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 Chapter 3

Modelling and Control of Grid-connected Solar

Photovoltaic Systems

Marcelo Gustavo Molina

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/62578

Abstract

At present, photovoltaic (PV) systems are taking a leading role as a solar-based renewa‐
ble energy source (RES) because of their unique advantages. This trend is being increased
especially in grid-connected applications because of the many benefits of using RESs in
distributed generation (DG) systems. This new scenario imposes the requirement for an
effective evaluation tool of grid-connected PV systems so as to predict accurately their
dynamic performance under different operating conditions in order to make a compre‐
hensive decision on the feasibility of incorporating this technology into the electric utility
grid. This implies not only to identify the characteristics curves of PV modules or arrays,
but also the dynamic behaviour of the electronic power conditioning system (PCS) for
connecting to the utility grid. To this aim, this chapter discusses the full detailed model‐
ling and the control design of a three-phase grid-connected photovoltaic generator
(PVG). The PV array model allows predicting with high precision the I-V and P-V curves
of the PV panels/arrays. Moreover, the control scheme is presented with capabilities of
simultaneously and independently regulating both active and reactive power exchange
with the electric grid. The modelling and control of the three-phase grid-connected PVG
are implemented in the MATLAB/Simulink environment and validated by experimental
tests.

Keywords: Photovoltaic System, Distributed Generation, Modeling, Simulation, Control

1. Introduction

The worldwide growth of energy demand and the finite reserves of fossil fuel resources have
led to the intensive use of renewable energy sources (RESs). Other major issues that have
driven strongly the RES development are the ever-increasing impact of energy technolo‐
gies on the environment and the fact that RESs have become today a mature technology. The

© 2016 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons
Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution,
and reproduction in any medium, provided the original work is properly cited.
54 Renewable Energy - Utilisation and System Integration

necessity for having available sustainable energy systems for substituting gradually conven‐
tional ones requires changing the paradigm of energy supply by utilizing clean and renewa‐
ble resources of energy. Among renewables, solar energy characterizes as a clean, pollution-
free and inexhaustible energy source, which is also abundantly available anywhere in the
world. These factors have contributed to make solar energy the fastest growing renewable
technology in the world [1]. At present, photovoltaic (PV) generation is playing a crucial role
as a solar-based RES application because of unique benefits such as absence of fuel cost, high
reliability, simplicity of allocation, low maintenance and lack of noise and wear because of
the absence of moving parts. In addition to these factors are the decreasing cost of PV panels,
the growing efficiency of solar PV cells, manufacturing-technology improvements and
economies of scale [2-3].

The integration of photovoltaic systems into the grid is becoming today the most important
application of PV systems, gaining interest over traditional stand-alone autonomous systems.
This trend is being increased due to the many benefits of using RES in distributed (also known
as dispersed, embedded or decentralized) generation (DG) power systems [4-5]. These
advantages include the favourable fiscal and regulatory incentives established in many
countries that influence straightforwardly on the commercial acceptance of grid-connected PV
systems. In this sense, the growing number of distributed PV systems brings new challenges
to the operation and management of the power grid, especially when this variable and
intermittent energy source constitutes a significant part of the total system generation capacity
[6]. This new scenario imposes the need for an effective design and performance assessment
tool of grid-connected PV systems, so as to predict accurately their dynamic performance
under different operating conditions in order to make a sound decision on whether or not to
incorporate this technology into the electric utility grid. This implies not only to identify the
current-voltage (I-V) characteristics of PV modules or arrays, but also the dynamic behaviour
of the power electronics interface with the utility grid, also known as photovoltaic power
conditioning system (PCS) or PV PCS, required to convert the energy produced into useful
electricity and to provide requirements for connection to the grid. This PV PCS is the key
component that enables to provide a more cost-effective harvest of energy from the sun and
to meet specific grid code requirements. These requirements include the provision of high
levels of security, quality, reliability, availability and efficiency of the electric power. Moreover,
modern DG applications are increasingly incorporating new dynamic compensation issues,
simultaneously and independently of the conventional active power exchange with the utility
grid, including voltage control, power oscillations damping, power factor correction and
harmonics filtering among others. This tendency is estimated to augment even more in future
DG applications [7].

This chapter presents a full detailed mathematical model of a three-phase grid-connected

photovoltaic generator (PVG), including the PV array and the electronic power conditioning
system, based on the MATLAB/Simulink software package [8]. The model of the PV array
proposed uses theoretical and empirical equations together with data provided by the
manufacturer, and meteorological data (solar radiation and cell temperature among others) in
order to predict with high precision the I-V and P-V curves of the PV panels/arrays. Since the
 Modelling and Control of Grid-connected Solar Photovoltaic Systems 55
http://dx.doi.org/10.5772/62578

PV PCS addresses integration issues from both the distributed PV generating system side and
from the utility side, numerous topologies varying in cost and complexity have been widely
employed for integrating PV solar systems into the electric grid. Thus, the document includes
a discussion of major PCS topologies. Moreover, the control scheme is presented with
capabilities of simultaneously and independently regulating both active and reactive power
exchange with the electric grid [9].

The modelling and simulation of the three-phase grid-connected PV generating system in the
MATLAB/Simulink environment allows design engineers taking advantage of the capabilities
for control design and electric power systems modelling already built-up in specialized
toolboxes and blocksets of MATLAB, and in dedicated block libraries of Simulink. These
features allows assessing the dynamic performance of detailed models of grid-connected PV
generating systems used as DG, including power electronics devices and advanced control
techniques for active power generation using maximum power point tracking (MPPT) and for
reactive power compensation of the electric grid.

2. Photovoltaic Generator (PVG) model

The building block of the PV generator is the solar cell, which is basically a P-N semiconductor
junction that directly converts solar radiation into DC current using the photovoltaic effect.
The most common model used to predict energy production in photovoltaic cells is the single
diode lumped circuit model, which is derived from physical principles, as depicted in Fig. 1.
In this model, the PV cell is usually represented by an equivalent circuit composed of a light-
generated current source, a single diode representing the nonlinear impedance of the P-N
junction, and series and parallel intrinsic resistances accounting for resistive losses [10-11].

RS

ID I

IPh D RP V

Figure 1. Equivalent circuit of a PV cell

PV cells are grouped together in larger units called modules (also known as panels), and
modules are grouped together in larger units known as PV arrays (or often generalized as PV
generator), which are combined in series and parallel to provide the desired output voltage
56 Renewable Energy - Utilisation and System Integration

and current. The equivalent circuit for the solar cells arranged in NP-parallel and NS-series is
shown in Fig. 2.

NP NS
R
NP S

IA

NS
NP IPh NS R VA
NP P

Figure 2. Equivalent circuit of a generalized PV generator

The mathematical model that predicts the power production of the PV generator becomes an
algebraically simply model, being the current-voltage relationship defined in Eq. (1).

