmean ( G ) , σ resB ( G ) )
Mod C : P ( G,T ) Mod D : P ( G,T )
N ( Pcalcm ( G,T ) , σ C ( G ) )
(13)
N ( Pcalcm ( G,T ) , σ resD ( G ) )
Mod E: P ( G,T ) = Pcalcm ( G,T ) + ∆P ( G,T ) ; ∆P ( G,T ) Mod F : P ( G )
N ( Pmean ( G ) , σ F )
N ( PdiffEm ( G,T ) , σ diffE ( G,T ) )
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where PdiffEm and σdiffE are the mean and standard deviation of the measured and calculated power differences (10), respectively.
Fig. 3. shows measured and mapped (7) PV system generated powers, from which it can be
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again seen that after considering the PV power temperature dependence there are dispersion of the power values for same (or very close) irradiance values. It should be noted from Fig. 5 that
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there are significant differences between measured PV generations and those calculated according to (6). Once again, this indicates that there are some influences in PV system power apart from the irradiance and PV panel temperature values (taking into account by (6)). and that
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these influences have a stochastic character.
Fig.3.: Measured and mapped (7) power values generated by PV system
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Fig.4.: Residuals of PV power generation for data fitted by (1)
Fig.5.: Differences between measured and according to (6) estimated PV powers
3
ACCEPTED MANUSCRIPT Simulation results– PV plant case study on micro location in Slovenia
The proposed methods are applied on a case study of a specific PV system; it is the same as described in [22]. The PV system used in the case study is installed in the town of Krško (45.95915° N, 15.49167° E), Slovenia, at the School Centre Krško-Sevnica. The rated power of
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the PV system is 1.05 kWp. The PV panel used in the system is a Solar World SW175 monocrystalline panel. The irradiance and panel temperature are measured by Sunny SensorBox. The irradiance sensor is placed on the same angle as the PV panels. The mathematical
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implementation of the proposed method on the computer is performed in the Python programming environment using SciPy [34] and numpy [35] packages. The numpy function
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“numpy.polyfit” is used to find coefficients in polynomial functions in equations (1) and (5) in case of the proposed stochastical models. The measured data are collected in 2013. The measured, mapped (according to (7)) and calculated according to (6) PV system power
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EP
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productions are given in Fig. 5 and 6.
Fig. 5.: Measured, calculated (6) and mapped (7) power values generated by PV system
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Fig. 6.: Measured and calculated (6) power values generated by PV system and polynomials of their means
The width of the irradiance range (Fig.2.) is set to 10 W/m². The polynomials coefficients
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obtained for the proposed stochastically models are given in Table 1.
Table 1.: Polynomials coefficients in (1) and (5) obtained for the proposed stochastically models Mod A
Mod B
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coefficient
Mod C
Mod E
Mod F
Mod D (for (10))
- 10.5
- 10.5
–
-3.043e-12
a1
0.7501
0.7501
–
7.546e-09
- 0.4847
0.7501
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a0
- 6.117
- 10.5
a2
0.001757
0.001757
–
- 6.981e-06
0.002999
0.001757
a3
- 4.145e-06
- 4.145e-06
–
0.002867
- 7.296e-06
- 4.145e-06
a4
3.728e-09
3.728e-09
–
0.5831
7.864e-09
3.728e-09
a5
-1.217e-12
-1.217e-12
–
- 5.678
-3.154e-12
-1.217e-12
Mod E
Mod F
coefficient
Mod A
Mod B
Mod C
Mod D (for (10)
b0
3.6
3.504
1.902
1.449
1.732
18.89
b1
0.04318
0.03311
0.09564
0.1017
0.1189
–
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0.0005321
0.0006302
- 0.0001733
–
b3
- 3.196e-06
- 3.568e-06
- 9.491e-07
- 5.059e-07
- 8.613e-08
–
b4
7.6e-09
8.276e-09
2.683e-09
1.648e-09
8.131e-10
–
b5
- 8.214e-12
- 8.799e-12
- 3.298e-12
- 2.207e-12
- 1.344e-12
–
b6
3.27e-15
3.464e-15
1.479e-15
1.054e-15
7.096e-16
–
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b2
The graphical overview of the calculated standard deviations over the irradiance ranges and their
Mod B
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Mod A
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fitting curves is given in Fig. 7.
Mod C
Mod D
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Mod E
Mod E – residuals mean
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Fig.7.: The distribution standard deviations over the irradiance ranges and their polynomials for proposed models
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After the stochastic model parameters are calculated based on the measured data, the power generated by the PV system can be generated as random numbers according to (13) for each proposed model. The PV system powers measured over the period that differ from the period used to determine the stochastic model parameters are used to validate the models. The data
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collected during 2013 are used to calculate the normal distribution parameters and data measured over 2014 are used for model validations. Figures 9–13 show two data set of PV power values one random generated by the stochastic models and other measured. The deterministically
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calculated PV powers (according to (6)) are shown in Fig. 14.
