Parameters and characteristics of different diode models.
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These books synthesize perspectives of renowned scientists from the world’s most prestigious institutions - from Fukushima Renewable Energy Institute in Japan to Stanford University in the United States, including Columbia University (US), University of Sidney (AU), University of Miami (USA), Cardiff University (UK), and many others.
\\n\\nThis collaboration embodied the true essence of Open Access by simplifying the approach to OA publishing for Academic editors and authors who contributed their research and allowed the new research to be made available free and open to anyone anywhere in the world.
\\n\\nTo celebrate the 50 books published, we have gathered them at one location - just one click away, so that you can easily browse the subjects of your interest, download the content directly, share it or read online.
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IntechOpen and Knowledge Unlatched formed a partnership to support researchers working in engineering sciences by enabling an easier approach to publishing Open Access content. Using the Knowledge Unlatched crowdfunding model to raise the publishing costs through libraries around the world, Open Access Publishing Fee (OAPF) was not required from the authors.
\n\nInitially, the partnership supported engineering research, but it soon grew to include physical and life sciences, attracting more researchers to the advantages of Open Access publishing.
\n\n\n\nThese books synthesize perspectives of renowned scientists from the world’s most prestigious institutions - from Fukushima Renewable Energy Institute in Japan to Stanford University in the United States, including Columbia University (US), University of Sidney (AU), University of Miami (USA), Cardiff University (UK), and many others.
\n\nThis collaboration embodied the true essence of Open Access by simplifying the approach to OA publishing for Academic editors and authors who contributed their research and allowed the new research to be made available free and open to anyone anywhere in the world.
\n\nTo celebrate the 50 books published, we have gathered them at one location - just one click away, so that you can easily browse the subjects of your interest, download the content directly, share it or read online.
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It includes 22 chapters, organized in two sections. The first book section: "Feedstocks for Biodiesel Production" covers issues associated with the utilization of cost effective non-edible raw materials and wastes, and the development of biomass feedstock with physical and chemical properties that facilitate it processing to biodiesel. These include Brassicaceae spp., cooking oils, animal fat wastes, oleaginous fungi, and algae. The second book section: "Biodiesel Production Methods" is devoted to the advanced techniques for biodiesel synthesis: supercritical transesterification, microwaves, radio frequency and ultrasound techniques, reactive distillation, and optimized transesterification processes making use of solid catalysts and immobilized enzymes. The adequate and up-to-date information provided in this book should be of interest for research scientist, students, and technologists, involved in biodiesel production.',isbn:null,printIsbn:"978-953-307-713-0",pdfIsbn:"978-953-51-4407-6",doi:"10.5772/1094",price:139,priceEur:155,priceUsd:179,slug:"biodiesel-feedstocks-and-processing-technologies",numberOfPages:472,isOpenForSubmission:!1,isInWos:1,isInBkci:!0,hash:"6515b40c0b7f5abd126e0325263a581c",bookSignature:"Margarita Stoytcheva and Gisela Montero",publishedDate:"November 9th 2011",coverURL:"https://cdn.intechopen.com/books/images_new/582.jpg",numberOfDownloads:129956,numberOfWosCitations:273,numberOfCrossrefCitations:93,numberOfCrossrefCitationsByBook:17,numberOfDimensionsCitations:265,numberOfDimensionsCitationsByBook:29,hasAltmetrics:1,numberOfTotalCitations:631,isAvailableForWebshopOrdering:!0,dateEndFirstStepPublish:"January 19th 2011",dateEndSecondStepPublish:"February 16th 2011",dateEndThirdStepPublish:"June 23rd 2011",dateEndFourthStepPublish:"July 23rd 2011",dateEndFifthStepPublish:"November 20th 2011",currentStepOfPublishingProcess:5,indexedIn:"1,2,3,4,5,6,7,8",editedByType:"Edited by",kuFlag:!1,featuredMarkup:null,editors:[{id:"6375",title:"Prof.",name:"Margarita",middleName:null,surname:"Stoytcheva",slug:"margarita-stoytcheva",fullName:"Margarita Stoytcheva",profilePictureURL:"https://mts.intechopen.com/storage/users/6375/images/1631_n.jpg",biography:"Professor Margarita Stoytcheva