Simulation Modelling Practice and Theory 35 (2013) 50–68
Contents lists available at SciVerse ScienceDirect
Simulation Modelling Practice and Theory
journal homepage: www.elsevier.com/locate/simpat
Reconfiguration analysis of photovoltaic arrays based
on parameters estimation
Juan David Bastidas-Rodriguez a,⇑, Carlos Andres Ramos-Paja b,
Andres Julian Saavedra-Montes b
a
b
Escuela de Ingenierı́a Eléctrica y Electrónica, Universidad del Valle, Cali, Colombia
Departamento de Energı´a Eléctrica y Automática, Universidad Nacional de Colombia, Medellín, Colombia
a r t i c l e
i n f o
Article history:
Received 10 January 2013
Received in revised form 2 March 2013
Accepted 13 March 2013
Available online 16 April 2013
Keywords:
Photovoltaic
Reconfiguration
Model
Parameter estimation
Series–parallel
Mismatching
Shadowing
a b s t r a c t
A method to determine the photovoltaic (PV) series–parallel array configuration that provides the highest Global Maximum Power Point (GMPP) is proposed in this paper. Such a
procedure was designed to only require measurements of voltage and current of each
string, which avoids to perform experiments in each module. The ideal single-diode model
parameters of each module in the string are obtained from the analysis of the voltage vs.
current characteristics of the string. Using the estimated parameters, all feasible PV array
configurations are evaluated to determine the array configuration that provides the highest
GMPP. Finally, the proposed solution is validated using simulations and experimental data.
Ó 2013 Elsevier B.V. All rights reserved.
1. Introduction
Nowadays there is a generalized interest in the use of renewable energy sources as alternative to supply the increasing
energy demand in different kind of applications in a clean and sustainable way. In particular, the use of PhotoVoltaic (PV)
power plants grows year by year [1] not only for all the advantages of this kind of energy but also for the financial incentives
and regulations implemented for different countries [2]. In such a way, during 2011 about 28 GW were installed in the International Energy Agency (IEA) countries for a total installed capacity of 63.6 GW, where more than 62 GW correspond to distributed and centralized grid-connected applications [3].
Depending on the place characteristics where a PV system is installed it may operates in mismatching conditions due to
the surrounding objects (trees, buildings, antennas, dust, etc.), the differences between the parameter of the panels (manufacturer tolerances) and even different orientations [4,5]. Such mismatching conditions produce large drops in the generated
electrical power since the power vs. voltage curve of the PV array exhibits more than one maximum power point (MPP),
where each one of them is lower than the aggregated power of all the modules [6–10]. The amount of electrical power drop
depends on the shape of the power vs. voltage curve, which is defined by the irradiance and temperature distribution over
the array and the configuration adopted to interconnect the panels [11].
In a PV system the modules can be connected in different configurations, the most used ones are Series-Parallel (SP) and
Total Cross-Tied (TCT), although in the literature is possible to find other configurations like Bridge-Linked (BL) or Honey
⇑ Corresponding author. Address: Calle 13, No. 100-00. Edif. 335. Ofi. 2012, Cali, Colombia. Tel.: +57 2 3212168.
E-mail address: juan.d.bastidas@correounivalle.edu.co (J.D. Bastidas-Rodriguez).
1569-190X/$ - see front matter Ó 2013 Elsevier B.V. All rights reserved.