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AC C
EP
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Fig.8.: Generated by Mod A and measured PV system powers on year level
Fig.9.: Generated by Mod B and measured PV system powers on year level
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AC C
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Fig.10.: Generated by Mod C and measured PV system powers on year level
Fig.11.: Generated by Mod D and measured PV system powers on year level
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AC C
EP
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Fig.12.: Generated by Mod E and measured PV system powers on year level
Fig.13.: Generated by Mod F and measured PV system powers on year level
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Fig.14.: Generated according to (6) and measured PV system powers on year level To compare the estimation quality of the measured PV generations by using the proposed methods, the different metrics used for measuring distance between probabilistic sets, some
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metrics from the statistic tests and some errors are calculated. The statistical distances used for measuring distances between two discrete probabilistic density functions used here are: Hellinger distance [36], [37], Chi–squared distance [36,37], Wasserstein distance [38,39], and
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energy distance[40,41]. The built-in functions in the SciPy Python package are used to apply Wasserstein and energy distances and Hellinger and Chi-squared distances are calculated
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applying expressions given in [36]. The Kolmogorov-Smirnov 2–sample statistic [42] and the Anderson–Darling k–sample test [42] are used from SciPy Python package as the statistical test metrics. Furthermore, the error metrics are used as follows. Mean absolute error (MAE) and MAE in percent of the PV system nominal power: N
EMEA =
∑ (P i =1
meas
N
− Pcalc )
,EMEA% = 100 ⋅
EMEA Pnom
Mean absolute percentage error (MAPE):
(13)
N
∑ EMAPE = 100 ⋅
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( Pmeas − Pcalc )
(14)
Pmeas N
i =1
Root mean square error (RMSE): N
∑(P
ERMSE =
meas
i =1
− Pcalc )
2
(14)
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N
Normalized mean square error (NMSE): N
E NMSE =
∑(P i =1
meas
− Pcalc )
2
(15)
N
∑ Pmeas 2
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i =1
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The values of these metrics are given in Table 2. In the last column in Table 2. data metrics for deterministically calculated PV powers according to (6) are given. Furthermore, the measured and calculated yearly produced energy and their percent errors are given. Table 2.: Comparison of proposed stochastic methods considering different metrics
the proposed statistical
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Different metrics for
Model
Mod A
Mod B
Mod C
Mod D
Mod E
Mod F
(6)
Hellinger
9.232
8.600
20.44
21.00
8.761
11.63
21.11
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models
Wasserstein
2.865
2.602
26.11
26.32
1.976
3.558
26.45
energy
0.1750
0.1558
1.740
1.746
0.1218
0.2357
1.777
χ2
128
150
424
426
100
162
408
k–sample statistic test
KS statistic
0.0128
0.0124
0.1235
0.1227
0.0130
0.0280
0.1261
metric
AD statistic
1.468
1.431
266.2
270.5
0.6452
8.674
283.8
EMAE
18.38
18.20
30.98
30.99
18.74
19.75
27.84
EMAE%
1.751
1.733
2.950
2.951
1.785
1.881
2.652
EMAPE
11.52
11.16
30.22
30.26
12.26
17.06
29.40
ERMSE
26.01
26.09
37.32
37.44
26.18
25.91
32.41
ENMSE
0.0055
0.0054
0.0113
0.0113
0.0057
0.0054
0.0085
samples distance
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metric
Error metric
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Measured yearly Wmeas [kWh]
797.07
797.07
797.07
797.07
797.07
797.07
797.07
Wcalc [kWh]
802.43
802.68
877.03
876.77
800.25
803.89
876.96
Werr [%]
0.6722
0.7032
10.03
10.00
0.3986
0.8553
10.02
energy Calculated yearly energy Estimation energy
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error
Based on the simulation results, some comments can be made. The proposed statistical models
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represent mathematical quantitation of the stochastic character of the PV system production. These models can be used primarily for application in alkalizing and predicting of the PV system
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impact on the distribution network. The proposed models can be very easily implemented in probabilistic power flow calculation because the probability density function parameters dependences on irradiance and PV panel temperature are given by analytical expressions. For this purpose, the range of the PV power production is important. In other words, the model is of
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better quality for this if the predicted power values are close to the overlap range of the measured PV powers. Based on Fig.9–13, it can be concluded that models Mod A and B show the best results for this purpose. This is interesting because these two models do not consider
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impact of the PV panel temperature on PV power. The explanation for this can be found in the fact that panel temperature is in strong correlation with the irradiance value. As is well-known,
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the panel temperature can be analytically estimated from the irradiance value, for example the expression in [27]. These two models indicate that for a statistical model based on in situ measurements, the dependence of distribution parameters only on irradiance can be considered. Furthermore, good overlapping of the measured by estimated powers is shown by Mod E. For this model, some worse overlapping is seen for the high irradiance values (>800 W/m²). If the prediction of the PV system produced energy is analysed on the year level, Mod A, B, E, and F estimate the yearly energy production with very high accuracy. At this point it is noteworthy that energy estimated according to deterministic expression (6) has a high error. The reason of this is
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visible in Fig. 6, where can be seen that there are differences between fitted mean values of measured and calculated powers. This indicated that some statistical but not only deterministic relationships should be considered for planning PV energy production. From the statistic distance, statistic test, and error metrics it can be concluded that Mods A, B and E are the best.