graduated from the University of Chemical Technology and Metallurgy of Sofia, Bulgaria, with titles of Chemical Engineer and Master of Electrochemical Technologies. She has a Ph.D. and DSc. degrees in chemistry and technical sciences. She has acted in research and teaching in several Universities in Bulgaria, Algeria and France. From 2006. to the present she has participated in activities of scientific research, technological development and teaching in Mexico at the University of Baja California, Institute of Engineering, Mexicali, as a full time researcher. Since 2008. she has been a member of the National System of Researchers of Mexico. Her interests and areas of research are analytical chemistry and biotechnology.",institutionString:null,position:null,outsideEditionCount:0,totalCites:0,totalAuthoredChapters:"6",totalChapterViews:"0",totalEditedBooks:"10",institution:{name:"Autonomous University of Baja California",institutionURL:null,country:{name:"Mexico"}}}],equalEditorOne:null,equalEditorTwo:null,equalEditorThree:null,coeditorOne:{id:"65519",title:"Dr.",name:"Gisela",middleName:null,surname:"Montero",slug:"gisela-montero",fullName:"Gisela Montero",profilePictureURL:"https://mts.intechopen.com/storage/users/65519/images/5399_n.jpg",biography:"Chemical engineer graduated from National Autonomous University \nof Mexico, (UNAM), specialized in heat transfer engineering projects \nin Mexican Petroleum Institute (IMP), Master in thermodynamic engineering by Autonomous University of Baja California (UABC), obtaining the maximum honors. Doctorate in Chemical Sciences (chemical engineering) by UNAM. She \nhas written several technical papers and chapters related to energy, biofuels and exergy analysis. She worked as a specialist in the Combustion Engineering Division in IMP; after she worked as a researcher in Exergy group in the same institute. She was coordinator of Geothermal energy program in UABC. Currently, working as fulltime researcher in Engineering Institute of UABC on energy-saving projects, exergy analysis, simulation and adaptation of processes to obtain biofuels.",institutionString:null,position:null,outsideEditionCount:0,totalCites:0,totalAuthoredChapters:"1",totalChapterViews:"0",totalEditedBooks:"0",institution:{name:"Autonomous University of Baja California",institutionURL:null,country:{name:"Mexico"}}},coeditorTwo:null,coeditorThree:null,coeditorFour:null,coeditorFive:null,topics:[{id:"763",title:"Bioresource Engineering",slug:"bioresource-engineering"}],chapters:[{id:"22991",title:"Non Edible Oils: Raw Materials for Sustainable Biodiesel",doi:"10.5772/25397",slug:"non-edible-oils-raw-materials-for-sustainable-biodiesel",totalDownloads:3922,totalCrossrefCites:3,totalDimensionsCites:7,hasAltmetrics:0,abstract:null,signatures:"C.L. Bianchi, C. Pirola, D.C. Boffito, A. Di Fronzo, G. Carvoli, D. Barnabè, R. Bucchi and A. 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In general, the potential of the confined region is lower than the surroundings (Figure 1) [1, 2].
Infinite potential well.
The potential of the system is defined as
The one dimensional Schrödinger equation in Cartesian coordinate is given as
In the infinite potential well, the confined particle is present in the well region (Region-II) for an infinitely long time. So the solution of the Schrödinger equation in the Region-II and Region-III can be omitted for our discussion right now. The Schrödinger equation in the Region-II is written as
The solution of the Eq. (2) is
At
Similarly, at
The addition and subtraction of these equations give two different solutions.
i.
The eigen function is
According to the normalization condition,
Hence the normalized eigen function for
ii.
The eigen function is
In Summary, the eigen value is
The integer “n” is the quantum number and it denotes the discrete energy states in the quantum well. We can extract some physical information from the eigen solutions.
The minimum energy state can be calculated by setting
This is known as zero-point energy in the case of the potential well. The excited state energies are
The energy difference between the successive states is simply the difference between the energy eigen value of the corresponding state. For example,
Though the eigen functions for odd and even values of “n” are different, the energy eigen value remains the same.
If the potential well is chosen in the limit
Step potential is a problem that has two different finite potentials [3]. Classically, the tunneling probability is 1 when the energy of the particle is greater than the height of the barrier. But the result is not true based on wave mechanics (Figure 2).
Step potential.
