Photovoltaic Sources Modeling E-Book

Giovanni Petrone

• Herausgeber: John Wiley & Sons
• Kategorie: Wissenschaft und neue Technologien
• Serie: Wiley - IEEE
• Sprache: Englisch

Seitenzahl: 319

Veröffentlichungsjahr: 2017

Beschreibung

A practical reference to support choosing, customising and handling the best PV simulation solution

This comprehensive guide surveys all available models for simulating a photovoltaic (PV) generator at different levels of granularity, from cell to system level, in uniform as well as in mismatched conditions. Providing a thorough comparison among the models, engineers have all the elements needed to choose the right PV array model for specific applications or environmental conditions matched with the model of the electronic circuit used to maximize the PV power production.

Key features:

• Multiple mathematical models are given for different application requirements.
• The shading effect is taken into account to improve the model accuracy.
• Procedures for parameter identification of the PV model are analysed and compared.
• Mathematical manipulations are introduced to some models to reduce their calculation time.
• The electronic interface effect on the power chain is analysed.
• Analytical expressions are used to design and control the power converter driving the PV field.

The book is an essential reference for R&D in the PV industry; designers of power converters for PV; PV systems designers; and practicing engineers.

Leseprobe

Table of Contents

Cover

Title Page

Acknowledgements

Introduction

Tables of Symbols and Acronyms

1 PV Models

1.1 Introduction

1.2 Modeling: Granularity and Accuracy

1.3 The Double‐diode Model

1.4 The Single‐diode Model

1.5 Models of PV Array for Circuit Simulator

1.6 PV Dynamic Models

1.7 PV Small‐signal Models and Dynamic‐resistance Modelling

References

2 Single‐diode Model Parameter Identification

2.1 Introduction

2.2 PV Parameter Identification from Datasheet Information

2.3 Single‐diode Model Simplification

2.4 Improved Models for Amorphous and Organic PV Technologies

References

3 PV Simulation under Homogeneous Conditions

3.1 Introduction

3.2 Irradiance‐ and Temperature‐dependence of the PV Model

3.3 Simplified PV Models for Long‐term Simulations

3.4 Real‐time Simulation of PV Arrays

3.5 Summary of PV Models

References

4 PV Arrays in Non‐homogeneous Conditions

4.1 Mismatching Effects: Sources and Consequences

4.2 Bypass Diode Failure

4.3 Hot Spots and Bypass Diodes

4.4 Effect of Aging Failures and Malfunctioning on the PV Energy Yield

References

5 Models of PV Arrays under Non‐homogeneous Conditions

5.1 The use of the Lambert W‐Function

5.2 Application Examples

5.3 Guess Solution by Inflection‐point Detection

5.4 Real‐time Simulation of Mismatched PV Arrays

5.5 Estimation of the Energy Production of Mismatched PV Arrays

References

6 PV Array Modeling at Cell Level under Non‐homogeneous Conditions

6.1 PV Cell Modeling at Negative Voltage Values

6.2 Cell and Subcell Modeling: Occurrence of Hot Spots

6.3 Simulation Example

6.4 Subcell PV Model

6.5 Concluding Remarks on PV String Modeling

References

7 Modeling the PV Power Conversion Chain

7.1 Introduction

7.2 Review of Basic Concepts for Modeling Power Converters

7.3 Effects of the Converter in the Power Conversion Chain

7.4 Modelling the Dynamics of the Power Conversion Chain

7.5 Additional Examples

7.6 Summary

References

8 Control of the Power Conversion Chain

8.1 Introduction

8.2 Linear Controller

8.3 Sliding‐mode Controller

8.4 Summary

References

Index

End User License Agreement

List of Tables

Tables of Symbols and Acronyms

Table 1 Acronyms

Table 2 Symbols

Chapter 1

Table 1.1 Scaling up the SDM parameter for a PV panel of 36 cells in series.

Chapter 2

Table 2.1 Energy bandgap at

.

Table 2.2 Datasheet information at STC.

Table 2.3 Values of the five parameters of Kyocera and Sanyo panels at STC.