http://dx.doi.org/10.1016/j.simpat.2013.03.001
J.D. Bastidas-Rodriguez et al. / Simulation Modelling Practice and Theory 35 (2013) 50–68
Nomenclature
Abbreviations
MPP
Maximum Power Point
SP
series–parallel connection
TCT
total-cross tied connection
GMPP
Global MPP
STC
Standard Test Conditions
IV
current vs. voltage
SCP
short-circuit point
OCP
open-circuit point
INP
inflection point
IP
point of interest
Subidexes
j
number of the string
i
position of the module in the string j
Variables
IPV
VPV
A
B
BSTC
ISC
VOC
Impp
Vmpp
ISTC
VocSTC
module current
module voltage
inverse saturation current
inverse of thermal voltage
B in STC conditions
module short-circuit current
module open-circuit voltage
current at MPP
voltage at MPP
ISC in STC conditions
VOC in STC conditions
aI
temperature coefficient of ISC
aV
temperature coefficient of VOC
Ns
number of cells in the module
GPV
module irradiance
TPV
module temperature
GPV in STC conditions
GSTC
TSTC
TPV in STC conditions
Ist
string current
inflection voltage of the kth module of string j
Vok,j
Vxm,k
contribution of mth module to Vok,j
Vst
voltage imposed to the string
Ndata
number of experimental measurements
voltage in OCP
VOCP
NINP
number of INPs
NINPmax maximum number of INPs
NIP
number of IPs
N
number of modules in each string
M
number of strings in the array
Nac
number of active modules
NamIP,k,k+1 number of active modules between IPk and IPk + 1
VIP,k,k+1 voltage data between IPk and IPk + 1
VIP
vector with voltages of IPs
VMIP,k,k+1 voltage produced by NamIP,k,k+1 modules
Nd,k,k+1 number of data in VIP,k,k+1
Nam,k
number of active modules up to NamIP,k,k+1
Vident,k
voltage data of one active module between IPk and IPk+1
Iident,k
current data of one active module between IPk and IPk+1
vector with currents of IPs
IIP
VIend
final point Vident,i and Iident,i
DIPV
error in IPV
DISC
error in ISC
51
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DA
DB
NSE
NSSE
y
^
y
H
NRSP
NRTCT
error in A
error in B
normalized squared error
normalized sum of squared errors
reference value to calculate NSE and NSSE
estimated value to calculate NSE and NSSE
number of data to calculate NSSE
number of feasible configurations in SP
number of feasible configurations in TCT
Comb (HB) [12]. In the SP configuration the PV modules are connected in series forming groups named strings, which connected in parallel form the PV array. In the TCT configuration the PV modules are connected in parallel forming groups
named rows, which connected in series form the PV array.
In the traditional SP and TCT configurations the interconnection of the PV modules are fixed. However, such connections
can be changed through a matrix of switches in order to reduce the effect of the mismatching conditions [13–15]. Such systems are named reconfigurable arrays and have gained popularity in the last years due to the increase in the maximum
power available at the output of the PV generator under mismatching conditions [16,17].
There are different methods to find the best configuration of the PV array reported in the literature [11,13–15,18,19]. In
[15] the authors propose a method to reconfigure a TCT array by balancing, or reducing as much as possible, the difference
between average irradiance in each row. In such a solution the irradiance of each module of the array is estimated to test all
feasible configurations in order to determine which of them provide the highest Global Maximum Power Point (GMPP).
Although the algorithm is simple and can be implemented in a micro-controller, the number of feasible possibilities, when
the number of rows in the array is larger than the number of columns, is very high with respect to the number of possibilities
in SP configuration. Moreover, the solution requires to measure the voltage and current in each module to estimate the irradiance by using the ideal single-diode model, but it does not provide a method to calculate the parameters of the modules.
The system proposed in [14,16] divides the PV array into a fixed and an adaptive parts. The fixed part is connected in TCT
configuration, while the modules of the adaptive part can be connected in parallel to any row of the fixed array to compensate the current drops produced by the mismatched modules in each row. In [14] two algorithms to compensate the mismatching conditions in the fixed part of the array are presented. The first one assumes that the open-circuit voltages of
the adaptive modules are proportional to their irradiance and connects them one-by-one to the mismatched rows to balance
the rows of the array. The second method is similar to the first one but uses the single-diode model to estimate the PV current of the fixed and adaptive parts to determine the best configuration. Nevertheless, the capability of mismatching compensation is limited by the number of the adaptive modules and the authors do not provide a method to determine the
model parameters of the modules. Besides, the first method requires one voltage sensor in each reconfigurable module
and the second method requires the voltage and current measurements of all modules of the array.
The technique proposed in [19] classifies the modules as shaded or unshaded. The shaded modules are bypassed by
switches and the unshaded ones are connected in series to form strings. If one string is not complete it is connected to a
step-up converter to reach the voltage level of the complete strings. The strings are connected in parallel and feed an inverter. This system requires a reduced number of switches and the algorithm to control the system is simple. However, the mismatching compensation capability is low and the energy of the mismatched modules is wasted. Moreover, it requires the
measurements of voltage, current and temperature of each module and it is not clear how to determine if a module is mismatched or not.