The potential of the system
The Schrödinger equation in the Region-I and Region-II is given, respectively as,
Case (i): when
where
According to admissibility conditions of wave functions, at
From these two equations,
The reflection coefficient R is given as
It is interesting to note that all the particles that encounter the step potential are reflected back. This is due to the fact that the width of the step potential is infinite. The number of particles in the process is conserved, which leads that
Case (ii): when
where
From these equations,
The reflection coefficient R and the transmission coefficient T, respectively, are given as
From these easily one can show that
The results again indicate that the total number of particles which encounters the step potential is conserved.
This problem clearly explains the wave-mechanical tunneling [3, 4]. The potential of the system is given as (Figure 3)
Potential barrier.
In the Region-I, the Schrödinger equation is
In Region-II, if
The Schrödinger equation in the Region-III is
At
At
Solving the equations from (27) to (30), one can find the coefficients in the equations. The reflection coefficient is R is found as
The transmission coefficient T is found as
From Eqs. (31) and (32), one can show that
When
As
When the length of the barrier is an integral multiple of
The tunneling probability depends on the height and width of the barrier.
Later, Kronig and Penney extended this idea to explain the motion of a charge carrier in a periodic potential which is nothing but the one-dimensional lattices.
The Dirac delta potential is infinitesimally narrow potential only at some point (generally at the origin, for convenience) [3]. The potential of the system
Here
Dirac delta potential.
The Schrödinger equation is
The solution of the Schrödinger equation is given as
where
This gives us
Equating the value of
The energy eigen value expression does not have any integer like in the case of the potential well. Hence there is only one bound state which is available for a particular value of “m.”
The eigen function can be evaluated as follows: The eigen function is always continuous. At
To normalize
This gives us
Simple harmonic oscillator, damped harmonic oscillator, and force harmonic oscillator are the few famous problems in classical physics. But if one looks into the atomic world, the atoms are vibrating even at 0 K. Such atomic oscillations need the tool of quantum physics to understand its nature. In all the previous examples, the potential is constant in any particular region. But in this case, the potential is a function of the position coordinate “x.”
The potential of the linear harmonic oscillator as a function of “x” is given as (Figure 5) [4, 5, 6]:
Potential energy of the linear harmonic oscillator.
The time-independent Schrödinger equation is given as
The potential is not constant since it is a function of “x”; Eq. (40) cannot solve directly as the previous problems. Let
Using the new constant
The asymptotic Schrödinger equation
The general solution of the equation is
The normalized eigen function is
The solution given in Eq. (43) is valid if the condition
The important results are given as follows:
The integer
The ground state normalized eigen function is
The energy difference between any two successive levels is
The operator method is also one of the convenient methods to solve the exactly solvable problem as well as approximation methods in quantum mechanics [5]. The Hamiltonian of the linear harmonic oscillator is given as,
Let us define the operator “a,” lowering operator, in such a way that
and the corresponding Hermitian adjoint, raising operator, is
Here,
In the same way, one can find the
Adding Eqs. (50) and (51) gives us the Hamiltonian in terms of the operators.
Subtracting Eq. (50) from (51) gives,
The Hamiltonian H acts on any state
The expectation value of
Let us consider the ground state
Since
Similarly, the energy of the first excited state is found as follows:
In the same way,
The uncertainties in position and momentum, respectively, are given as
In order to evaluate the uncertainties
a. The expectation value of ‘x’ is given as,
Since the states
b. The expectation value of momentum is
Not only position, the expectation value of momentum in any state is also zero.
c.
d.
From Eq. (58) and (59), the uncertainty in position and momentum, respectively are given as,
The minimum uncertainty state is the ground state. In this state,
Hence the minimum uncertainty product is
Since
Integrating the above equation gives,
The normalized eigen function is given as
One can see that this result is identical to Eq. (45).
The other eigen states can be evaluated using the equation,
The confinement of a particle in a three-dimensional potential is discussed in this section [4, 6]. The potential is defined as (Figure 6)
Three-dimensional potential box.
The three dimensional time-independent Schrödinger equation is given as
Let the eigen function
Divide the above equation by
Now we can boldly write E as
Now the equation can be separated as follows:
The normalized eigen function
In the same way,
Hence, the eigen function
The energy given values are given as
The total energy E is
Some of the results are summarized here:
In a cubical potential box,
The minimum energy that corresponds to the ground state is
Different states with different quantum numbers may have the same energy. This phenomenon is known as degeneracy. For example, the states (i)
The states (111), (222), (333), (444),…. has no degeneracy.