Table 2.4 Values of the five parameters of Kyocera KC175GHT‐2 for different irradiance conditions.

Table 2.5 Suntech STP280‐24 V datasheet information at STC.

Table 2.6 Parameters of the SDM for the Suntech STP280‐24Vd PV panel.

Table 2.7 Datasheet information for Solarex MST‐56MV at STC.

Table 2.8 Calculated SDM parameters for the Solarex MST‐56MV.

Table 2.9 Parameters of the two‐diode model for an organic PV cell.

Chapter 3

Table 3.1 Empirical coefficients for estimating cell temperature

T

and temperature at the rear of the PV panel.

Table 3.2 Values of the five parameters of Kyocera KC175GHT‐2.

Table 3.3 Errors in the MPP estimate for the Kyocera KC175GHT‐2 panel.

Table 3.4 Parameters of the King model.

Table 3.5 Polynomial coefficients required for modeling spectral response.

Table 3.6 Approaches for modeling the electrical behavior of a PV field and calculating the

parameters.

Chapter 4

Table 4.1 Typical values of derating factors.

Table 4.2 Frequency of tickets and associated energy loss for each failure area.

Table 4.3 Frequency of tickets for specific failures of PV modules.

Chapter 5

Table 5.1 Irradiation and temperature profile for the simulated PV string.

Chapter 6

Table 6.1 Cell and bypass diode parameters used.

List of Illustrations

Chapter 1

Figure 1.1 Equivalent circuit of the double‐diode model.

Figure 1.2 Equivalent circuit of the single‐diode model.

Figure 1.3 Equivalent circuits of (a) the ideal single‐diode model; (b) and (c) two simplified single‐diode models.

Figure 1.4 Fill factor identification on a I–V curve.

Figure 1.5 I–V curve modification due to SDM parameter variation.

Figure 1.6 I–V curve modification due to

R

s

and

R

h

variation.

Figure 1.7 The scalable SDM can be used to model a single PV cell or a large PV field operating in homogeneous and stationary environmental conditions.

Figure 1.8 PV electrical schemes based on computational blocks suitable for simulating PV fields in presence of irradiance (

G

) and ambient temperature (

T

a

) variations.

Figure 1.9 PV electrical schemes based on the Lambert W‐function.

Figure 1.10 Dynamic model of the PV generator.

Figure 1.11 PV frequency behavior in the impedance plane;

k

Ω

,

,

F.

Figure 1.12 Comparison of SDM and 1oT models.

Figure 1.13 Linear small‐signal models.

Figure 1.14 Comparison of the SDM, Norton, and Thevenin models.

Figure 1.15 Comparison of SMD, Norton, and Thevenin models with accurate MPP approximation.

Chapter 2

Figure 2.1 PV curves obtained using the approximated set of parameters (

I

ph

,

I

s

,

η

,

R

s

,

R

h

) and using an exact method [1] for the two panels under study.

Figure 2.2 Error in the PV current obtained with the approximated set of parameters (

I

ph

,

I

s

,

η

,

R

s

,

R

h

) and the exact parameters [1] for the two panels under study.

Figure 2.3 P–V and I–V curves for the Kyocera KC175GHT‐2 PV panel; approximated parameters and experimental data extracted from the datasheet.

Figure 2.4 Error in PV current between value obtained with the approximated set of parameters and value extracted from the datasheet for Kyocera KC175GHT‐2 panel.

Figure 2.5 PV curves of the Suntech STP280‐24Vd PV panel obtained using the two sets of approximated parameters.

Figure 2.6 Percentage error in the PV current obtained with the two approximated sets of parameters (

I

ph

,

I

s

,

η

,

R

s

,

R

h

) with respect to the PV current extracted from the datasheet of the Suntech STP280‐24Vd PV panel.

Figure 2.7 SPR indicator as a function of FF: (A)

SPR

A

at

; (B)

SPR

B

at

; (C)

SPR

C

at

.

Figure 2.8 Single‐diode equivalent circuit for amorphous PV panels.

Figure 2.9 PV curves of the Solarex MST‐56MV PV panel obtained by using the standard SDM and the improved model [13].