ìï é 1 æ V A I A RS ö ù ïü N P æ V A I A RS ö
I A = N P I Ph - N P I RS íexp ê çç + ÷ ú - 1ý - ç + ÷, (1)
ïî ëê A VTh è N s N P ø÷ úû ïþ RP çè N s N P ÷ø

where:
IA: PV array output current, in A
VA: PV array output voltage, in V
IPh: Solar cell photocurrent, in A
IRS: Solar cell diode reverse saturation current (aka dark current), in A
A: Solar cell diode P-N junction ideality factor, between 1 and 5 (dimensionless)
RS: Cell intrinsic series resistance, in Ω
RP: Cell intrinsic shunt or parallel resistance, in Ω
VTh: Cell thermal voltage, in V, determined as VTh= k TC/q
k: Boltzmann's constant, 1.380658e-23 J/K
TC: Solar cell absolute operating temperature, in K
q: Electron charge, 1.60217733e-19 Cb
This nonlinear equation can be solved using the Newton Raphson iterative method. The
parameters IPh, IRS, RS, RP, and A are commonly referred to as “the five parameters” from which
the term “five-parameter model” originates. These five parameters must be known in order to
 Modelling and Control of Grid-connected Solar Photovoltaic Systems 57
http://dx.doi.org/10.5772/62578

determine the current and voltage characteristic, and therefore the power generation of the PV
generator for different operating conditions. Thus, in order to obtain a complete model for the
electrical performance of the PV generator over all solar radiation and temperature conditions,
Eq. (1) is supplemented with equations that define how each of the five parameters changes
with solar radiation and/or cell temperature. These equations introduce additional parameters
and thus complexity to the model.
The five parameters in Eq. (1) depend on the incident solar radiation, the cell temperature, and
on their reference values. Manufacturers of PV modules normally provide these reference
values for specified operating conditions known as Standard Test Conditions (STC), which
make it possible to conduct uniform comparisons of photovoltaic modules by different
manufacturers. These uniform test conditions are defined with a solar radiation of 1000 W/m²,
a solar cell temperature of 25 °C and an air mass AM (a measure of the amount of atmosphere
the sun rays have to pass through) of 1.5.
Actual operating conditions, especially for outdoor conditions, are always different from STC,
and mismatch effects can affect the real values of these reference parameters. Consequently,
the evaluation of the five parameters in real operating conditions is of major interest in order
to provide an accurate mathematical model of the PV generator.

2.1. Dependence of the PV array photocurrent on the operating conditions

The photocurrent IPh for any operating conditions of the PV array is assumed to be related to
the photocurrent at standard test conditions as follows:

S
I Ph = f AMa f IA é ISC + a ISC (TC - TR ) ù , (2)
ë ûS
R

where:
f AM a : Absolute air mass function describing the solar spectral influence on the photocurrent
IPh, dimensionless
f IA : Incidence angle function describing the influence on the photocurrent IPh, dimensionless

ISC: Solar cell short-circuit current at STC, in A

αI SC : Solar cell temperature coefficient of the short-circuit current, in A/module/diff. temp (in
K or °C)
TR: Solar cell absolute reference temperature at STC, in K
S: Total solar radiation absorbed at the plane-of-array (POA), in W/m2
SR: Total solar reference radiation at STC, i.e. 1000 W/m2
The absolute air mass function f AM a accounting for the solar spectral influence on the “effec‐
tive” irradiance absorbed on the PV array surface is described through an empirical polyno‐
mial function, as expressed in Eq. (3) [10].
58 Renewable Energy - Utilisation and System Integration

4 4
f AMa = åai ( AM a ) = M P åai ( AM ) ,
i i
(3)
i =0 i =0

where:

a0 − a4 : Polynomial coefficients for fitting the absolute air mass function of the analysed cell
material, dimensionless

AMa: Absolute air mass, corrected by pressure, dimensionless

AM: Atmospheric optical air mass, dimensionless

MP: Pressure modifier, dimensionless

The pressure modifier corrects the air mass by the site pressure in order to yield the absolute
air mass. This factor is computed as the ratio of the site pressure to the standard pressure at
sea level.

The air mass AM is the term used to describe the path length that the solar radiation beam has
to pass through the atmosphere before reaching the earth, relative to its overhead path length.
This ratio measures the attenuation of solar radiation by scattering and absorption in atmos‐
phere; the more atmosphere the light travels through, the greater the attenuation. As can be
noted, the air mass indicates a relative measurement and is calculated from the solar zenith
angle which is a function of time.

The incidence angle function f IA describes the optical effects related to the solar incidence angle
(IA) on the radiation effectively transmitted to the PV array surface and converted to electricity
through the panel photocurrent. This modifier accounts for the effect of reflection and
absorption of solar radiation and is defined as the ratio of the radiation absorbed by the solar
cell at some incident angle θI to the radiation absorbed at normal incidence. The incidence
angle is defined between the solar radiation beam direction (or direct radiation) and the normal
to the PV array surface (or POA), as can be seen from Fig. 3. By using the geometric relation‐
ships between the plane at any particular orientation relative to the earth and the beam solar
radiation, both the incidence angle and the zenith angle can be accurately computed at any
time.

An algorithm for computing the solar incidence angle for both fixed and solar-tracking
modules has been documented in [11]. In the same way, the optical influence of the PV module
surface, typically glass, was empirically described through the incidence angle function [10],
as shown in Eq. (3) for different incident angle θI (in degrees).

5
f IA = 1 - åbi (q I ) ,
i
(4)
i =1

where:
 Modelling and Control of Grid-connected Solar Photovoltaic Systems 59
http://dx.doi.org/10.5772/62578

Zenith

Sun Normal to the

horizontal surface

Normal to the
PV array surface qzt: True zenith angle
qI qzt qI: Incidence angle
as: Solar altitude angle
W b
gs: Solar azimuth angle
as N
g: Surface azimuth angle
b: PV array slope angle
gs PV Array

S g

E
Earth
Surface

Figure 3. Zenith angle and other major angles for a tilted PV array surface

b1- b5: Polynomial coefficients for fitting the incidence angle function of the analysed PV cell
material, dimensionless
Even though the correlation of Eq. (4) is cell material-dependant, most modules with glass
front surfaces share approximately the same fIA function, so that no extra experimentation is
required for a specific module [11]. An alternative theoretical form for estimating the incidence
angle function without requiring specific experimental information was proposed in [12].

2.2. Dependence of the PV array reverse saturation current on the operating conditions
The solar cell reverse saturation current IRS varies with temperature according to the following
equation [13]:

3
éT ù éq E æ 1 1 öù
I RS = I RR ê C ú exp ê G çç - ÷÷ ú (5)
ë TR û êë A k è TR TC ø ûú

where:
IRR: Solar cell reverse saturation current at STC, in A
EG: Energy band-gap of the PV cell semiconductor at absolute temperature (TC), in eV
The energy band-gap of the PV array semiconductor, EG is a temperature-dependence
parameter. The band-gap tends to decrease as the temperature increases. This behaviour can
be better understood when it is considered that the interatomic spacing increases as the
60 Renewable Energy - Utilisation and System Integration

amplitude of the atomic vibrations augments due to the increased thermal energy. The
increased interatomic spacing decreases the average potential seen by the electrons in the
material, which in turn reduces the size of the energy band-gap.

2.3. Dependence of the PV array series and shunt resistances on the operating conditions

Series and shunt resistances are very significant in evaluating the solar array performance since
they have direct effect on the PV module fill factor (FF). The fill factor is defined as the ratio
of the power at the maximum power point (MPP) divided by the short-circuit current (Isc) and
the open-circuit voltage (Voc). In this way, the FF serves as a quantifier of the shape of the I-V
characteristic curve and consequently of the degradation of the PV array efficiency.