In this problem, the state may have zero-fold degeneracy, 3-fold degeneracy or 6-fold degeneracy.
The current trends in photovoltaic (PV) deployments worldwide are a clear indication that PV energy will play a big role in the near future energy mix. For example, the global solar photovoltaic (PV) capacity is projected to increase by 37.5% from 2019 to 2030 (i.e from 593.9 GW in 2019 to 1,582.9 in 2030) [1]. This rapid growth in grid penetration of PV calls for more accurate methods to forecast the performance and reliability of PV. Additionally, PV projects policy and investment decisions rely on PV performance and reliability forecasts. Therefore, to reduce the risks of PV investments, more reliable methods to forecast the performance and reliability of PV power are a prerequisite.
Different methods have been proposed for PV power forecasting. These methods can be classified as: physical, heuristic, statistical and machine learning methods [2, 3]. Each method might have different conceptual design, implementation, application and accuracy. In this chapter, the application and accuracy of the different methods are assessed using measured PV module power and weather data.
PV power forecasting can range from different temporal horizons depending on ones need. At a moment there is no standard classification criterion of the temporal horizon. General classification can be made as: very short term (Intra-hour: 15 minutes to 2 hours ahead), short-term (hour ahead: 1 to 6 hours ahead, day ahead: 1–3 days ahead), Medium-term (week to months ahead), long-term (one to several years) and lifetime forecast (until PV expected lifetime).
Achieving high-accuracy forecasts at each of these temporal horizons is influenced by different variables such as: solar radiation models, PV performance models, data availability, data quality and forecasting methods. The accuracy highly deteriorates with increasing forecasting temporal horizon. This is because, more input parameters such as: seasonal variations, soiling effects, degradation and many other performance reducing effects need to be taken into consideration [4]. These factors are not easy to evaluate precisely, therefore, they are simply approximated which increase the uncertainties in long-term and lifetime PV forecast.
The main influencing factor of PV production is the amount of global solar irradiation incident on the PV panels. As shown in Figure 1, there is a quasi-linear correlation of power and irradiance. This property means that the accuracy of power prediction is highly determined by the accuracy of the solar irradiation forecast.
Measured irradiance versus measured power. Data corresponds to six month measurements of irradiance and module power in Gran Canaria (Spain).
In this chapter, the different methods used in PV power forecasting are presented. The chapter is organised as follows: In Section 1, a brief introduction on power forecasting is presented. In Section 2, different PV forecasting methods are presented, for some methods a practical example of their application and their accuracy are evaluated using real measured PV module power. Section 3, is dedicated to lifetime PV power forecasting. In this section, several effects affecting lifetime PV power forecasting are stated and a more elaborative discussion of the degradation effect on lifetime PV power forecasting is presented. Lastly, in Section 4, a summary of the different aspects within the chapter is presented.
Different methods: physical, heuristic, statistical and machine learning are commonly used in PV power forecasting [2, 3]. The methods are based on two main approaches to generate the PV power forecasting. One is the physical approach, which requires prior knowledge of PV material properties and the metadata of a PV system, together with the need of weather data. The second ones is the data-driven approach, which requires operational data to train/calibrate coefficients of the models which are then used to generate the predictions. This means that, a data-driven approach can only be applied after a given PV module or system has been exposed and enough data is available to train/calibrate the models. On-contrary, a physical approach can be applied even when the PV system is not yet commissioned. Hence PV power forecasting methods based on a physical approach are the mostly used methods by PV stakeholders to evaluate the economic viability of PV projects during the initial phases.
Figure 2 illustrates the required inputs and general steps to generate PV power forecast by the two approaches. What is clear is that both approaches require weather data (mainly solar irradiation) as input. Therefore, solar irradiation forecast is highly essential step for PV power forecast using both approaches. Unlike data-driven approach, physical approach is based on physical assumptions and therefore, knowledge of the physical parameters influencing PV generation is required.
Schematic of data-driven and physical approaches for PV power prediction. Data-driven approach requires historical data (in green) to train and physical approach requires physical parameters as inputs.
Physical models calculate the PV power using the equivalent electrical circuits [5]. The equivalent circuit model developed for a single cell can be used to derive equivalent circuit models for a PV module as well as a PV system [6]. They are the commonly applied models in the PV power forecasting commercial software packages (such as PVSyst [7] PVWATTS [8]).