Figure 2.10 Equivalent circuit for organic solarcells.

Figure 2.11 PV curves of the organic PV cell obtained using standard SDM and the improved version [23].

Chapter 3

Figure 3.1 I–V curves of the Kyocera KC175GHT‐2 panel obtained using De Soto’s translated parameters and explicit equations.

Figure 3.2 Measured fill factors of PV modules on a sunny day.

Figure 3.3 I–V curves of the Kyocera KC175GHT‐2 panel obtained using modified Picault and De Soto translated parameters compared with the experimental data from the datasheet.

Figure 3.4 I–V curve showing the five points provided by the King’s model.

Figure 3.5 I–V curves of the Kyocera KC175GHT‐2 panel obtained using the King translated parameters.

Figure 3.6 I–V curves of the Kyocera KC175GHT‐2 panel obtained using the King translated parameters.

Figure 3.7 Parameter drift calculated using the King equation for different irradiance and ambient temperature levels: left, BP Solar BP4175; right, Sanyo HIP‐190BA2.

Figure 3.8 Parameter drift calculated using the King equation for different values of irradiance and ambient temperature: left column, BP Solar BP4175; right column, Sanyo HIP‐190BA2.

Figure 3.9 Neural network configuration for parameter estimation (

).

Figure 3.10 MPPT architecture for ANN with fuzzy logic controller.

Figure 3.11 Grid‐connected PV utility with DC/DC power converter and intermediate DC link.

Figure 3.12 Averaged model of the PV array and the DC/DC power interface operating in continuous conduction mode.

Figure 3.13 Comparison of measured and simulated waveforms from 5:38 am to 8:22 pm, 8 May 2010 in Vancouver, Canada.

Chapter 4

Figure 4.1 Snail trails affecting a PV panel.

Figure 4.2 Dirt affecting a PV panel, visible as dark speckles in the bottom part of the upper side panels and in the upper part of the panels in the row below.

Figure 4.3 Shadowing of one string of PV panels by another.

Figure 4.4 Shadowing due to an obstacle, such as a tree.

Figure 4.5 PV string with one cell shadowed.

Figure 4.6 Solder melt at hot‐spot site.

Figure 4.7 Hot spot temperature and shadowing area.

Figure 4.8 Cable shadowing affecting some PV panels.

Figure 4.9 Photovoltaic panel degradation rate.

Chapter 5

Figure 5.1 String of

M

series‐connected PV modules including a blocking diode to prevent current back‐flow.

Figure 5.2 Circuit model of a PV module including the bypass diode

D

b

.

Figure 5.3 I–V curve of the PV string obtained by using the three methods. Differences are indistinguishable.

Figure 5.4 Inflection points for four PV series modules.

Chapter 6

Figure 6.1 Single‐diode model including the reverse polarization term.

Figure 6.2 Complete I–V curve of a PV cell. This plot has been obtained by using the following parameter values:

,

,

,

,

,

,

,

.

Figure 6.3 Effect of the shunt resistance on the I–V curve of a PV cell. Parameter values as per Figure 6.2, and with

(continuous line) and with

(dashed line).

Figure 6.4 PV array including

M

modules, each one made from

N

cells.

Figure 6.5 Clustering of Jacobian matrix

J

.

Matrix 6.1 Jacobian matrix for the system of equations describing the string behavior at cell level.

Figure 6.6 I–V curve for PV system with a shadowed cell.

Chapter 7

Figure 7.1 General structure of a module integrated unit.

Figure 7.2 Boost converter operation.

Figure 7.3 Boost converter simulation.

Figure 7.4 Inductor current ripple in the boost converter.

Figure 7.5 Block diagram for the steady‐state simulation.

Figure 7.6 MIU based on a boost converter.

Figure 7.7 Power curves of a MIU based on a BP‐585 PV module and a boost converter.

Figure 7.8 Effect of converter efficiency in the optimal PV voltage.

Figure 7.9 Flowchart to test a P&O algorithm using the steady‐state model.

Figure 7.10 Simulation of the P&O algorithm using the steady‐state model.

Figure 7.11 Power profile generated by a MIU.