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Mod E has slightly better distance metrics and AD statistic value. Based on the simulation results, the conclusion is that Mod A, B and E can be the models for the stochastic prediction of the power production of the PV system. In Fig. 15–18, prediction of the PV power generation at
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day level for Mod A, B and E and deterministically calculated (6) power are given.
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Fig.15.: Generated by Mod A and measured PV system powers for five days
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Fig.16.: Generated by Mod B and measured PV system powers for five days
Fig.17.: Generated by Mod C and measured PV system powers for five days
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Fig.18.: PV system powers calculated according to (6) for five days
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Conclusion
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The statistic models considering different PV production under the same irradiance and PV panel temperature values are researched in the paper. Very simple stochastic models for PV system power production are proposed in the paper. The proposed models are not
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mathematically demanding and can be performed by using built-in functions in common computer simulation tools. Three of the six researched models have potential to be used for
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modelling the PV power production considering the stochastic character of the power generation caused by different reasons. The proposed models are applicable for the probability power flow in the distribution networks with installed PV systems. One drawback of the proposed model is that historical PV system production data are required. The further research will be about possibility of using the model obtained for the specific PV system to estimate the production of other PV systems (with different sizes, PV panel technologies, etc.) at the same micro location. The proposed models A, B and F, dependent only on irradiance, have similar error metrics as
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model E, dependent on irradiance and panel temperature. Root mean square errors are 26.01 W, 26.09 W, 26.18 W and 25.91 W, normalized mean square errors are 0.0055, 0.0054, 0.0057 and 0.0054 and mean absolute errors in percent are 1.751%, 1.733%, 1.785% and 1.881% for models A, B, E, and F respectively. Because models A and B have better overlapping between the
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measured and calculated data (Fig. 8, 9, 12 and 13) than models E and F. it can be concluded that model A and B (dependent on irradiance only) can be successfully used for the stochastic prediction of the PV power production.
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The presented research based on obtained results indicates that approach of building the statistical model of PV production prediction considering the irradiance only can be used to take
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into account the stochastic character of PV production. Such an approach can be attractive because it required only irradiance measurement, which is widely used on PV plants. The presented stochastic model can be used for prediction of PV energy production with high accuracy (as can be seen from Table 2). Beside this, such a stochastic approach makes possible the quantification of the stochastic occurrences that impact PV production. The stochastic
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character of PV production can be considered in other simulations in the power network analysis (such as the power flow calculations) using the proposed approach.
Acknowledgment
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The authors are thankful to Tin Benšić assistant at Faculty of Electrical Engineering, Computer Science and Information Technology Osijek for his useful comments and discussions during the research.
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The statistic models considering different PV production under same irradiance and PV panel temperature values are researched. The model parameters are determined from historical data of PV generation
•
Dependence on radiance and temperature of probability distribution functions is proposed.
The probabilistic models are based on the normal distribution have irradiance and
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temperature dependent parameters.
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S0360-5442(18)31904-2
DOI:
10.1016/j.energy.2018.09.134
Reference:
EGY 13828
To appear in:
Energy
Received Date: 17 December 2017 Revised Date:
14 September 2018
Accepted Date: 19 September 2018
Please cite this article as: Barukčić M, Hederić Ž, Hadžiselimović M, Seme S, A simple stochastic method for modelling the uncertainty of photovoltaic power production based on measured data, Energy (2018), doi: https://doi.org/10.1016/j.energy.2018.09.134. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
ACCEPTED MANUSCRIPT A Simple Stochastic Method for Modelling the Uncertainty of Photovoltaic
Power Production Based on Measured Data
M.Barukčić*a, Ž. Hederićb, M. Hadžiselimovićc, S. Semed J. J. Strossmayer University of Osijek, Faculty of Electrical Engineering, Computer Science
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a,b
and Information Technology Osijek, Kneza Trpimira 2B, 31 000 Osijek, Croatia c,d
University of Maribor, Faculty of Energy Technology, Hočevarjev trg 1, 8270 Krško, Slovenia
a
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* Corresponding author
[email protected], (Telephone: +385 31 224 685, Fax: +385 31 224 605) [email protected], c [email protected], d [email protected]
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Abstract: This paper describes statistical quantification tools for predicting photovoltaic (PV) production considering uncertainty in PV production at same irradiation levels and PV panel
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temperatures. When analysing measured data, it is observed that there are different PV power production levels for the same irradiation levels and panel temperatures. These PV power spread out can be caused by different causes, such as dust deposition over the panel, non-ideal working
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of maximal power point tracking devices, device efficiencies’ dependence on power, different temperatures over the PV panel, and others. Due to the stochastic character of these occurrences,
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they can be challenging when considered in the deterministic mathematical models usually used for PV power prediction. The probabilistic method for PV power production is proposed based on the probability density function with respect to the solar irradiation and the panel temperature. The simulation results are compared among the different models based on the probability density function with respect to the solar irradiation and panel temperature. The best overlapping between measured and calculated PV power production gives the proposed stochastic models dependent only on irradiance. The proposed stochastic model gives a PV energy prediction on a yearly basis with an error of less than 1%.