In addition, there are different methods to calculate the parameters of a PV module. Some of them [20,21] pose a system
of non-linear equations and perform some simplifications of in order to obtain a set of explicit functions to calculate the
parameters of the single-diode model from the experimental data of the complete current vs. voltage curve of a PV module.
Refs. [22–26] perform simplifications in order to propose a system of non-linear equations, which is solved in the operating points provided by the manufacturer datasheet, or experimental IV curve, to obtain the parameters of the model. However, such parameters may be different from the real values due to the manufacturer tolerances and environmental
disturbances [27].
Other authors [8,27–29] use explicit equations to find some parameters (PV current, inverse saturation current and thermal voltage) and use iterative methods to adjust the values of series and parallel resistance to fit the model with the information taken from the datasheet. Although such methods could be applied to PV modules operating in real conditions, it is
difficult to use them for real-time applications due to the complexity of the equations that need to be solved in each iteration. Moreover, the previous solutions are intended to estimate the parameters from data of each module instead of string or
array data.
The solution proposed in [30] considers a method to calculate the reference values of the parameters by using experimental and datasheet information, then it uses the equations to update the parameters values depending on the values of the
irradiance and cell temperature. Unless this method provides the parameters for different environmental conditions with
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J.D. Bastidas-Rodriguez et al. / Simulation Modelling Practice and Theory 35 (2013) 50–68
explicit equations, it would require one temperature and irradiance sensor in each module, which increases considerably the
implementation costs of the PV system.
Other approach given in [31] performs a large number of experiments in a wide range of irradiance and temperature conditions to obtain the IV curves of a module. For each current vs. voltage curve it solves a system of five non-linear equations
to find the values of the single-diode model parameters. Using such information the authors train an Artificial Neural Network to predict the parameters values depending on the irradiance and temperature conditions.
In [32] the authors perform a continuous identification of the parameters of a fourth order polynomial model of the PV
module from few pairs of voltage/current measurements. The authors fit the model equation to the experimental measurements by modifying the coefficients using the recursive least-square method. Then the Newton–Raphson method is adopted
to find the MPP from the model. Nonetheless such a method is designed for PV arrays in uniform conditions, therefore it is
not applicable in mismatching conditions.
This paper proposes a procedure to detect the SP array configuration that provides the highest maximum power, in uniform or mismatching conditions, requiring current and voltage measurements of strings only. To evaluate all feasible configurations in order to find the one that provides the highest GMPP, the ideal single-diode model parameters of the modules
in each string are obtained. Despite the TCT configuration has been adopted in literature for reconfigurable PV arrays [14,15],
it exhibits a larger amount of feasible configurations, in comparison with SP arrays, when the number of rows (number of
modules in each column) is higher than the number of columns (number of modules in each row). This condition suggests
that SP arrays require lower calculation time to find the optimal configuration in high-voltage applications, where larger
strings are required (e.g. grid-connected PV systems). Therefore, this paper is focused on SP configuration, but the method
is also applicable to TCT configuration with some modifications.
The paper is organized as follows: Section 2 presents a model to calculate the array current by using the inflection point
concept, which defines the operating condition in which a bypass diode becomes active. Section 3 introduces a method to
detect the inflection points from PV string measurements. Then, Section 4 introduces a procedure to calculate the PV modules parameters from measurements of the string only, without requiring individual experiments for each module. Finally,
Section 5 presents the application of the proposed solution to reconfigure a PV array. Conclusions close the paper.
2. Calculation of PV array current by using inflection points
In a PV array connected in SP configuration multiple PV modules are connected in series to form strings, which are connected in parallel to form the array as shown in Fig. 1. Each string has a blocking diode to avoid back-flow currents from the
other strings and each PV module has a bypass diode connected in anti-parallel to avoid thermal overload and hot spots [33].