The series resistance RS describes the semiconductor layer internal losses and losses due to
contacts. It influences straightforwardly the shape of the PV array I-V characteristic curve
around the MPP and thus the fill factor. As the series resistance increases, its deteriorative
effects on the short-circuit current will be increased, especially at high intensities of radiation,
while not affecting the open-circuit voltage. This unwanted feature causes a reduction of the
peak power and thus the degradation of the PV array efficiency. The dependence of the PV
array series resistance on the cell temperature can be characterized by Eq. (6).

RS = RSR éë1 + a SR (TC - TR ) ùû (6)

where:

RSR : Solar cell series resistance at STC, in Ω

αSR : PV array temperature coefficient of the series resistance, Ω/module/diff. temp. (in K or
°C)

The shunt (or parallel) resistance RP accounts for leakage currents on the PV cell surface or in
PN junctions. It influences the slope of the I-V characteristic curve near the short-circuit current
point and therefore the FF, although its practical effect on the PV array performance is less
noticeable than the series resistance. As the shunt resistance decreases, its degrading effects
on the open-circuit current voltage will be increased, especially at the low voltages region,
while not affecting the short-circuit current. The shunt resistance is dependent upon the
absorbed solar radiation. As indicated in [14], the shunt resistance is approximately inversely
proportional to the short-circuit current, and thus to the absorbed radiation, at very low
intensities. As the absorbed radiation increases, the slope of the I-V characteristic curve near
the short-circuit current point decreases and then the effective shunt resistance proportionally
decreases. In this way, this phenomenon can be empirically characterized by Eq. (7).

RPR S
= , (7)
RP SR
 Modelling and Control of Grid-connected Solar Photovoltaic Systems 61
http://dx.doi.org/10.5772/62578

2.4. Dependence of the PV array material ideality factor on the operating conditions
The P-N junction ideality factor A of PV cells is generally assumed to be constant and inde‐
pendent of temperature. However, as reported by [15] the ideality factor varies with temper‐
ature for most semiconductor materials by the following general expression, as given in Eq. (8).

é a T2 ù
A = AR - ê A C ú (8)
ëê TC + b D ûú

where:
AR : Ideality factor of the PV cell semiconductor at absolute zero temperature, 0 K (-273.15°C),
dimensionless, assumed 1.9 for silicon cells
αA : Temperature coefficient of the ideality factor, for silicon 0.789e-3 K-1

βD : Temperature constant approximately equal to the 0 K Debye’s temperature, for silicon 636
K
The analysis of the five parameters IPh, IRS, RS, RP, and A has permitted to complete the detailed
five-parameter model representative of the PV solar array for different operating conditions.

3. Photovoltaic Power Conditioning System (PCS) model

Usually, one of the major challenges of grid-connected or utility-scale solar photovoltaic

systems is to attain an optimal compatibility of PV arrays with the electricity grid. Since a PV
array produces an output DC voltage with variable amplitude, an additional conditioning
circuit is required to meet the amplitude and frequency requirements of the stiff utility AC
grid and inject synchronized power into the grid. As the output of PV panels are direct current,
the PV PCS is typically a DC-AC converter (or inverter) which inverts the DC output current
generated by the PV arrays into a synchronized sinusoidal waveform. This PV interface must
generate high quality electric power and at the same time be flexible, efficient and reliable.
Another key challenge of grid-connected PV systems is the procedure employed for power
extraction from solar radiation and is mostly related to the nature of PV arrays. Each PV module
is a nonlinear system with an output power mostly influenced by atmospheric conditions, such
as solar radiation and temperature. To transfer the maximum solar array power into the utility
grid for all operating conditions, a maximum power point tracking (MPPT) technique is
usually implemented. Therefore, each grid-connected PV generating system has to perform
two essential functions, i.e. to extract the maximum output power from the PV array, and to
inject a sinusoidal current into the grid.
The photovoltaic PCS can be classified with respect to the number of power stages of its
structure into three classes, known as single-stage, dual-stage and multi-stage topologies, as
depicted in Fig. 4 [16].
62 Renewable Energy - Utilisation and System Integration

The first structure of the PV PCS connects the PV array directly to the DC bus of a power
inverter. Consequently, the maximum power point tracking of the PV modules and the inverter
control loops (current and voltage control loops) are handled all in one single stage. The second
topology employs a DC-DC converter (or chopper) as interface between the PV array and the
static inverter. In this case, the additional DC-DC converter connecting the PV panels and the
inverter handles the MPPT control. The third arrangement uses one DC-DC converter for
connecting each string of PV modules to the inverter. For these multi-stage inverters, a DC-
DC converter implements the maximum MPPT control of each string and one power inverter
handles the current and voltage control loops.

The two distinct categories of the inverter are known as voltage source inverter (VSI) and
current source inverter (CSI). Voltage source inverters are named so because the independently
controlled output is a voltage waveform. In this structure, the VSI is fed from a DC-link
capacitor, which is connected in parallel with the PV panels. Similarly, current source inverters
control the AC current waveform. In this arrangement, the inverter is fed from a large DC-link
inductor. In industrial markets, the VSI design has proven to be more efficient and to have
higher reliability and faster dynamic response.

Since applications of modern distributed energy resources introduce new constraints of high
quality electric power, flexibility and reliability to the PV-based distributed generator, a two-
stage PV PCS topology using voltage source inverters has been mostly applied in the literature.
This configuration of two cascade stages offers an additional degree of freedom in the
operation of the grid-connected PV system when compared with the one-stage configuration.
Hence, by including the DC-DC boost converter, various control objectives, as reactive power
compensation, voltage control, and power oscillations damping among others, are possible to
be pursued simultaneously with the typical PV system operation without changing the PCS
topology [17].

The detailed model of a grid-connected PV system is illustrated in Fig. 5, and consists of the
solar PV arrangement and its PCS to the electric utility grid [8]. PV panels are electrically
combined in series to form a string (and sometimes stacked in parallel) in order to provide the
desired output power required for the DG application. The PV array is implemented using the
aggregated model previously described, by directly computing the total internal resistances,
non-linear integrated characteristic and total generated solar cell photocurrent according to
the series and parallel contribution of each parameter. A three-phase DC-AC voltage source
inverter is employed for connecting to the grid. This three-phase static device is shunt-
connected to the distribution network by means of a coupling transformer and the corre‐
sponding line sinusoidal filter. The output voltage control of this VSI can be efficiently
performed using pulse width modulation (PWM) techniques [18].

3.1. Voltage source inverter

Since the DC-DC converter acts as a buffer between the PV array and the power static inverter
by turning the highly nonlinear radiation and temperature-dependent I-V characteristic curve
of the PV system into a quasi-ideal atmospheric factors-controlled voltage source characteris‐
 Modelling and Control of Grid-connected Solar Photovoltaic Systems 63
http://dx.doi.org/10.5772/62578

Utility
Power Conditioning System Grid
PV Array

DC
AC

MPPT and Control Loops

Control System

(a)
Utility
Grid
Power Conditioning System
PV Array
DC DC
DC AC

MPPT Control Loops

Control System

(b)

Power Conditioning System

PV String DC Utility
DC Grid

PV String DC DC
DC AC

PV String DC
DC

MPPTs Control Loops

Control System

(c)

Figure 4. PV PCS configurations: (a) single-stage inverter, (b) dual-stage inverter, and (c) multi-stage inverter.
64 Renewable Energy - Utilisation and System Integration

tic, the natural selection for the inverter topology is the voltage source-type. This solution is
more cost-effective than alternatives like hybrid current source inverters (HCSI).