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Introduction
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Many factors impact the production of photovoltaic (PV) panels; solar irradiance and PV panel
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temperature have the greatest influence. As is well-known, these two impact factors are modeled with a deterministic mathematical model. However, due to the stochastic character of the energy
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source (solar irradiance), the probabilistic approach is often employed in this case. In [1], the PV production is predicted based on the solar irradiance estimated by using the probability density function of the clearness index. The probability density function is modelled based on the Bayesian approach here. The Bayesian approach to prediction PV energy production is used in
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[2] with aim of finding daily pattern profiles of the PV production. The time series method for PV power forecasting depends on the day of year and the time of day is presented in [3]. In [4], the authors use an Artificial Neural Network (ANN) with ambient and panel temperatures and
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irradiance as input data to obtain PV power as output from ANN. ANN is also used in [5–9] for PV power estimation. In [10], a combination of different time series and ANN models tuned by
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using Genetic Algorithm (GA) is used for PV power prediction. The Genetic Programming (GP) to evolve the fuzzy inference system for PV power production is presented in [11]. The historical data processed by data mining technique are used in [12] to estimate PV power production. In [13] and [14], support vector machines (SVM) using the numerically forecasting weather data is used for PV energy and power prediction. The methods for short-term prediction of the PV power generation are presented in [15]. The random generated PV power with respect to time of the day is presented in [16]. This literature was also one of the motivation for the authors to conduct the research presented here. Most of the literature mentioned used a deterministic model
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of PV power production considering the stochastic character of the intensity of the energy source. Such an example of PV production prediction using the deterministic calculations can be found in the very recent literature [17,18] also. In [19], the software tools using deterministic calculations are also used for PV energy prediction. The inverse model for prediction of the PV
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production based on measuring the weather conditions is given in [20]. The analytic expression for PV power production is used in [21], in which models for estimation of PV panel temperature are researched. Typically, the stochastic model of irradiance and temperature is
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researched with respect to weather conditions, the time of day, or the cloudiness index, but PV power generation is calculated by using a deterministic expression from stochastically
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determined irradiance and temperature. Some references research the stochastic in PV power itself as it is presented in [22–24]. The above given overview of the literature on the research prediction of the PV system production can be summarized as follows: prediction of the weather conditions (irradiance and temperature mainly) is researched in [1,3,4,15]; estimations of PV power using the weather conditions as input variables are presented in [5–10,13,14]; prediction
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of the PV production in the form of time series is presented in [10,11]. None of these sources researched the stochastic character of PV power for the same irradiance and temperature values.
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In other words, for same the irradiance and PV panel temperature, the output of PV system will always be equal, because the calculation of the PV power is deterministic. Distinct from the
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other studies, the research presented in this paper deals with PV production uncertainty in the case of the same irradiance and temperature input values, and the calculation of the PV power is stochastic. The research dealing with the uncertainty of the PV power production is presented in [12,16,23,24]. A stochastic model of PV production based on irradiance condition patterns (sunny, overcast, partly cloudy) is presented in [24]. In [12], [16], and [22], the uncertainty of the PV production is given depending on the period of the day (time). In [12] and [16], the parameters of the probability density function are time-dependent while linear combinations of the normal density functions (the Gaussian mixture method) with constant parameters are used
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in [23]. The stochastic model proposed here has irradiance and temperature dependent parameters of the probability density function, which is different from models presented in the cited literature. The impact of taking into account the PV production and other uncertainties on energy management in microgrids is presented in [25].
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During the analysis of the measured data of the PV plant, the authors observed that there are cases in which different PV powers were measured even though values of irradiance and PV panel temperature were the same (or very close each other). The causes of this can be due to dust
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deposition on PV panel surface [26], non-ideal work of MPPT, different spectral response of PV panel and PV cell used for the solar irradiance measurement, different temperatures over the PV
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panel, different air mass over time. Because of the stochastic character of these possible causes, they bring uncertainty to the estimation of PV production. This was the motivation to conduct this research, and it is presented in Fig. 1 and Fig. 5. As can be seen in Fig. 1, there are different levels of PV power generation even in case of the same the irradiance and temperature values. Furthermore, the shading effect can have a significant impact on PV production, as can be seen
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in [27]. Because this can be modeled by a deterministic model, as presented in [27] this impact is not considered here. The hypothesis is that the uncertainty of the PV power production can be
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quantified by using a probability density function that has parameters dependent on the solar irradiance. The proposed models can be used in probabilistic power flow in a distribution
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network containing PV systems to address uncertainty in PV system generation [28]. The statistical models of PV production can be of interest in case of the different optimization in networks with installed PV such is optimization considering the reactive power control presented in [29] or optimal control in hybrid microgrid [30]. Also, the importance of mathematical models that include uncertainty of PV power production can be found in [31–33]. The stochastic models proposed here are based on those presented in [22], and this is a continuation of the research presented in [22]. The probability density function parameters are determined based on historical measured PV plant data on a micro-location in Slovenia. In the present paper, the
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parameters of the probability density function are dependent on the irradiance and PV panel temperature, unlike the research presented in [22], in which only the dependence on the irradiance was considered. The contribution of the paper to the topic of PV power prediction is in proposing the new stochastic (probabilistic) model, which is distinct from much of the
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literature considering the deterministic calculation of PV power. The stochastic models of PV power estimation are presented in only small number of the literature, as can be seen from above overview. This paper also makes a contribution tp the topic of the stochastic models of PV
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production prediction. The innovation of the presented method is in proposing a stochastic model that has probability parameters depending on irradiance (and temperature), which is
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different than in the above-mentioned literature. The rest of the presented research is organized as follows: in Section 2, an overview of the proposed model is given; in Section 3. simulation results are presented and compared among the proposed models; in the last section a conclusion is given.