Figure 7.12 Model of the connection between the PV module and the converter.

Figure 7.13 Impedance of the PV module at the optimal condition of the MIU in Figure 7.5.

Figure 7.14 PV voltage ripple in a MIU.

Figure 7.15 PV system including the DC/DC converter effect in the output I–V curve.

Figure 7.16 Effect of the DC/DC converter on the output I–V curve.

Figure 7.17 Current‐vs‐voltage curves of equivalent PV modules.

Figure 7.18 Comparison between circuit simulation, switched equations and averaged equations for the MIU in Figure 7.5.

Figure 7.19 Values of

for a BP‐585 PV module.

Figure 7.20 Frequency response of circuit and model representing the MIU in Figure 7.6.

Figure 7.21 Natural frequency and damping of the PV voltage in the MIU in Figure 7.6.

Figure 7.22 Settling time and time response of the PV voltage.

Figure 7.23 Effect of environmental and load conditions on the dynamic behavior of the PV voltage.

Figure 7.24 Effect on the dynamic behavior of the PV voltage of variations in the converter components (

L

and

C

).

Figure 7.25 Effect on the dynamic behavior of the PV voltage of variations in the converter components (

R

L

).

Figure 7.26 MIU based on a buck converter.

Figure 7.27 MIU based on a buck–boost converter.

Chapter 8

Figure 8.1 General structure of a closed‐loop MIU.

Figure 8.2 MIU based on a boost converter using linear control of the PV voltage.

Figure 8.3 Effect of negative‐gain plant in feedback control systems.

Figure 8.4 MATLAB®

sisotool

interface for PID controller design.

Figure 8.5 Performance of the linear controller.

Figure 8.6 Simulation of the linear controller.

Figure 8.7 Scheme of a PV voltage SMC based on inductor current control.

Figure 8.8 Implementation of the SMC for

.

Figure 8.9 Simulation of the SMC for

.

Figure 8.10 Simulation of PV voltage control including the SMC for

.

Figure 8.11 Scheme of a PV voltage SMC based on capacitor current control.

Figure 8.12 Implementation of the SMC for

.

Figure 8.13 Simulation of PV voltage control including the SMC for

.

Guide

Cover

Table of Contents

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Photovoltaic Sources Modeling

Giovanni Petrone

University of Salerno, Italy

Carlos Andrés Ramos‐Paja

National University of Colombia

Giovanni Spagnuolo

University of Salerno, Italy

This edition first published 2017© 2017 John Wiley & Sons Ltd

All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, except as permitted by law. Advice on how to obtain permission to reuse material from this title is available at http://www.wiley.com/go/permissions.

The right of Giovanni Petrone, Carlos Andrés Ramos‐Paja and Giovanni Spagnuolo to be identified as the authors of this work has been asserted in accordance with law.

Registered OfficesJohn Wiley & Sons, Inc., 111 River Street, Hoboken, NJ 07030, USAJohn Wiley & Sons Ltd, The Atrium, Southern Gate, Chichester, West Sussex, PO19 8SQ, UK

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Library of Congress Cataloging‐in‐Publication Data

Names: Petrone, Giovanni, author. | Ramos‐Paja, Carlos Andrés, author. | Spagnuolo, Giovanni, author.Title: Photovoltaic sources modeling/Giovanni Petrone, University of Salerno, Italy, Carlos Andrés Ramos‐Paja, Universidad Nacional de Colombia, Giovanni Spagnuolo, University of Salerno, Italy.Description: Chichester, West Sussex, United Kingdom : John Wiley & Sons, Inc., [2017] | Includes bibliographical references and index.Identifiers: LCCN 2016038240 (print) | LCCN 2016054257 (ebook) | ISBN 9781118679036 (cloth) | ISBN 9781118756126 (pdf) | ISBN 9781118756492 (epub)Subjects: LCSH: Photovoltaic power generation–Mathematical models.Classification: LCC TK1087 .P43 2017 (print) | LCC TK1087 (ebook) | DDC 621.31/244011–dc23LC record available at https://lccn.loc.gov/2016038240

Cover image: Tetra Images/GettyimagesCover design by Wiley

To Patrizia, Nicola and Marco AndreaGiovanni Petrone

To Claudia Patricia and AlejandroCarlos Andrés Ramos‐Paja

To Simonetta and ValeriaGiovanni Spagnuolo

Acknowledgements

The authors wish to acknowledge the research groups they belong to, the University of Salerno and the Universidad Nacional de Colombia, for the moral and technical support in achieving the research results and the book writing.