Electric Utility Grid

DC-DC Boost Three-Phase Three-Level
PV Array Converter Voltage Source Inverter
Step-Up PCC
Line Filter Coupling
NP +VPV /2 Lb Db +Vd /2 Transformer
LF1
PV PV +
Cd1 LF2
–
PV PV
Cb +
0
Tbst Dfd NP LF3
NS –
+
CF1
Cd2
PV PV –
CF2

–VPV /2 –Vd /2 CF3

1 : N (Δ:Yg)

Figure 5. Detailed model of the proposed three-phase grid-connected photovoltaic system

The voltage source inverter presented in Fig. 5 consists of a multi-level DC-AC power inverter
built with insulated-gate bipolar transistors (IGBTs) technology. This semiconductor device
offers a cost-effective solution for distributed generation applications since it has lower
conduction and switching losses with reduced size than other switching devices. Furthermore,
as the power of the inverter is in the range of low to medium level for the proposed application,
it can be efficiently driven by sinusoidal pulse width modulation (SPWM) techniques.
The VSI utilizes a diode-clamped multi-level (DCML) inverter topology, also commonly called
neutral point clamped (NPC), instead of a standard inverter structure with two levels and six
pulses. The three-level twelve-pulse VSI structure employed is very popular especially in high
power and medium voltage applications. Each one of the three-phase outputs of the inverter
shares a common DC bus voltage that has been divided into three levels over two DC bus
capacitors. The middle point of the two capacitors constitute the neutral point of inverter and
output voltages have three voltage states referring to this neutral point. The general concept
of this multi-level inverter is to synthesize a sinusoidal voltage from several levels of voltages.
Thus, the three-level structure attempts to address some restrictions of the standard two-level
one by providing the flexibility of an extra level in the output voltage, which can be controlled
in duration to vary the fundamental output voltage or to assist in the output waveform
construction. This extra feature allows generating a more sinusoidal output voltage waveform
than conventional structures without increasing the switching frequency. In this way, the
voltage stress on the switching devices is reduced and the output harmonics distortion is
minimized [19].
The connection of the inverter to the distribution network in the so-called point of common
coupling (PCC) is made by means of a typical step-up Δ-Y power transformer with line
sinusoidal filters. The design of this single three-phase coupling transformer employs a delta-
connected windings on its primary and a wye/star connected windings with neutral wire on
its secondary. The delta winding allows third-harmonic currents to be effectively absorbed in
the winding and prevents from propagating them onto the power supply. In the same way,
 Modelling and Control of Grid-connected Solar Photovoltaic Systems 65
http://dx.doi.org/10.5772/62578

high frequency switching harmonics generated by the PWM control of the VSI are attenuated
by providing second-order low-pass sine wave filters. Since there are two possibilities of fit
ting the filters, i.e. placing them in the primary and in the secondary of the coupling trans‐
former, it is normally preferred the first option because it reduces notably the harmonics
contents in the transformer windings, thus reducing losses as heat and avoiding its overrating.

The mathematical equations describing and representing the operation of the VSI can be
derived from the detailed model shown in Fig. 5 by taking into account some assumptions
with respect to the operating conditions of the inverter [20]. For this purpose, a simplified
scheme of the VSI connected to the electric system is developed, also referred to as the averaged
model, which is presented in Fig. 6. The power inverter operation under balanced conditions
is considered as ideal, i.e. the VSI is seen as an ideal sinusoidal voltage source operating at
fundamental frequency. This consideration is valid since the high-frequency harmonics
produced by the inverter as result of the sinusoidal PWM control techniques are mostly filtered
by the low pass sine wave filters and the net instantaneous output voltages at the point of
common coupling resembles three sinusoidal waveforms spaced 120° apart. At the output
terminals of the low pass filters, the voltage total harmonic distortion (VTHD) is reduced to
as low as 1%, decreasing this quantity to even a half at the coupling transformer output
terminals (or PCC).

Voltage Source Inverter Transformer Utility

System
Id vinva ia Rs Ls va
PCC
M
vinvb ib Rs Ls M
Ideal vb
Vd Cd Rp
VSI
vinvc ic Rs Ls M
vc

Figure 6. Equivalent circuit diagram of the VSI connected to the utility grid

The equivalent ideal inverter depicted in Fig. 6 is shunt-connected to the AC network through
the inductance LS, accounting for the equivalent leakage of the step-up coupling transformer
and the series resistance RS, representing the transformer winding resistance and VSI IGBTs
conduction losses. The magnetizing inductance of the step-up transformer takes account of
the mutual equivalent inductance M. On the DC side, the equivalent capacitance of the two
DC bus capacitors, C1 and C2 (C1=C2), is described through C=C1/2=C2/2 whereas the switching
losses of the VSI and power loss in the DC capacitors are represented by Rp.

The dynamic equations governing the instantaneous values of the three-phase output voltages
on the AC side of the VSI and the current exchanged with the utility grid can be directly derived
by applying Kirchhoff’s voltage law (KVL) as follows:
66 Renewable Energy - Utilisation and System Integration

é vinv ù é v ù é ia ù
ê a ú ê aú ê ú
ê vinvb ú - ê vb ú = ( R s + s Ls ) êib ú , (9)
ê ú ê ú ê ic ú
êë vinvc úû ë vc û ë û

where:

s: Laplace variable, being s = d / dt for t > 0

é Rs 0 0 ù é Ls M Mù
ê ú ê ú
Rs = ê 0 Rs 0 L
ú s êM
= Ls Mú (10)
ê0 0 Rs úû êM M Ls úû
ë ë

Under the assumption that the system has no zero sequence components, all currents and
voltages can be uniquely transformed into the synchronous-rotating orthogonal two-axes
reference frame, in which each vector is described by means of its d and q components, instead
of its three a, b, c components. Consequently, as depicted in Fig. 7, the d-axis always coincides
with the instantaneous voltage vector and thus vd equates |v|, while vq is set at zero. Conse‐
quently, the d-axis current component contributes to the instantaneous active power and the
q-axis current component to the instantaneous reactive power. This operation allows for a
simpler and more accurate dynamic model of the VSI.

q-axis

ω
B-axis

i |vinv|
iq d-axis
vinvq v
α vd= v
vinvd
0
vq=0 id
A-axis

C-axis

Figure 7. VSI vectors in the synchronous rotating d-q reference frame

 Modelling and Control of Grid-connected Solar Photovoltaic Systems 67
http://dx.doi.org/10.5772/62578

By applying Park’s transformation stated by Eq. (11), Eqs. 9 through 10 can be transformed
into the synchronous rotating d-q reference frame as follows (Eqs. (12) through (14)):

é æ 2p ö æ 2p ö ù
ê cosq cos ç q - ÷ cos ç q + ÷ú
ê è 3 ø è 3 øú
2ê æ 2p ö æ 2p ö ú
K s = ê - sin q - sin ç q - ÷ - sin ç q + ÷ú (11)
3ê è 3 ø è 3 øú
ê 1 1 1 ú
ê ú
ëê 2 2 2 ûú

with:

∫
t

θ = ω (ξ )dξ + θ (0) angle between the d-axis and the reference phase axis, beingξ an integration
0

variable

ω: synchronous angular speed of the network voltage at the fundamental system frequency f
(50 Hz in this chapter)

Thus,

év - v ù é vinv - va ù éi ù é ia ù
ê invd d
ú ê a ú ê dú
ê vinv - vq ú = K s ê vinv - vb ú , ê iq ú = K s êêib úú , (12)
ê q ú ê b ú ê ú ê ic ú
ê vinv - v0 ú
ë 0 û êë vinvc - vc úû ëi0 û ë û

By neglecting the zero sequence components, Eqs. (13) and (14) are obtained.