To build the physical model one need to know the basic photo-to-voltage theory. The diode model is used to develop the PV cell model to calculate the PV output power. The diode model can be characterized as: one-, two- and three-diode models [5, 7] (see Figure 3). The choice of the model selection depend on charge carrier recombination mechanisms one need to take into consideration. Because of its simplest, the one-diode model is the most commonly used to model the operating principles of a PV cell. The one-diode model can consist of three, four or five parameters (see Table 1) .
a, four parameters (4-p) one-diode model. b, five parameters (5-p) one-diode model. c, two-diode model (seven parameters) and d, three diode model (nine parameters).
Model | Parameters | Characteristic |
---|---|---|
3-p One-diode model | Basic model No series and shunt resistive losses | |
4-p One-diode model | Includes series losses No shunt losses | |
5-p One-diode model | Includes both series and shunt shunt resistive losses | |
Two-diode model | Two diodes to represent junction recombination Relevant at low irradiance operation of a PV cell | |
Three-diode model | Takes into account grain boundaries and leakage current |
Parameters and characteristics of different diode models.
Photocurrent.
Reverse saturation current, the subscripts (1, 2 and 3) represents the diode number respectively.
diode ideality factor the subscripts (1, 2 and 3) represents the diode number respectively.
Series resistance.
Shunt resistance.
The three parameter (3-P) one-diode model is only used to demonstrate the basic working principles of a PV cell but not as a representative of the real operating condition. To simulate the actual working conditions of a PV cell, the five parameter (5-P) one-diode model is commonly used because it takes into account both the series and shunt resistive losses. It is expressed as [9, 10]:
where
To derive the equivalent circuit model for a PV module, the basic assumption that a PV module comprises of identical PV cells in series is usually taken [6]. This assumption implies that under similar conditions (irradiance and temperature), all the PV cells should generate equal current and voltage. According to [6], the PV module equivalent model is derived from a PV cell diode model (Eq. (1)) as:
where
The photo-generated current
where
The reverse saturation current (
where
Readers are referred to [6, 10, 11, 12] for extended knowledge on how to derive and evaluate the different PV cell and module model parameters. The functions are also implement is freely available pvlib simulation packages [13].
Addition to solar irradiation, the I-V curve characteristics also depend on the operating temperatures of the cell
where
Figure 4 shows the effect of irradiance and module temperature on
Effect of irradiance and cell temperature on I-V characteristics. (I-V curve simulated using Pvlib one-diode model).
According to module mounting (e.g Open rack, close to the roof, insulated rack) and module construction (e.g glass–glass or glass-backsheet), the cell temperature might be some degrees hotter than module temperature [15]. In [15] the cell temperature (
where
The module temperature is calculated from solar irradiation, ambient temperatures and/or wind speed using different methods [16]. The commonly used models are: Standard NOCT model (Eq. (8)), the Faiman model (Eq. (9)) [17] and the Kings model also known as Sandia model (Eq. (10)) [15].
where NOCT is the nominal operating cell temperature at given conditions (
In this example, we demonstrate a practical application of the described physical model to predict four days PV module power using measured irradiance. The predicted power is then compared with real measured power of the PV module. The properties, electrical and thermal parameters of the PV module are presented in Table 2. The module is exposed in Gran Canaria, Spain at tilt angle of 23
Module properties | |
---|---|
Design and cell technology | Glass–Glass and Poly-crystalline silicon cells |
Number of cells | 80 |
Maximum power rating ( | 283 [Wp] |
Rated current ( | 7.2 [A] |
Rated voltage ( | 39.3 [V] |
Short-circuit current ( | 7.8 [A] |
Open-circuit voltage ( | 48.9 [V] |
Temperature coefficient of power ( | −0.47 [ |
Temperature coefficient of short-circuit current ( | 2.39 [mA/K] |
Temperature coefficient of open-circuit voltage ( | −161 [mV/K] |
Normal operating cell temperature | 48 [ |
Module properties and manufacturer datasheet electrical and thermal parameters.
The values are measured at STC (i.e. 1000 W/m2 irradiance, 1.5 air mass and at 25°C temperature).