G. Petrone and G. Spagnuolo also acknowledge the University of Salerno and the Italian Ministry of Education for having funded, through many projects, the research activities the results of which are outlined in this book.

C.A. Ramos‐Paja also acknowledges the Universidad Nacional de Colombia and Colciencias (Fondo Nacional de Financiamiento para Ciencia, la Tecnología y la Innovación Francisco José de Caldas) for having funded project MicroRENIZ‐25439 (Code 1118‐669‐46197) the results of which are outlined in this book.

G.Petrone and G.Spagnuolo acknowledge Prof. N.Femia and Prof.L.Egiziano, from the University of Salerno, for the joint research activity in photovoltaics over the last sixteen years and for their support since the PhD course studies.

SolarTechLab1 of Politecnico di Milano (Italy) is acknowledged for some photos in Chapter 4.

The contribution of Dr. Martha Lucia Orozco Gutierrez of the Universidad del Valle (Colombia) is warmly acknowledged for her contributions to Chapters 5 and 6.

Note

1

http://www.solartech.polimi.it

.

Introduction

The idea of writing this book was born ten years ago, when the authors, who before then used the simple electrical models that had been reported in the – very scant – literature at that time, published their first paper on this topic. At that time, the values of the parameters of the widely used single‐diode circuit model (SDM) were identified through approximate procedures, often consisting of two steps, the first neglecting the loss mechanisms and the following one introducing them by fixing the values of the other parameters. Mismatching phenomena, and especially partial shadowing, were accounted for by simple multiplication of the array power production in standard test conditions by a loss factor. Events producing degradation of the cells and bypass diodes were only observed or simulated in simplified cases. Also, the need to simulate the array, including bypass and blocking diodes, in unconventional operating conditions and in real time using an embedded system, did not arise at that time, so that the problem of having a fast, accurate, efficient and reliable simulation of even large arrays was treated in literature only marginally. Ten years later, the authors have gained an improved knowledge of these matters, which has been condensed in this book with the main aim of driving the reader towards the best and more suitable array model for a given application.

The SDM is used first, in Chapter 1, for fast simulations of even large arrays working in standard conditions through a suitable symbolic manipulation of the non‐linear equation describing it at the terminals through the current and voltage variables. This chapter gives the background for understanding almost all the advanced approaches presented in the book. In order to use the aforementioned model in practice, and also the others in the book, the reader has to turn the data available in the datasheet provided by any panel manufacturer into the parameters involved into the circuit model. This problem is addressed in the second chapter of the book, where methods – both approximations using explicit formulae and accurate ones – requiring the solution of a non‐linear system of equations, are compared. The SDM presented in the first chapter and the parameters values determined by the methods described in Chapter 2 are thus ready for simulating arrays working in the so‐called “standard test conditions” the datasheet measurements refer to. Chapter 3 allows the reader to extend the simulation capabilities of the SDM by suggesting a number of additional equations allowing simulation of the array in any operating conditions, both in terms of irradiance and temperature values, under the assumption that all the cells in the array work in exactly the same operating conditions and are described by the SDM with the same parameter values.

Mismatched conditions, and especially the presence of the partial shadowing phenomenon affecting the array, need more sophisticated simulation models and tools. Chapter 4 gives an overview of the main sources of degradation and mismatching, pointing out that in many cases the effect is at cell level, and so the modeling of such conditions may require models with different levels of detail and granularity. Chapter 5 introduces a first possibility, describing a computationally efficient simulation model that works at the granularity level of the module. Thus each panel can be simulated by accounting for up to two or three different operating conditions affecting the modules from which it is made. The proposed approach allows for very fast simulation of the mismatched array, with more accuracy than when introducing a simple power reduction. Nevertheless, cell‐level phenomena such as hot spots and a detailed string behavior can be achieved only by modeling the array at cell level. For fast simulation of arrays consisting of thousands of cells, this requires more detailed models and the use of algorithmic approaches, and also the consideration of cell reverse biasing.