é vinv ù é v ù éi ù é -w 0 ù é iq ù
ê d ú - ê d ú = R s + s L´s ê d ú + ê ú L´s ê ú , (13)
v v
êë invq úû ëê q ûú i
ëê q ûú ë 0 w û êëid ûú

where:

éR 0ù é L´s 0 ù é Ls - M 0 ù
Rs = ê s ú L´ = ê ú=ê ú (14)
ë0 Rs û s ë 0 L´ss û ë 0 Ls - M û

It is to be noted that the coupling of phases abc through the term M in matrix Ls (Eq. (10)), was
eliminated in the dq reference frame when the inverter transformer is magnetically symmetric,
68 Renewable Energy - Utilisation and System Integration

as is usually the case. This decoupling of phases in the synchronous-rotating system simplifies
the control system design.

By rewriting Eq. (13), the state-space representation of the inverter is obtained as follows:

é - Rs ù
w ú
éid ù ê L´ év - v ù
ú éêid ùú + 1 ê invd
sê ú = ê s ú, (15)
ê - Rs ú ëê iq ûú L´s ê vinvq ú
ëê iq ûú ê -w ú ë û
ëê L´s ûú

A further major issue of the dq transformation is its frequency dependence (ω). In this way,
with appropriate synchronization to the network (through angle θ), the control variables in
steady state are transformed into DC quantities. This feature is quite useful to develop an
efficient decoupled control system of the two current components. Although the model is
fundamental frequency-dependent, the instantaneous variables in the dq reference frame
contain all the information concerning the three-phase variables, including steady-state
unbalance, harmonic distortions and transient components.

The AC and DC sides of the VSI are related by the power balance between the input and the
output on an instantaneous basis. In this way, the ac power should be equal to the sum of the
DC resistance power and to the charging rate of the DC capacitors, as described by Eq. (16).

3
2
( ) C V2
vinvd id + vinvq iq = - d Vd sVd - d ,
2 Rp
(16)

The VSI basically generates the AC voltage vinv from the DC voltage Vd, in such a way that the
connection between the DC-side voltage and the generated AC voltage can be described by
using the average switching function matrix in the dq reference frameSav,dq, as given by Eqs.
(17) through (19). This relation assumes that the DC capacitors voltages are equal to Vd/2.

é vinv ù
ê d ú = Sav , dq Vd , (17)
êë vinvq úû

and the average switching function matrix in dq coordinates is computed as:

éSav , d ù 1 éS d ù
Sav , dq = ê ú = mi a ê ú , (18)
S
ëê av , q ûú 2 êë Sq ûú

where,
 Modelling and Control of Grid-connected Solar Photovoltaic Systems 69
http://dx.doi.org/10.5772/62578

mi: modulation index of inverter, mi∈ [0, 1]

α: phase-shift of the STATCOM output voltage from the reference position
3 n2
a= : voltage ratio of the step-up coupling transformer turns ratio of the step-up Δ-Y
2 n1
coupling transformer and the average switching factor matrix for the dq reference frame,

éS d ù écos a ù
ê ú=ê ú, (19)
êë Sq úû ë sin a û

with α being the phase-shift of the VSI output voltage from the reference position
Essentially, Eqs. (12) through (19) can be summarized in the state-space as stated by Eq. (20).
This continuous state-space averaged mathematical model describes the steady-state dynam‐
ics of the VSI in the dq reference frame.

é - Rs maSd ù
ê w ú é v ù
L´s 2 L´s ú ê ú
é id ù ê éi ù L´
ê ú ê - Rs maSq ú ê d ú êê s úú
s ê iq ú = ê -w ú ê iq ú - 0 , (20)
ê ú ê L´s 2 L´s ú ê ú ê ú
ëVd û ê ú V ê 0 ú
ê - 3 maS 3 2 úë dû ê ú
d
- maSq - êë ûú
ê 2 Cd 2 Cd RpCd úû
ë

Inspection of Eq. (20) shows a cross-coupling of both components of the VSI output current
through the term ω. This issue of the d-q reference frame modelling approach must be
counteracted by the control system. Furthermore, it can be observed an additional coupling
resulting from the DC capacitors voltage Vd. Moreover, average switching functions (Sd and
Sq) introduce nonlinear responses in the inverter states id, iq and Vd when α is regarded as an
input variable. This difficulty demands to keep the DC bus voltage as constant as possible, in
order to decrease the influence of the dynamics of Vd. There are two ways of dealing with this
problem. One way is to have a large capacitance for the DC capacitors, since bigger capacitors
value results in slower variation of the capacitors voltage. However, this solution makes the
compensator larger and more expensive. Another way is to design a controller of the DC bus
voltage. In this fashion, the capacitors can be kept relatively small. This last solution is
employed here for the control scheme.

3.2. DC-DC boost converter

The intermediate DC-DC boost converter fitted between the PV array and the inverter acts as
an interface between the output DC voltage of the PV modules and the DC link voltage at the
input of the voltage source inverter. The voltage of the PV array is variable with unpredictable
atmospheric factors, while the VSI DC bus voltage is controlled to be kept constant at all load
70 Renewable Energy - Utilisation and System Integration

conditions. In this way, in order to deliver the required output DC voltage to the VSI link, a
standard unidirectional topology of a DC-DC boost converter (also known as step-up converter
or chopper) is employed. This switching-mode power device contains basically two semicon‐
ductor switches (a rectifier diode and a power transistor) and two energy storage devices (an
inductor and a smoothing capacitor) for producing an output DC voltage at a level greater
than its input DC voltage. The basic structure of the DC-DC boost converter, using an IGBT as
the main power switch, is shown in Fig. 5.

The converter produces a chopped output voltage through pulse-width modulation (PWM)
control techniques in order to control the average DC voltage relation between its input and
output. Thus, the chopper is capable of continuously matching the characteristic of the PV
system to the equivalent impedance presented by the DC bus of the inverter. In this way, by
varying the duty cycle of the DC-DC converter it is feasibly to operate the PV system near the
MPP at any atmospheric conditions and load.

The operation of the converter in the continuous (current) conduction mode (CCM), i.e. with
the current flowing continuously through the inductor during the entire switching cycle,
facilitates the development of the state-space model. The reason for this is that only two switch
states are possible during a switching cycle, namely, (i) the power switch Tb is on and the diode
Db is off, or (ii) Tb is off and Db is on. In steady-state CCM operation and neglecting the parasitic
components, the state-space equation that describes the dynamics of the DC-DC boost
converter is given by Eq. (21) [21].