Defining Uncertainty Parameters from Measured Data
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The proposed stochastic model is based on next hypothesis: the uncertainties in PV panel
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production can be represented by random numbers. The random number is generated by using normal (Gaussian) distribution that has variable (no constant) distribution mean and standard
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deviations. The stochastic character of the PV system power generation can be observed from the presentation of the measured powers given in Fig. 1. As mentioned before, even in the case of the same values of the irradiance and PV panel temperature, the PV power production can be different.
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T = 36.83°C T = 36.23°C
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T = 39.63°C
Fig. 1.: Measured power generation of PV system over a year and detail for an irradiance value
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(mapped data are calculated according to (7) and represent normalized data with respect to the reference temperature)
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As mentioned before, in [22], the stochastic models considering the distribution parameters dependent only on irradiance are presented. Because those models are compared with the proposed method, the models are briefly represented here and details can be found in [22]. The mean Pmean of the normal distribution dependent on the solar irradiance G is approximated by a polynomial of the H order in both approaches in [22]: H
Pmean ( G ) = ∑ ah ⋅ G h h=0
where ah are polynomial coefficients of the power mean.
(1)
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Dependence according to (1) is stated based on the experimentally measured PV generated power. In addition, to determine the standard deviation dependent on the irradiance, the measured data set of the PV power is divided in a number of irradiance ranges according to Fig.2 [22] with
Gavr ,i =
Gmin,i + Gmax,i 2
; i ∈ (1L N r )
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range average irradiance as: (2)
where Gavr,i is the average value of the irradiance for each irradiance range, Gmin,i and Gmax,i are
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range bounds of the i–th range, and Nr is number of the ranges.
In the first approach used in [22] (hereinafter denoted as Mod A), the standard deviation in each
σ A,i =
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irradiance range is calculated as: Ni 2 1 ⋅ ∑ ( Pmeas ,i , j − Pmi ) N i − 1 j =1
(3)
where, Pmeas,i , j is measured the j-th PV power within the i-th irradiance range, Pmi is average
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values (mean) of the measured PV powers of the i-th irradiance range and Ni is number of measured PV powers in i-th irradiance range.
The second approach (hereinafter denoted as Mod B) is based on the standard deviation of the
Ni 2 1 ⋅ ∑ ( PdiffB ,i , j − PdiffB ,i ) N i − 1 j =1
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σ resB ,i =
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differences (residuals) between measured and fitted PV powers (Fig.4.): ;PdiffB = Pmeas − Pmean ( G )
where Pmean in (4) is calculated according to (1).
(4)
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i-th irradiance range
i+1 irradiance range
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j-th power in i-th range
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Gavr,i
Gavr,i+1
Fig.2.: Grouping measured data in irradiance ranges [22]
Based on the range standard deviations calculated according to (3) (in Mod A) or to (4) (in Mod
(
)
K
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B), the standard deviation dependent on the irradiance is estimated as [22]: σ G,Gavr ,1:N r , σ1:N r or σ res ,1:N r = ∑ bk ⋅ G k k =0
(5)
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where bk are polynomial coefficients of the power standard deviations. As is well-known, there are deterministic models of PV power generation dependent on the
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panel temperature and the solar irradiation. This model is incorporated into the stochastic models proposed here. According to this deterministic model, the mean Pcalcm of the normal distribution dependent on the solar irradiance G and panel temperature T is approximated by deterministic PV power dependent on the irradiance and panel temperature as: Pcalcm ( G,T ) = PSTC ⋅
G ⋅ 1 + γ ⋅ (T − TSTC ) GSTC
(6)
where are: PSTC, GSTC and TSTC referent panel power, irradiance, and panel temperature, respectively (here these values are referred to Standard Test Conditions (STC) because these
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data are easy available from the PV panel datasheet), γ power temperature coefficient (available from the datasheet). The first step in defining the standard deviation of this model (with irradiance and temperature dependent mean) is the mapping of measured PV power values (Pmeas) into new values (Pmapp)
Pmapp =
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as:
Pmeas 1 + γ ⋅ (T − TSTC )
(7)
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This mapping (this is the normalization of the power values with respect to a reference temperature) is done to eliminate influence of panel temperature on estimation of the stochastic
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model parameters because the influence can be deterministically modelled. The normalized values of the PV power generation with respect to the reference temperature are obtained by data mapping according to (7). As can be seen from Fig. 1, based on the presentation of the mapped power values, the temperature influence (PV power temperature dependence considered by using
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(7) as the common PV power temperature dependence) is not the only cause of the power dispersion at some irradiance value. Thera are some impacts that are not mathematically described through the common expression according to (6). The hypothesis here is that this