Chapter 6 gives the modeling and algorithmic recipes for approaching this task in a proper way: some examples and comparisons are presented in order to show the reader that a suitable organization of the equations in the overall non‐linear model allows for a saving of hours of simulation time compared to the use of the standard algorithms that are built into commercial mathematical software packages. Once the reader has chosen the right array model for the analysis that he or she has to perform, it is merged with the model of the power processing system that allows harvesting of the maximum PV power at any given operating condition. Indeed, the literature and market trends clearly indicate that, in the near future, electronics will be embedded into PV modules with the aim of maximizing the power output and to make its operating point independent of other modules connected in series or in parallel to it. Thus the module and its dedicated electronics will no longer be distinguishable and their integrated modeling will be required.

Chapter 7 shows that a suitable array model is needed for the design of the switching converter that controls the power flux. Static and dynamic performance can be optimized provided that a suitable simulation of the whole system, including both the array and the power‐processing system, is performed. Once this has been achieved, different control techniques can be used, as summarized in Chapter 8.

Tables of Symbols and Acronyms

Table 1 Acronyms

Acronym

Definition

DDM

Double‐diode model

DMPPT

Distributed maximum power point tracking

FF

Fill factor

GCR

Ground‐cover ratio

ISDM

Ideal single‐diode model

I–V

Current vs voltage

MIU

Module integrated unit

MPP

Maximum power point

MPPT

Maximum power point tracking

P–V

Power vs voltage

PV

Photovoltaic

P&O

Perturb and observe MPPT algorithm

SDM

Single‐diode model

SIF

Shade‐impact factor

SSDM

Simplified single‐diode model

STC

Standard test conditions

Table 2 Symbols

Symbol

Definition

Value or units

C

0

Temperature coefficient

E

g

Silicon energy gap

J

G

Irradiance

W/m

2

G

0

Irradiance reference condition

W/m

2

G

STC

Irradiance in STC

1000 W/m

2

I

MPP

Current at the maximum power point

A

i

MPP

Small signal current at the maximum power point

A

I

ph

PV photo‐induced current

A

I

sc

PV short‐circuit current

A

I

s

PV diode saturation current

A

I

s

,

db

Bypass diode saturation current

A

I

s

,

blk

Blocking diode saturation current

A

I

STC

DC current in STC

A

K

Boltzman constant

M

(

D

)

Voltage conversion ratio of a power converter

n/a

P

MPP

Power at the maximum power point

W

q

Electron charge

R

s

Series resistance

Ω

R

h

Shunt resistance

Ω

T

Cell temperature

K

T

0

Cell temperature reference condition

K

T

a

Ambient temperature

K

T

m

Temperature at the backside of the PV modules

K

T

STC

Cell temperature in STC

298 K

V

MPP

Voltage at the maximum power point

V

v

MPP

Small signal voltage across the maximum power point

V

V

oc

PV open‐circuit voltage

V

V

STC

DC voltage in STC

V

V

t

Thermal voltage of PN junction

V

V

t

,

db

Thermal voltage of bypass diode

V

V

t

,

blk

Thermal voltage of blocking diode

V

W

(

θ

)

Lambert W function with argument

θ

n/a

α

i

Temperature coefficient of the short circuit PV current

(%)/

°

C

α

v

Temperature coefficient of the open circuit PV voltage

(%)/

°

C

η

Diode ideality factor

n/a

η

pc

Efficiency of a power converter

n/a

1PV Models

1.1 Introduction

As for any physical system, PV cell modeling can be done with different levels of accuracy, depending on the user’s purposes. In this book the PV generator is always modeled through an equivalent circuit and by using concentrated parameters and variables. Indeed, the aim is to give the reader tools for implementing the PV array model in simulation environments, allowing them to analyze and design the whole PV generator, including the power‐processing system feeding a load or the grid. For users interested in this kind of study, access to data concerning the physical properties of the semiconductor material involved or parameter values that depend on the cells’ manufacture, such as dopant concentrations or material response to the radiation spectrum, is either not easy or the figures are not readily translated into circuit parameters. Instead, the use of laboratory measurements at the PV terminals, in terms of current and voltage values, or use of experimental data from the product datasheet, are more viable ways of proceeding.