é 1 - Sdc ù é1 ù
é IA ù ê 0 - ú
L ú é I A ù + êê L
0 ú éV ù
sê ú = ê ê ú ú - ê Aú, (21)
ëVd û ê - 1 - Sdc ú V
0 úë dû ê0
ê 1 ú ë Id û
- ú
êë C û ë Cû

where:

IA: Chopper input current, matching the PV array output current, in A

VA: Chopper input voltage, the same as the PV array output voltage, in V

Vd: Chopper output voltage, coinciding with the DC bus voltage, in V

Id: Chopper output current, in A

Sdc: Switching function of the boost converter

The switching function is a two-level waveform characterizing the signal that drives the power
switch Tb of the DC-DC boost converter, defined as follows:

ìï 0, for the switchTboff

Sdc í (22)
ïî 1, for the switchTbon
 Modelling and Control of Grid-connected Solar Photovoltaic Systems 71
http://dx.doi.org/10.5772/62578

If the switching frequency of Tb is significantly higher than the natural frequencies of the DC-
DC boost converter, this discontinuous model can be approximated by a continuous state-
space averaged (SSA) model, where a new variable D is introduced. In the [0, 1] subinterval,
D is a continuous function and represents the duty cycle D of the DC-DC converter. It is defined
as the ratio of time during which the power switch Tb is turned-on to the period of one complete
switching cycle, TS. This variable is used for replacing the switching function of the power
converter in Eq. (21), yielding the following SSA expression:

é 1- Dù é1 ù
éIA ù ê 0 -
L úú é I A ù + êê L
0 ú éV ù
sê ú = ê ê ú ú - ê Aú, (23)
ëVd û ê - 1 - D V
0 úú ë d û êê 0
1 ú ë Id û
- ú
êë C û ë Cû

The DC-DC converter produces a chopped output voltage for controlling the average DC
voltage relation between its input and output. In this way, it is significant to derive the steady-
state input-to-output conversion relationship of the boost converter in the CCM. Since in
steady-state conditions the inductor current variation during on and off times of the switch Tb
are essentially equal, and assuming a constant DC output voltage of the boost converter, the
voltage conversion relationship can be easily derived. To this aim, the state-derivative vector
in Eq. (23) is set to zero, yielding the following expression:

VA
Vd = (24)
(1 - D )

In the same way, by assuming analogous considerations, the current conversion relationship
of the boost converter in the CCM is given by Eq. (25).

Id = (1 - D ) I A (25)

4. PVG control strategy

The hierarchical control of the three-phase grid-connected PV generator consists of an external,

middle and internal level, as depicted in Fig. 8 [21].

4.1. External level control

The external level control, which is outlined in the left part of Fig. 8 in a simplified form, is
responsible for determining the active and reactive power exchange between the PV generator
72 Renewable Energy - Utilisation and System Integration

and the utility electric system. This control strategy is designed for performing two major
control objectives, namely the voltage control mode (VCM) with only reactive power com‐
pensation capabilities and the active power control mode (APCM) for dynamic active power
exchange between the PV array and the electric power system. To this aim, the instantaneous
voltage at the PCC is computed by employing a synchronous-rotating reference frame. As a
consequence, the instantaneous values of the three-phase AC bus voltages are transformed
into d-q components, vd and vq respectively. Since the d-axis is always synchronized with the
instantaneous voltage vector vm, the d-axis current component of the VSI contributes to the
instantaneous active power p while the q-axis current component represents the instantaneous
reactive power q. Thus, to achieve a decoupled active and reactive power control, it is required
to provide a decoupled control strategy for id and iq. In this way, only vd is used for computing
the resultant current reference signals required for the desired PV output active and reactive
power flows. Additionally, the instantaneous actual output currents of the PV system, id and
iq, are obtained and used in the middle level control.

va vb vc
External Level Control Middle Level Control

vr VCM iqr
PCC Voltage PLL
Simplified state-space
vm Controller
θs
mathematical model
vinv d θs
idr of the VSI in the
dq0
dq0 ref. frame, Eq. (20)
Vd Maximum Power Pr idr* vinv q abc
IA Point Tracking vinva vinv b vinvc
(MPPT) System id iq
VA vd idr´ VSI

D D VSI SPWM IGBTs

APCM Generator Control
Pulses
vd id iq VSI DC Link DC-DC
dq0 θs dq0 Voltage Chopper PWM Converter
abc abc Controller Generator IGBTs
Control
Pulses

va vb vc θs ia ib ic Vd Vdr Internal Level Control

Figure 8. Multi-level control scheme for the three-phase grid-connected PV generator

In many modern electricity grids with high integration of intermittent renewable-based

distributed generation, voltage regulation is becoming a necessary task at the distribution
level. Since the inverter-interfaced sources are deployed to regulate the voltage at the point of
common coupling of each inverter, the PVG can easily perform this control action and
participate in the voltage control of the grid. To this aim, a control loop of the external level is
the VCM, also called Automatic Voltage Regulation (AVR). It controls (supports and regulates)
the voltage at the PCC through the modulation of the reactive component of the inverter output
current, iq. Since only reactive power is exchanged with the grid in this control mode, there is
no need for the PV array or any other external energy source. In fact, this reactive power is
locally generated just by the inverter and can be controlled simultaneously and independently
 Modelling and Control of Grid-connected Solar Photovoltaic Systems 73
http://dx.doi.org/10.5772/62578

of the active power generated by the PV array. The design of the control loop in the rotating
frame employs a standard proportional-integral (PI) compensator including an anti-windup
system. This control mode eliminates the steady-state voltage offset via the PI compensator.
A voltage regulation droop is included in order to allow the terminal voltage of the PV inverter
(PCC) to vary in proportion with the compensating reactive current. Thus, the PI controller
with droop characteristics becomes a simple phase-lag compensator.

The main objective of the grid-connected solar photovoltaic generating system is to exchange
with the electric utility grid the maximum available power for the given atmospheric condi‐
tions, independently of the reactive power generated by the inverter. In this way, the APCM
allows dynamically controlling the active power flow by constantly matching the active power
exchanged by the inverter with the maximum instant power generated by the PV array. This
implies a continuous knowledge of not only the PV panel internal resistances but also the
voltage generated by the PV array. This requirement is very difficult to meet in practice and
would increase complexity and costs to the DG application. It would require additional sensing
of the cell temperature and solar radiation jointly with precise data of its characteristic curve.
Even more, PV parameters vary with time, making it difficult for real-time prediction.

Many MPPT methods have been reported in literature. These methods can be classified into
three main categories: lookup table methods, computational methods (neural networks, fuzzy
logic, etc.) and hill climbing methods [22-25]. These vary in the degree of sophistication,
processing time and memory requirements. Among them, hill climbing methods are indirect
methods with a good combination of flexibility, accuracy and simplicity. They are efficient and
robust in tracking the MPP of solar photovoltaic systems and have the additional advantages
of control flexibility and easiness of application with different types of PV arrays. The power
efficiency of these techniques relies on the control algorithm that tracks the MPP by measuring
some array quantities.