The PV module specific parameters and the weather variables (irradiance and cell temperature) are used as input variable in Pvlib to simulate the five parameter (5-P) one-diode model (Eq. (2)). Figure 5 shows the simulated I-V curves at different irradiance and temperature levels for a period of four days. 15 minutes aggregated data of temperature and irradiance are used hence the curves are evaluated every 15 minutes. From each I-V curve the power at maximum power point can be computed using:
I-V curves of simulated solar module at different incident irradiance and temperature levels. Irradiance range from 0
In this example, the uncertainty of power prediction due to temperature models are evaluated. The three commonly used temperature models presented in (Eqs. (8), (9) & (10)) are used to model the module temperature. The parameters of the models are: a = −3.87 & b = −0.0594 for Kings model and
To evaluate the uncertainty in prediction, the normalized root mean square error (NRMSE) (Eq. (12)) is used. Additionally, the normalized mean bias error (NMBE) (Eq. (13)) is also evaluated as a metric to capture the average bias in the prediction (i.e, to check whether the predictions are overestimated or underestimated).
Where
Figure 6 shows the plot of measured and modeled temperature (a), measured and predicted power (b). The uncertainty in temperature models as well as the corresponding uncertainty in power predictions are presented in (c). It is clearly visible that, for each temperature model, the prediction is different and hence the uncertainty value. Generally, the Kings model showed the best performance both in temperature modeling as well as power prediction depending on the NRMSE and
a, Measured module (black) and ambient (yellow) temperatures, modeled temperature with standard NOCT (green), Faiman (blue) and Kings (red) models. b, Measured (black) and predicted power with module temperature calculated using NOCT (green), Faiman (blue) and Kings (red) models. c, Evaluated NRMSE and NMBE for module temperature modeling (blue) and power prediction (orange). d, Measured versus predicted power with module temperature calculated using NOCT, Faiman and Kings models.
Data-driven methods can be categorized into: data-driven heuristic methods, statistical methods and machine learning methods.
The physical models described in Section 2.1 have one big drawback that they require too many input variables which are not usually directly available. In this case, heuristic models are proposed to reduce the number of required inputs. They are heuristic models because they are not developed from physical assumptions/theories. Therefore, they have no physical dependencies/interpretations. They are classified as data-driven models because they are derived from correlation between weather and power output data. In [19], several heuristic models are presented and compared. They are developed on similar principles of generating power forecast from irradiance and module temperature but only differs in the numbers of fitting model parameters. The basic advantage of heuristic models is their simplicity to derive the model parameters from PV power historic data. Here we present the two- (Eq. (14)) and three- (Eq. (15)) parameter models described in [19, 20] respectively.
Where
In this example, the described heuristic models are applied to predict the power of the same PV module described in subSection
Model | Parameter | ||||
---|---|---|---|---|---|
a | b | c | x | y | |
1. parameter model (Eq. (14)) | — | — | — | 0.0255 | −0.03016 |
2. parameter model (Eq. (15)) | 0.2842 | −2.935e-5 | −0.0106 | — | — |
3. parameter- | 0.2802 | −8.985e-7 | −0.0111 | — | — |
Extracted model parameters of the heuristic models.
Tcorr is the temperature correction.
In Table 3 3-parameter model corresponds to calibration without a temperature term and 3-parameter-
Figure 7a shows a plot of the four days measured power and the predict power using 2-parameters and the 3-parameters model. Figure 7b shows the uncertainty in model calibration and the corresponding uncertainty in prediction using the different models. According to the NRMSE and NMBE values, the 3-parameter model with temperature correction term showed the best performance. From the same figure it can also be concluded that it is important to include a temperature correction in power prediction since the same model showed the least performance when applied without a temperature correction.
a, Measured power (black), predicted power using 2-parameter model (blue) and with 3-parameter model without (green) and with (red) temperature correction. b, Evaluated NRMSE and NMBE for the models during calibration (blue) and prediction (orange).
Like heuristic models, statistical and machine learning (ML) methods are also based on historical data to generate PV power forecast. While heuristic models focus on an in-deep formulation of mathematical operations, statistical models require selecting a model that considers previous knowledge of the system. ML methods require the selection of a predictive algorithm by relying on its empirical capabilities. Statistical models aim to “inference” the outcome of a model, while ML approaches aim to find generalizable predictive patterns [21]. Both statistical and ML methods are data-driven approaches that rely on the availability and accuracy of existing operational data to generate the forecasting. Usually, the larger the historical data, the better the PV system can be understood in terms of operational behaviour under different weather conditions and hence the better the forecasting accuracy.