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dispersion of the PV power production can be described stochastically. After the mapping
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according to (7) is done, the standard deviation according to (3) and (4) are calculated as (using mapped PV powers from (7) and the mean fitted on the mapped data according to (1)), PMmapp(G): σ C ,i =
Ni 2 1 ⋅ ∑ ( Pmapp ,i , j − Pmmapp ,i ) N i − 1 j =1
(8)
where Pmmapp,i is average values (mean) of the mapped PV powers of the i-th irradiance range σ resD ,i =
Ni 2 1 ⋅ ∑ ( PdiffD ,i , j − PdiffD ,i ) N i − 1 j =1
;PdiffD ( G ) = PMmapp ( G ) − Pmapp ( G )
(9)
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The proposed model using (7) and (8) is denoted here as Mod C, and the model using (7) and (9) is denoted as Mod D. For both models, the irradiance dependent distribution standard deviation is estimated according to (5). One more stochastic model of PV power generation is proposed including the power distribution
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mean dependent on the irradiance and panel temperature. In this model, the differences between the measured and according to (6) estimated PV powers are calculated (Fig.5.): PdiffE = Pmeas − Pcalcm
(10)
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The difference mean Pdiffm (G) and standard deviation are fitted according (1) and (5), respectively, to obtain their dependences on irradiance. The standard deviations of the irradiance
σ diffE ,i =
Ni 2 1 ⋅ ∑ ( PdiffE ,i , j − PmdifEf ,i ) N i − 1 j =1
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ranges are determined by using the irradiance range deviations as:
(11)
Because the mean of these differences (Fig. 5.) is not zero in this model, the PV power is estimated as summation of the calculated value (6) and random generated value with the mean
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and standard deviations of the differences (13). This stochastic model is here denoted as Mod E (11).
In addition to these models, one simpler model is proposed, named Mod F, in which the
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irradiance-dependent normal distribution mean is the same as in Mod A and B. The standard
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deviation in this case is constant, and it is calculated according to (8), considering all differences (residuals) between the Pmean (1) and measured values (residuals for PV powers fitted according to (1) are given on Fig. 4.) as: σF =
N 2 1 ⋅ ∑ ( PdiffB ,k − PmdiffB ) N − 1 k =1
(12)
where N is number of measured data over the year. After the normal distribution parameters are defined for each proposed stochastic model, the PV system power is estimated as the normal distributed random values according to:
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Mod A : P ( G ) Mod B : P ( G )
N ( Pmean ( G ) , σ A ( G ) )
N ( Pmean ( G ) , σ resB ( G ) )
Mod C : P ( G,T ) Mod D : P ( G,T )
N ( Pcalcm ( G,T ) , σ C ( G ) )
(13)
N ( Pcalcm ( G,T ) , σ resD ( G ) )
Mod E: P ( G,T ) = Pcalcm ( G,T ) + ∆P ( G,T ) ; ∆P ( G,T ) Mod F : P ( G )
N ( Pmean ( G ) , σ F )
N ( PdiffEm ( G,T ) , σ diffE ( G,T ) )
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where PdiffEm and σdiffE are the mean and standard deviation of the measured and calculated power differences (10), respectively.
Fig. 3. shows measured and mapped (7) PV system generated powers, from which it can be
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again seen that after considering the PV power temperature dependence there are dispersion of the power values for same (or very close) irradiance values. It should be noted from Fig. 5 that
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there are significant differences between measured PV generations and those calculated according to (6). Once again, this indicates that there are some influences in PV system power apart from the irradiance and PV panel temperature values (taking into account by (6)). and that
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these influences have a stochastic character.
Fig.3.: Measured and mapped (7) power values generated by PV system
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Fig.4.: Residuals of PV power generation for data fitted by (1)
Fig.5.: Differences between measured and according to (6) estimated PV powers
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The proposed methods are applied on a case study of a specific PV system; it is the same as described in [22]. The PV system used in the case study is installed in the town of Krško (45.95915° N, 15.49167° E), Slovenia, at the School Centre Krško-Sevnica. The rated power of
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the PV system is 1.05 kWp. The PV panel used in the system is a Solar World SW175 monocrystalline panel. The irradiance and panel temperature are measured by Sunny SensorBox. The irradiance sensor is placed on the same angle as the PV panels. The mathematical
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implementation of the proposed method on the computer is performed in the Python programming environment using SciPy [34] and numpy [35] packages. The numpy function
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“numpy.polyfit” is used to find coefficients in polynomial functions in equations (1) and (5) in case of the proposed stochastical models. The measured data are collected in 2013. The measured, mapped (according to (7)) and calculated according to (6) PV system power
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productions are given in Fig. 5 and 6.
Fig. 5.: Measured, calculated (6) and mapped (7) power values generated by PV system
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Fig. 6.: Measured and calculated (6) power values generated by PV system and polynomials of their means
The width of the irradiance range (Fig.2.) is set to 10 W/m². The polynomials coefficients
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obtained for the proposed stochastically models are given in Table 1.