The simplest MPPT using climbing methods is the “Perturbation and Observation” (P&O)
method. This MPPT strategy uses a simple structure and few measured variables for imple‐
menting an iterative method that permits matching the load with the output impedance of the
PV array by continuously adjusting the DC-DC converter duty cycle. This MPPT algorithm
operates by constantly perturbing, i.e. increasing or decreasing, the output voltage of the PV
array via the DC-DC boost converter duty cycle D and comparing the actual output power
with the previous perturbation sample, as depicted in Fig. 9. If the power is increasing, the
perturbation will continue in the same direction in the following cycle, otherwise the pertur‐
bation direction will be inverted. This means that the PV output voltage is perturbed every
MPPT iteration cycle k at sample intervals Ttrck, while maintaining always constant the VSI DC
bus voltage by means of the middle level control. Therefore, when the optimal power for the
specific operating conditions is reached, the P&O algorithm will have tracked the MPP (the
climb of the PV array output power curve) and then will settle at this point but oscillating
slightly around it. The output power measured in every iteration step is employed as a
reference power signal Pr and then converted to a direct current reference idr for the middle
level control.
74 Renewable Energy - Utilisation and System Integration

Start

Program Variables Initialization:

D(k-1)=D ini , ΔD=C1

Read Vpv(k-1), Ipv(k-1)

Compute Ppv(k-1)=Vpv(k-1) Ipv(k-1)

D(k)=D(k-1)+ΔD

Read Vpv(k), Ipv(k)

Compute Ppv(k)=Vpv(k) Ipv(k)

Yes No
Ppv(k) ≥ Ppv(k-1)
Increase D Decrease D

D(k+1)=D(k)+ΔD D(k+1)=D(k)-ΔD

Yes No No Yes
D(k+1) >0.9 D(k+1) <0.1

D(k+1)=0.9 Upper Boundary Lower Boundary D(k+1)=0.1

Read Vpv(k+1), Ipv(k+1)

Compute Ppv (k+1)=Vpv(k+1) Ipv(k+1)

Yes No
Ppv(k+1) ≥ Ppv(k)

Figure 9. Flowchart for the P&O MPPT algorithm

4.2. Middle level control

The middle level control generates the expected output, particularly the actual active and
reactive power exchange between the PV VSI and the AC system, to dynamically track the
reference values set by the external level. This level control, which is depicted in the middle
part of Fig. 8, is based on a linearization of the state-space mathematical model of the PV system
in the d-q reference frame, described in Eq. (20).

In order to achieve a decoupled active and reactive power control, it is required to provide a
decoupled current control strategy for id and iq. Inspection of Eq. (20) shows a cross-coupling
of both components of the PV VSI output current through ω. Therefore, appropriate control
signals have to be generated. To this aim, it is proposed to use two control signals x1 and x2,
which are derived from assumption of zero derivatives of currents in the upper part (AC side)
of (20). In this way, using two conventional PI controllers with proper feedback of the VSI
output current components allows eliminating the cross-coupling effect in steady state. Eq.
(20) also shows an additional coupling of derivatives of id and iq with respect to the DC voltage
Vd. This issue requires maintaining the DC bus voltage constant in order to decrease the
 Modelling and Control of Grid-connected Solar Photovoltaic Systems 75
http://dx.doi.org/10.5772/62578

influence of Vd. The solution to this problem is obtained by using a DC bus voltage controller
via a PI controller for eliminating the steady-state voltage variations at the DC bus. This DC
bus voltage control is achieved by forcing a small active power exchange with the electric grid
for compensating the VSI switching losses and the transformer ones, through the contribution
of a corrective signal idr*.

4.3. Internal level control

The internal level provides dynamic control of input signals to the DC-DC and DC-AC
converters. This control level, which is depicted in the right part of Fig. 8, is responsible for
generating the switching signals for the twelve valves of the three-level VSI, according to the
control mode (PWM) and types of valves used (IGBTs). This level is mainly composed of a
three-phase three-level sinusoidal PWM generator for the VSI IGBTs, and a two-level PWM
generator for the single IGBT of the boost DC-DC converter. Furthermore, it includes a line
synchronization module, which consists mainly of a phase locked loop (PLL). This circuit is a
feedback control system used to automatically synchronize the converter switching pulses
with the positive sequence components of the AC voltage vector at the PCC. This is achieved
by using the phase θs of the inverse coordinate transformation from dq to abc components.

5. PVG model and control implementation in MATLAB/Simulink

The complete detailed model and control scheme of the three-phase grid-connected PVG is
implemented in the MATLAB/Simulink software environment using the SimPowerSystems
(SPS) [8], as depicted in Figs. 10 to 11. SPS was designed to provide a modern design tool that
allows scientists and engineers to rapidly and easily build models that simulate power systems.
SimPowerSystems uses the Simulink environment, which is a tool based on a graphical user
interface (GUI) that permits interactions between mechanical, thermal, control, and other
disciplines. This is possible because all the electrical parts of the simulation interact with the
extensive Simulink modelling library. These libraries contain models of typical power
equipment such as transformers, lines, machines, and power electronics among others. As
Simulink uses MATLAB as the computational engine, designers can also use MATLAB
toolboxes and other Simulink blocksets.
Since the detailed model of the proposed PVG application contains many states and non-linear
blocks such as power electronics switches, the discretization of the electrical system with fixed-
step is required so as to allow much faster simulation than using variable time-step methods.
Two sample times are employed in order to enhance the simulation, Ts_Power= 5 µs for the
simulation of the power system, the VSI and the DC-DC converter, and Ts_Control= 100 µs
for the simulation of the multi-level control blocks.
The three-phase grid-connected PV energy conversion system is implemented basically with
the Three-Level Bridge block. The three-phase three-level Voltage Source Inverter makes uses
of three arms of power switching devices, being IGBTs in this work. In the same way, the DC-
DC converter is implemented through the Three-Level Bridge but using only one arm of IGBTs.
76 Renewable Energy - Utilisation and System Integration

Fig. 10. Detailed model and dynamic control of the grid-connected PVG in the
Figure 10. Detailed model and dynamic control of the grid-connected PVG in the MATLAB/Simulink environment
MATLAB/Simulink environment

From the four power switching devices

The three-phase of each PV
grid-connected arm, just one
energy device issystem
conversion activated for accom‐ basically with
is implemented
plishing the chopping
the Three-Level Bridge block. The three-phase three-level VoltageWith
function while the other three are kept off all the time. SourcethisInverter makes
implementation approach, the turn-on and turn-off times (Fall time, Tail time) of the power
uses of three arms of power switching devices, being IGBTs in this work. In the same way,
switching devicethe
are DC-DC
not modelled, resulting
converter in a faster simulation
is implemented through thewhen compared
Three-Level to a single
Bridge but using only one
IGBT mask usingarman increased state-space model.
of IGBTs. From the four power switching devices of each arm, just one device is
Fig. 11 shows the detailedfor
activated model of the PV the
accomplishing array in the MATLAB/Simulink
chopping environment,
function while the other three are kept off all the
including the implementation of the
time. With this equivalent circuit
implementation of the PV
approach, thegenerator by using
turn-on and controlled
turn-off times (Fall time, Tail
current sources and resistances,
time) andswitching
of the power control blocks
deviceforare
implementing
not modelled,Eqs. 1 through
resulting in a 8faster
[26]. simulation when
compared to a single IGBT mask using an increased state-space model.

Fig. 11 shows the detailed model of the PV array in the MATLAB/Simulink environment,
6. Simulationincluding
and experimental results of
the implementation the equivalent circuit of the PV generator by using
controlled current sources and resistances, and control blocks for implementing Eqs. 1
In order to analyze the effectiveness
through 8 [26]. of the proposed models and control algorithms of the
three-phase grid-connected PV system, time-discrete dynamic simulation tests have been
performed in the MATLAB/Simulink environment. To this aim, the simulation of a 250 Wp
6. Simulation and Experimental Results
(peak power) PVG has been compared with experimental data collected from a laboratory-
scale prototype, which is presented in Fig. 12.
In order to analyze the effectiveness of the proposed models and control algorithms of the
The PV array implemented
three-phaseconsists of a single
grid-connected PVstring of 5time-discrete
system, high-efficiency polycrystalline
dynamic simulationPVtests have been
modules (NS=5, performed
NP=1) of 50inWp (Solartec KS50T, built with Kyocera cells) [27]. This array
the MATLAB/Simulink environment. To this aim, the simulation of a 250 Wp
makes up a peak(peak
installed power
power) PVG ofhas
250been
W and is linkedwith
compared to aexperimental
110 V DC busdata of acollected
three-phase
from a laboratory-
three-level PWM scale
voltage source inverter through a DC-DC
prototype, which is presented in Fig. 12.boost converter. The resulting dual-
stage converter is connected to a 380 V/50 Hz three-phase electric system using three 60 V/220
V step-up coupling transformers connected in a Δ-Yg configuration. The VSI has been rated
at 1 KVA and designed to operate at 5 kHz with sinusoidal PWM. It is built with IGBTs with
internal anti-parallel diodes and fast clamping diodes, and includes an output inductive-
capacitive low pass filter. The DC-DC boost converter interfacing the PV string with the DC
bus of the VSI is also built with IGBTs and fast diodes, and has been designed to operate at 5
kHz.
 Modelling and Control of Grid-connected Solar Photovoltaic Systems 77
http://dx.doi.org/10.5772/62578