The list of published methods is extensive and a case-to-case benchmark is usually needed. Statistical methods such as Naive method, ARIMA (Autoregressive Integrated Moving Average), SARIMA (seasonal ARIMA), Ordinary Least Squares (OLS) or Facebook Prophet (FbP) are typically applied for PV forecasting with and without regressors [22, 23, 24]. Below is a basic description of the commonly applied statistical methods. A detailed description of each method can be found on the respective cited reference.
More advanced methods, the so-called machine learning (ML) methods, can provide better results [29, 30], however, in most cases they require more computational efforts. Some examples of machine learning methods used in PV power forecasting include [2, 29, 30, 31]: k-nearest neighbors (k-NN), artificial neural networks (ANN), support vector machine (SVM), random forests (RF) and light gradient boost machines (LightGBM). The basic description of these methods is presented below with the references for a detailed description.
ML algorithms can be classified as supervised and unsupervised. A supervised ML algorithm uses labelled training data. It is related to a standard fitting procedure to find the unknown function/relationship between the input and output variables. The unsupervised ML algorithm uses unlabelled training data to find the data patterns (e.g., in the samples’ clustering). For PV power prediction, the supervised algorithms are commonly used, due to weather forecasting availability. In general, the procedure to run a ML algorithm can be composed of the following stages:
Data collection: the available historical data (weather and PV operational) are collected and filtered. The collection of weather forecasting data is also considered.
Feature selection: identification of the most relevant variables with regard to the PV power output selected for further analysis.
Data augmentation: in this stage, the enhancement of the initial dataset is expected by typically applying mathematical operations (e.g., physical relationships) to one or more relevant input variables.
Dataset split: the input dataset is divided into a training and testing dataset. Also, a validation dataset is recommended. This task is typically applied over the sorted or random timestamps.
Accuracy improvement: statistical indicators such as the MBE, RMSE or R
A simple ANN network architecture is shown in Figure 8, where the layers of a multilayer perceptron (MLP) for PV power forecasting are presented. The input data for training can be the historical weather and PV power output, while for testing and forecasting, expected weather variables are the input to the expected PV power output in the future.
Multilayer perceptron (MLP) for PV power forecasting, where the input layer includes at least irradiance and temperature, while the output layer comprises the PV energy yield.
The statistical and ML approaches are applied to the same dataset used for physical and heuristic models. To train the models, the regressors selected are limited to the plane-of-array irradiance and PV module temperature, while the target variable is the PV output power. Most of these models are already implemented in open-source software packages (i.e., statsmodels [38], prophet [28], sklearn [39] and lightgbm [39]) and executed with Python scripts. Setting parameters for each statistical and machine learning method are given in Table 4.
Model | Parameter | ||
---|---|---|---|
Setup 1 | Setup 2 | Setup 3 | |
ARIMAa | p: 0 | q:0 | d: 0 |
SARIMAa | p: 1 | q:1 | d: 1 |
OLSa | — | — | — |
Prophetb | Daily Seasonality | Changepoint prior scale: 0.01 | — |
DNNc | Hidden layer size: (40,20,10) | relu activation | adam solver |
LightGBMd | Number of leaves: 10 | Min. data in leaf: 5 | Number iterations: 100 |
SVMc | Kernel: rbf | Degree: 3 | Regularization: 1e6 |
Setup parameters for different statistical and machine learning approaches.
Statsmodels.
Prophet.
Sklearn.
Lightgbm.
Figure 9a shows a plot of the four-day measured power and the predicted power using statistical and machine learning models. Figure 9b shows the uncertainty in model calibration and the corresponding uncertainty in prediction using the different models. According to the NRMSE, LightGBM model shows the best performance, followed by the SVM and Facebook Prophet.
A, Measured power (black), predicted power using statistical and machine learning approaches. B, Evaluated NRMSE and NMBE of statistical and machine learning approaches at calibration and predictions.
Generally, comparing the uncertainty values among the physical, heuristic, statistical and machine learning methods (see
To begin with, it is important to understand how the lifetime of a PV module is defined. Unlike other electrotechnical devices where the term lifetime is clearly defined [40], the definition of a PV module “lifetime” is somewhat complex. This is because, despite the catastrophic events (such as fire) it is unlikely that a PV module drops its power generation to zero. However, even though a PV module is still generating power, its output might be too low to be economically viable to continue its operation. Therefore, in general terms PV lifetime is defined in economical rather than technical terms.