This chapter introduces the two main circuit PV models used in the literature: the single‐diode and the double‐diode models. The first is the more widely used because of the reduced number of circuit parameters to be identified. The double‐diode model (DDM) has better accuracy, especially at low irradiance levels, but it requires a more involved identification of the parameter values. Thus while some space is dedicated in this chapter to the DDM, in the following chapters the single‐diode model (SDM) will be considered the reference one.

1.2 Modeling: Granularity and Accuracy

In the last five years a huge literature has been devoted to modeling PV sources, for two main purposes. The first is the reproduction of the I–V curve at the generator terminals through a suitable electrical model, regardless of the size of the PV source: from one PV cell, or even sub‐portions of it, up to large PV fields made of series‐connected modules forming strings that are in turn connected in parallel. The second purpose is performing energetic analyses concerning plant productivity, using models based on empirical or semi‐empirical equations.

The first set of models aims to describe the functional current–voltage relationship at the PV terminals on the basis of the equivalent electrical circuit of the PV source. Such models are usually scalable and the parameter values can be varied according to the operational weather conditions at the PV source. They are also useful for modeling unusual operating conditions, such as mismatches due to partial‐shading phenomena. The implementation in circuit‐oriented simulators such as PSPICE and PSIM, or in general‐purpose simulation environments such as MATLAB® and SCILAB, is almost straightforward. This allows study of the PV source together with dedicated controls, power‐processing systems, switching converters, and so on. The PV‐source non‐linearity is accurately reproduced by these models, but at the cost of a significant computational burden, especially if the granularity of the simulation is at the cell, or even sub‐cell level, in the context of a large PV field consisting of thousands of cells. As a consequence, PV equivalent circuit models allow simulations of systems to be performed over short time windows, usually of fractions of a minute.

The second set of models is aimed at performing long‐term analyses – over days or months – and they therefore cannot use detailed descriptions of the current–voltage relationship in any operating conditions. Instead, they use simplified equations in which the energy produced by the PV source is described as a function of the environmental conditions and parameters related to the installation type. These approaches usually require tuning, with correcting factors based on experimental measurements used to account for the real operating conditions of the PV field. Approaches based on fuzzy logic [1] and neural networks [2] are often used to take into account the historical meteorological data of the installation site when estimating the produced energy, in order to predict the pay‐back time and to evaluate the economic viability of the PV system.

The largest part of this book is focused on the first set of models. The next sections introduce the main circuit models used for PV source simulation in ordinary operating conditions.

1.3 The Double‐diode Model

The double‐diode model (DDM) describes the PN junction operation through the Shockley equation, and includes series and shunt resistances to incorporate the current‐dependent and the voltage‐dependent loss mechanisms. It has the important feature of modeling the carrier‐recombination losses in the depletion region.

The non‐linear equation giving the relationship between the current and the voltage at the PV source terminals is:

(1.1)

This equation can be represented in any circuit‐oriented simulator by the equivalent circuit shown in Figure 1.1. Id1 and Id2 are the second and the third additive terms in (1.1).

Figure 1.1 Equivalent circuit of the double‐diode model.

The photoinduced current Iph, the two diodes’ saturation currents Is1 and Is2, the two diodes’ ideality factors η1 and η2, and the two resistances Rs and Rh are unknown parameters. Their values have to be identified on the basis of measurements performed by the producer or by the user, in the laboratory or in the field, on the real PV unit that is to be simulated. is the thermal voltage of the PN junction in the PV cell, and it is calculated by considering the cell temperature. The value of the second saturation current Is2 is three to five times higher than the first:

(1.2)

Detailed expressions of the two saturation currents, including the temperature dependencies, are given in the literature [3, 4]. As for the two diodes’ ideality factors, the first is often given in the literature at a value of 1 and the second at 2 [4], although their ranges of variation are and [5]. Such findings, related to the physical properties of the semiconducting material, allow a reduction in the number of parameter values to be identified on the basis of measurements performed on the specific PV source that is being modeled. This means that a small set of measurements and data, sometimes even just those given in the manufacturer’s datasheet, can be used for the identification process.