Table 1.: Polynomials coefficients in (1) and (5) obtained for the proposed stochastically models Mod A
Mod B
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coefficient
Mod C
Mod E
Mod F
Mod D (for (10))
- 10.5
- 10.5
–
-3.043e-12
a1
0.7501
0.7501
–
7.546e-09
- 0.4847
0.7501
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a0
- 6.117
- 10.5
a2
0.001757
0.001757
–
- 6.981e-06
0.002999
0.001757
a3
- 4.145e-06
- 4.145e-06
–
0.002867
- 7.296e-06
- 4.145e-06
a4
3.728e-09
3.728e-09
–
0.5831
7.864e-09
3.728e-09
a5
-1.217e-12
-1.217e-12
–
- 5.678
-3.154e-12
-1.217e-12
Mod E
Mod F
coefficient
Mod A
Mod B
Mod C
Mod D (for (10)
b0
3.6
3.504
1.902
1.449
1.732
18.89
b1
0.04318
0.03311
0.09564
0.1017
0.1189
–
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0.0005321
0.0006302
- 0.0001733
–
b3
- 3.196e-06
- 3.568e-06
- 9.491e-07
- 5.059e-07
- 8.613e-08
–
b4
7.6e-09
8.276e-09
2.683e-09
1.648e-09
8.131e-10
–
b5
- 8.214e-12
- 8.799e-12
- 3.298e-12
- 2.207e-12
- 1.344e-12
–
b6
3.27e-15
3.464e-15
1.479e-15
1.054e-15
7.096e-16
–
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b2
The graphical overview of the calculated standard deviations over the irradiance ranges and their
Mod B
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Mod A
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fitting curves is given in Fig. 7.
Mod C
Mod D
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Mod E
Mod E – residuals mean
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Fig.7.: The distribution standard deviations over the irradiance ranges and their polynomials for proposed models
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After the stochastic model parameters are calculated based on the measured data, the power generated by the PV system can be generated as random numbers according to (13) for each proposed model. The PV system powers measured over the period that differ from the period used to determine the stochastic model parameters are used to validate the models. The data
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collected during 2013 are used to calculate the normal distribution parameters and data measured over 2014 are used for model validations. Figures 9–13 show two data set of PV power values one random generated by the stochastic models and other measured. The deterministically
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calculated PV powers (according to (6)) are shown in Fig. 14.
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Fig.8.: Generated by Mod A and measured PV system powers on year level
Fig.9.: Generated by Mod B and measured PV system powers on year level
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Fig.10.: Generated by Mod C and measured PV system powers on year level
Fig.11.: Generated by Mod D and measured PV system powers on year level
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Fig.12.: Generated by Mod E and measured PV system powers on year level
Fig.13.: Generated by Mod F and measured PV system powers on year level
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Fig.14.: Generated according to (6) and measured PV system powers on year level To compare the estimation quality of the measured PV generations by using the proposed methods, the different metrics used for measuring distance between probabilistic sets, some
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metrics from the statistic tests and some errors are calculated. The statistical distances used for measuring distances between two discrete probabilistic density functions used here are: Hellinger distance [36], [37], Chi–squared distance [36,37], Wasserstein distance [38,39], and
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energy distance[40,41]. The built-in functions in the SciPy Python package are used to apply Wasserstein and energy distances and Hellinger and Chi-squared distances are calculated
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applying expressions given in [36]. The Kolmogorov-Smirnov 2–sample statistic [42] and the Anderson–Darling k–sample test [42] are used from SciPy Python package as the statistical test metrics. Furthermore, the error metrics are used as follows. Mean absolute error (MAE) and MAE in percent of the PV system nominal power: N
EMEA =
∑ (P i =1
meas
N
− Pcalc )
,EMEA% = 100 ⋅
EMEA Pnom
Mean absolute percentage error (MAPE):
(13)
N
∑ EMAPE = 100 ⋅
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( Pmeas − Pcalc )
(14)
Pmeas N
i =1
Root mean square error (RMSE): N
∑(P
ERMSE =
meas
i =1
− Pcalc )
2
(14)
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N
Normalized mean square error (NMSE): N
E NMSE =
∑(P i =1
meas
− Pcalc )
2
(15)
N
∑ Pmeas 2
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i =1
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The values of these metrics are given in Table 2. In the last column in Table 2. data metrics for deterministically calculated PV powers according to (6) are given. Furthermore, the measured and calculated yearly produced energy and their percent errors are given. Table 2.: Comparison of proposed stochastic methods considering different metrics
the proposed statistical
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Different metrics for
Model
Mod A
Mod B
Mod C
Mod D
Mod E
Mod F
(6)
Hellinger
9.232
8.600
20.44
21.00
8.761
11.63
21.11
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models
Wasserstein
2.865
2.602
26.11
26.32
1.976
3.558
26.45
energy
0.1750
0.1558
1.740
1.746
0.1218
0.2357
1.777
χ2
128
150
424
426
100
162
408
k–sample statistic test
KS statistic
0.0128
0.0124
0.1235
0.1227
0.0130
0.0280
0.1261
metric
AD statistic
1.468
1.431
266.2
270.5
0.6452
8.674
283.8
EMAE
18.38
18.20
30.98
30.99
18.74
19.75
27.84
EMAE%
1.751
1.733
2.950
2.951
1.785
1.881
2.652
EMAPE
11.52
11.16
30.22
30.26
12.26
17.06
29.40
ERMSE
26.01
26.09
37.32
37.44
26.18
25.91
32.41
ENMSE
0.0055
0.0054
0.0113
0.0113
0.0057
0.0054
0.0085
samples distance
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metric
Error metric
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Measured yearly Wmeas [kWh]
797.07
797.07
797.07
797.07
797.07
797.07
797.07
Wcalc [kWh]
802.43
802.68
877.03
876.77
800.25
803.89
876.96
Werr [%]
0.6722
0.7032
10.03
10.00
0.3986
0.8553
10.02
energy Calculated yearly energy Estimation energy
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error
Based on the simulation results, some comments can be made. The proposed statistical models
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represent mathematical quantitation of the stochastic character of the PV system production. These models can be used primarily for application in alkalizing and predicting of the PV system