Figure 11. Detailed model of the PV array in the MATLAB/Simulink environment

250 Wp PV Array Power Conditioning System and Control System

Figure 12. Laboratory-scale prototype of the three-phase grid-connected PV system.

78 be  continuously 
Renewable operated 
Energy - Utilisation within 
and System the 
MPP  locus  (shaded  region)  for  an  optimized 
Integration
application of the system. In this way, a continuous adjustment of the array terminal voltage 
is required for providing maximum power to the electric grid. 

 
(a) 

 
(b) 

Fig. 13.  Simulated  and  measured  characteristic  curves  of  the  test  PV  string  for 
Figure 13. Simulated and measured characteristic curves of the test PV string for given climatic conditions: (a) I-V
curve. (b) P-V curve. given climatic conditions: (a) I‐V curve. (a) P‐V curve. 

TheFig. 14 
three-level control
presents  scheme was
a  comparison  of entirely implemented
actual,  measured  and  on a high-performance
simulated  output  power  32-bit fixed-
trajectory 
point digital signal processor (DSP) operating at 150 MHz (Texas Instruments TMS320F2812)
within  a  10‐hour  period  of  analysis  for  a  cloudy  day  with  high  fluctuations  of  solar 
[28].radiation, 
This processor includes 250 W 
for  the  proposed  an advanced
PV  system 12-bit
with analog-to-digital converter withof a the 
and  without  the  implementation  fast
conversion time which makes it possible real time sampling with high accuracy and real time
abc to synchronous dq frame coordinate transformation. The DSP is operated with a selected
sample rate of 160 ksps and low-pass filters were implemented using 5th order low-pass filters
based on a Sallen & key designs. The control pulses for the VSI and the DC-DC chopper has
been generated by employing two DSP integrated pattern generators (event managers). The
gate driver board of the IGBTs has been designed to adapt the wide differences of voltage and
 Modelling and Control of Grid-connected Solar Photovoltaic Systems 79
http://dx.doi.org/10.5772/62578

current levels with the DSP and to provide digital and analog isolation using optically coupled
isolators. All the source code was written in C++ by using the build-in highly efficient DSP
compiler.

Fig. 13 depicts the I-V and P-V characteristic curves of the 250 W PV array for given climatic
conditions, such as the level of solar radiation and the cell temperature. The characteristic curve
at 25°C and 200/600/1000 W/m2 have been evaluated using the proposed model (blue solid
line) with the software developed and measurements obtained from the experimental set-up
(blue dotted line). The experimental data have been obtained using a peak power measuring
device and I-V curve tracer for PV modules and strings (PVE PVPM 1000C40) [29]. As can be
observed, the proposed model of the PV array shows a very good agreement with measured
data at all the given levels of solar radiation.

As can be derived from both characteristic curves of the PV system, there exist a specific point
at which the generated power is maximized (i.e. MPP) and where the output I-V characteristic
curve is divided into two parts: the left part is defined as the current source region in which
the output current approximates to a constant, and the right part is the voltage source region
in which the output voltage hardly changes. Since the MPP changes with variations in solar
radiation and solar cell operating temperature, the PV array have to be continuously operated
within the MPP locus (shaded region) for an optimized application of the system. In this way,
a continuous adjustment of the array terminal voltage is required for providing maximum
power to the electric grid.

80
Actual Maximum Power
PV Generator Output Power (W)

70 Simulations with P&O MPPT Algorithm

Measurements with P&O MPPT Algorithm
60
Measurements with Constant Voltage (No MPPT)
50

40

30

20

10

0
11:36
07:51
08:36
09:21
10:06
10:51

12:21
13:06
13:51
14:36
15:21
16:06
16:51
17:36
18:21

Local Time

Figure 14. Comparison of actual, measured and simulated output power trajectory for the proposed 250 W PV system
with and without the MPPT algorithm implementation.
80 Renewable Energy - Utilisation and System Integration

Fig. 14 presents a comparison of actual, measured and simulated output power trajectory
within a 10-hour period of analysis for a cloudy day with high fluctuations of solar radiation,
for the proposed 250 W PV system with and without the implementation of the MPPT
algorithm. The time data series shown in light blue solid line represents the actual maximum
power available from the PV array for the specific climatic conditions, i.e. the MPP to be tracked
at all times by the MPPT. Simulation results obtained with the MPPT algorithm are shown in
red solid lines. In the same way, the two time data series shown in black and green solid lines,
respectively represents the measurements obtained from the experimental setup with the
control system with the MPPT activated (black) and with no MPPT (green). As can be observed,
the MPPT algorithm (measurements and simulation) follows accurately the maximum power
(actual available power) that is proportional to the solar radiation and temperature variations.
In this sense, it can be noted from measurements and simulation a very precise MPP tracking
when soft variations in the solar radiation take place, while differing slightly when these
variations are very fast and of a certain magnitude. It can be also derived that there is a good
correlation between the experimental and the simulation data. In addition, the deactivation of
the MPPT control results in a constant voltage operation of the PV array output at about 60 V
for the given prototype conditions. In this last case, a significant reduction of the installation
efficiency is obtained, which is worsen with the increase of the solar radiation. This preceding
feature validates the use of an efficient MPPT scheme for maximum exploitation of the PV
system.

7. Conclusion

This chapter has presented a full detailed mathematical model of a three-phase grid-connected
photovoltaic generator, including the PV array and the electronic power conditioning system.
The model of the PV array uses theoretical and empirical equations together with data
provided by manufacturers and meteorological data (solar radiation and cell temperature
among others) in order to accurately predict the PV array characteristic curve. Moreover, it
has presented the control scheme of the PVG with capabilities of simultaneously and inde‐
pendently regulating both active and reactive power exchange with the electric grid. The
control algorithms incorporate a maximum power point tracker (MPPT) for dynamic active
power generation jointly with reactive power compensation of distribution utility system. The
model and control strategy of the PVG have been implemented using the MATLAB/Simulink
environment and validated by experimental tests.

Acknowledgements

The author wishes to thank the CONICET (Argentinean National Council for Science and
Technology Research), the UNSJ (National University of San Juan) and the ANPCyT (National
Agency for Scientific and Technological Promotion) for the financial support of this work.
 Modelling and Control of Grid-connected Solar Photovoltaic Systems 81
http://dx.doi.org/10.5772/62578

Author details

Marcelo Gustavo Molina*

Address all correspondence to: mgmolina@iee.unsj.edu.ar

Institute of Electrical Energy, National University of San Juan−CONICET, Argentina

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