For economical viability of PV projects, most PV module manufacturers guarantee a power reduction of less than 20% within 25–30 years of operation. The 20% power reduction is usually referenced at standard test conditions (STC) (modules tested under 25
The actual performance of a PV module throughout its lifetime is very uncertain and difficult to accurately forecast. This is because many factors can influence the performance of a PV module. Some of these factors may include: solar resource, the quality of the PV components and the long-term variations in system performance (degradation). All these factors increases the uncertainty in PV lifetime forecast. The Internation Energy Agency- Photovoltaic Power Systems Programme (IEA-PVPS) -task 13 report [41] provides a detailed overview of the uncertainties in lifetime yield predictions. To improve the accuracy and to achieve reliable lifetime PV forecast, all these effects must be explored separately. Here, we asses the effect of degradation on lifetime PV power forecasting.
Degradation is defined as the gradual and non-reversible decrease in PV performance over time. Degradation is a crucial influencing factor to be taken into account during lifetime PV power forecast. This is because over time the PV components are ageing and deteriorating in their normal operation. Understanding how PV degrades is a very widely studied topic in the PV community but it is also among the not well understood topics. This can be explained by the numerous factors influencing PV degradation. These factors include: PV technology, bill of materials (BoM), climatic conditions [4], transportation, installation and operational conditions. More-so, since new materials are being proposed frequently, it increases the complexity to correlate the factors that influence PV degradation. Usually, different materials have different degradation kinetics and are influenced by different stress factors differently.
In lifetime power prediction models, degradation effect is included in a number of ways. For example, according to the survey carried out among PVPS-task 13 experts regarding degradation effects in lifetime energy yield prediction, the following assumptions are taken [41]: (a). A variable degradation during the first five year of operation and a fixed degradation from 5 to 30 years of operation. A degradation of 1–2% is assumed in the first year, 0.7% to 0.5% to year 5 and0.3% to 0.5% up to year 30. (b). Initial degradation of 0.3% to 1.0% in the first years to include the effects of initial degradation modes such as light induced degradation (LID). (c). Constant degradation over the years with the exception of the first year to take into account technology specific behaviour.
Generally, a constant degradation rate with linear performance loss is considered (see Eq. (16)). However, some authors [42, 43, 44], have evaluated and modeled the non-linearity in degradation rates and performance. For example in [43] a non-linear power degradation model (Eq. (17)) was proposed and applied in [44] with a time-dependent degradation rates to predict PV performance lifetime.
Where
In this example, the effect of degradation rate on lifetime power forecast is presented. Using, 30 years of historical weather data (global irradiance, ambient temperature and wind speed from ERA 5 reanalysis [45]), three different degradation scenarios are presented and their impact on lifetime power and energy yield prediction. The first scenario is using a non-linear performance degradation with a shape parameter (
Figure 10 shows the PV lifetime power and yield predictions using different scenarios. It can be seen that depending on the degradation scenario, the lifetime yield is significantly different. In numbers, when compared to the usually used linear scenario, a relative difference of over 5.0% is evaluated in respect with the non-linear scenarios.
A, Lifetime power prediction using different degradation scenarios: non-linear with shape parameter = 1.2 (red), linear (green) and non-linear with shape parameter = 0.2 (blue) as well as a no degradation scenario (cyan). B, Corresponding lifetime yield for all the respective scenarios.
PV power forecasting is important to stabilize the electrical grids, financing PV projects and also to plan operational and maintenance activities. In this regard, different methods are proposed to forecast the PV power generation. In this chapter, the different methods used in PV power forecasting are presented, applied and their accuracy in PV forecasting is evaluated using measured PV module and weather data. The degradation effect on lifetime PV power forecasting is also assessed using two main scenarios; linear degradation scenario and non-linear degradation scenario. The key observations in the chapter are:
The uncertainties in PV module temperature modelling affect the forecasting accuracy. In the chapter, three temperature models: Standard NOCT, Faiman and King’s are compared, the King model showed the best performance among the three models. The standard NOCT model, that neglects the impact of wind speed displayed huge uncertainty.
Data-driven models outperforms physical models in prediction accuracy. This can be explained by the fact that physical models are derived from to many assumptions and that they need too many input parameters that are usually approximated.
For lifetime PV power forecasting, a relative difference of over 5% is evaluated between the linear and non-linear degradation scenarios.
The authors declare no conflict of interest.
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