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impact on the distribution network. The proposed models can be very easily implemented in probabilistic power flow calculation because the probability density function parameters dependences on irradiance and PV panel temperature are given by analytical expressions. For this purpose, the range of the PV power production is important. In other words, the model is of
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better quality for this if the predicted power values are close to the overlap range of the measured PV powers. Based on Fig.9–13, it can be concluded that models Mod A and B show the best results for this purpose. This is interesting because these two models do not consider
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impact of the PV panel temperature on PV power. The explanation for this can be found in the fact that panel temperature is in strong correlation with the irradiance value. As is well-known,
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the panel temperature can be analytically estimated from the irradiance value, for example the expression in [27]. These two models indicate that for a statistical model based on in situ measurements, the dependence of distribution parameters only on irradiance can be considered. Furthermore, good overlapping of the measured by estimated powers is shown by Mod E. For this model, some worse overlapping is seen for the high irradiance values (>800 W/m²). If the prediction of the PV system produced energy is analysed on the year level, Mod A, B, E, and F estimate the yearly energy production with very high accuracy. At this point it is noteworthy that energy estimated according to deterministic expression (6) has a high error. The reason of this is
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visible in Fig. 6, where can be seen that there are differences between fitted mean values of measured and calculated powers. This indicated that some statistical but not only deterministic relationships should be considered for planning PV energy production. From the statistic distance, statistic test, and error metrics it can be concluded that Mods A, B and E are the best.
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Mod E has slightly better distance metrics and AD statistic value. Based on the simulation results, the conclusion is that Mod A, B and E can be the models for the stochastic prediction of the power production of the PV system. In Fig. 15–18, prediction of the PV power generation at
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day level for Mod A, B and E and deterministically calculated (6) power are given.
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Fig.15.: Generated by Mod A and measured PV system powers for five days
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Fig.16.: Generated by Mod B and measured PV system powers for five days
Fig.17.: Generated by Mod C and measured PV system powers for five days
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Fig.18.: PV system powers calculated according to (6) for five days
4
Conclusion
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The statistic models considering different PV production under the same irradiance and PV panel temperature values are researched in the paper. Very simple stochastic models for PV system power production are proposed in the paper. The proposed models are not
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mathematically demanding and can be performed by using built-in functions in common computer simulation tools. Three of the six researched models have potential to be used for
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modelling the PV power production considering the stochastic character of the power generation caused by different reasons. The proposed models are applicable for the probability power flow in the distribution networks with installed PV systems. One drawback of the proposed model is that historical PV system production data are required. The further research will be about possibility of using the model obtained for the specific PV system to estimate the production of other PV systems (with different sizes, PV panel technologies, etc.) at the same micro location. The proposed models A, B and F, dependent only on irradiance, have similar error metrics as
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model E, dependent on irradiance and panel temperature. Root mean square errors are 26.01 W, 26.09 W, 26.18 W and 25.91 W, normalized mean square errors are 0.0055, 0.0054, 0.0057 and 0.0054 and mean absolute errors in percent are 1.751%, 1.733%, 1.785% and 1.881% for models A, B, E, and F respectively. Because models A and B have better overlapping between the
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measured and calculated data (Fig. 8, 9, 12 and 13) than models E and F. it can be concluded that model A and B (dependent on irradiance only) can be successfully used for the stochastic prediction of the PV power production.
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The presented research based on obtained results indicates that approach of building the statistical model of PV production prediction considering the irradiance only can be used to take
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into account the stochastic character of PV production. Such an approach can be attractive because it required only irradiance measurement, which is widely used on PV plants. The presented stochastic model can be used for prediction of PV energy production with high accuracy (as can be seen from Table 2). Beside this, such a stochastic approach makes possible the quantification of the stochastic occurrences that impact PV production. The stochastic
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character of PV production can be considered in other simulations in the power network analysis (such as the power flow calculations) using the proposed approach.
Acknowledgment
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The authors are thankful to Tin Benšić assistant at Faculty of Electrical Engineering, Computer Science and Information Technology Osijek for his useful comments and discussions during the research.
6 [1]
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The statistic models considering different PV production under same irradiance and PV panel temperature values are researched. The model parameters are determined from historical data of PV generation
•
Dependence on radiance and temperature of probability distribution functions is proposed.
The probabilistic models are based on the normal distribution have irradiance and
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temperature dependent parameters.
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