Hejri et al. scaled down the number of parameter values to be identified from seven to five by fixing and [6]. This allowed them to use data available in the PV module or cell datasheet, so the problem became similar to using the SDM (see Section 2.2.1).

Besides the aforementioned one, other approaches in the literature are always based on a reduction of the number of parameter values to be identified because of the lack of seven significant sets of experimental data. For instance, the loss mechanisms might be neglected and the relationship between the two saturation currents might be used for constructing an iterative procedure converging to the optimal fitting values of the remaining parameters [7].

Such approaches, based on a preliminary symbolic manipulation of the equations, on fixing the value of some of the parameters at values taken from the literature, or on neglecting some of the terms, contrast with the adoption of an accurate model like the DDM. In the rigorous case, when all the values of the seven parameters occurring in the DDM have to be identified, fitting methods based on optimization algorithms, often based on stochastic approaches, need to be used [8]. They require large computational resources and contact with the physical problem under examination is lacking.

The DDM is able to describe accurately the behavior of thin‐film cells because of its ability to give a smoother curve in the region of the maximum power point (MPP). Indeed, a high value of the ratio matches the typical I–V curve of thin‐film cells, while gives a more abrupt transition from the constant‐current to the constant‐voltage branches of the I–V curve, which is much more typical of crystalline‐cell technology [9]. This is confirmed by literature data and studies on crystalline‐Si cells, which show that junction recombination is a mechanism that can be neglected, so that a SDM is sufficient for modeling this kind of PV cell. The usefulness of the double‐diode‐based modeling of crystalline cells is restricted to an improved accuracy at both high and low irradiance levels [4].

1.4 The Single‐diode Model

Although it has some inaccuracies at low irradiance levels, the SDM is very often used in the literature to model crystalline‐silicon‐based PV generators. The recombination current is neglected; it is fixed at in (1.1) so that the corresponding equation is given by (1.3). The simplified equivalent circuit is shown in Figure 1.2.

(1.3)

Figure 1.2 Equivalent circuit of the single‐diode model.

Depending on the desired simulation accuracy, the SDM can be simplified further. In the literature it is often considered a lossless model; see Figure 1.3a, where both the series and the shunt resistances are neglected, the former put to zero and the latter to infinity. This choice leads to inaccuracies:

when the numerical analysis involves very high irradiance levels, because the significant voltage drop across

R

s

is neglected (see

Figure 1.3

b)

in mismatched conditions, because the significant current flowing into

R

h

in these conditions is not modeled (see

Figure 1.3

c).

Figure 1.3 Equivalent circuits of (a) the ideal single‐diode model; (b) and (c) two simplified single‐diode models.

The non‐linear functions in (1.4)–(1.6) give the current–voltage relationships corresponding to the equivalent circuits shown in Figures 1.3a–c, respectively.

(1.4)

(1.5)

(1.6)

1.4.1 Effect of the SDM Parameters on the I–V Curve

Equation 1.3, or equivalently its approximate versions, allows us to know the I–V relationship for any PV source operating in homogeneous conditions. Indeed, under the assumption that all the cells are equal and subject to the same environmental conditions, the parameters (Iph, Is, η, Vt, Rs, Rh) can be scaled up or down by accounting for the number of the cells/panels connected in series to form a string, and the number of strings connected in parallel. In such conditions the voltages are multiplied by Ns, the number of the series‐connected cells, and all the currents are multiplied by Np, the number of parallel‐connected strings. Consequently, the series resistance Rs and the parallel resistance Rh increase by a factor Ns and are divided by a factor Np. The SDM parameters are scaled as follows:

(1.7)

(1.8)