The simplified single-diode model shown at the left of Fig. 1 is widely used in literature for modeling PV modules
[6,34,35]. The current source represents the electron flow produced by photon collisions in the semiconductor material,
while the non-linear behavior of the p–n junction is modeled through a diode in parallel. By using the Kirchhoff law (1)
is obtained, where the sub-index j represents the number of the string and the sub-index i illustrates the position of the
PV module in the string j. IPV,i,j and VPV,i,j are current and voltage of the PV module in position (i, j), respectively. ISC,i,j is
the short-circuit current, Ai,j is the inverse saturation current of the diode and Bi,j is the inverse of the diode thermal voltage.
IPV;i;j ¼ ISC;i;j Ai;j expðBi;j V PV;i;j Þ
ð1Þ
The three parameters (ISC,i,j, Ai,j and Bi,j) can be evaluated from datasheet information for a given irradiance (GPV,i,j) and semiconductor temperature (TPV,i,j) by using (2)–(5) [6], where ISTC and VocSTC are the short circuit current and open-circuit voltage
in Standard Test Conditions (STC), respectively. TSTC and GSTC are temperature and irradiance of the PV module in STC, respectively. BSTC is the value of parameter B in STC and, aI and aV are the current and voltage temperature coefficients, respectively.
For a given irradiance and temperature conditions Impp and Vmpp are the current and voltage of the PV module at the MPP,
respectively, while VOC is the open-circuit voltage.
PV module 1,1
Vbld,1
PV module 1,M
Vbld,M
Iarray
IPV1,1=Ist,1
VPV1,1
ISC1,1
ID1,1
VPV1,1
Ist,1
PV module N,1
VPV,M
Ist,M
PV module N,M
VPVN,1
Fig. 1. PV Field with M strings parallel connected of N modules each.
Vst
VPV,M
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J.D. Bastidas-Rodriguez et al. / Simulation Modelling Practice and Theory 35 (2013) 50–68
GPV;i;j
ð1 þ aI ðT PV;i;j T STC ÞÞ
GSTC
BSTC
Bi;j ¼
1 þ aV ðT PV;i;j T STC Þ
lnð1 ðImpp =ISTC ÞÞ
BSTC ¼
V mpp V ocSTC
ð3Þ
Ai;j ¼ ISTC expðBSTC V oc Þ
ð5Þ
ISC;i;j ¼ ISTC ð2Þ
ð4Þ
The bypass and blocking diodes are modeled as ideal switches in order to simplify the calculation of the voltage value at
which the bypass diode becomes active. Moreover, the effects on the characteristic curves of both bypass and blocking
diodes are lumped in the parameters of the PV modules. This paper considers one module (group of 18 cells with one bypass
diode) of a BP585 panel [36] for simulations. The main electrical characteristics available in the datasheet are: ISTC = 5.0 A,
VocSTC = 11.05 V, Impp = 4.72 A, Vmpp = 9.0 V, aI = 0.065 %/°K, and aV = 80 mV/°K.
2.1. Inflection points in mismatching conditions
In SP configuration all strings shares the same voltage; hence, the objective of the array model, under both uniform and
mismatching conditions, is to calculate the current of each string individually for a given array voltage. Then, the current of
the PV array is obtained by aggregating the currents of all strings. Therefore, this section analyzes a single string, and such a
procedure must be performed for each string of the array.
Fig. 2a illustrates one string of 2 PV modules, where parameters Ai,1 and Bi,1 and temperatures TPV,i,1 (i 2 [1, 2]) are considered equal (uniform conditions). In uniform conditions both modules receive the same irradiance (GPV,1,1 = GPV,2,1), which
implies that ISC,1,1 = ISC,2,1; therefore the voltages and currents of the two modules are equal and depend on the voltage imposed to the string. The current versus voltage (IV) curves of such modules and the string are shown in upper Fig. 3a. It is
noted that the characteristic curve of the string has the same shape of the curve of one module with the voltage scaled by
two due to the modules are connected in series. In this case exists a single MPP in the power versus voltage curve (bottom
part of Fig. 3a) due to the uniform operating conditions.
To illustrate the effect of the bypass diodes in mismatching conditions, a shade over the module 2 is considered. Hence,
the short-circuit currents of the modules are different (ISC,1,1 > ISC,2,1). When the string current (Ist) is lower than ISC,2,1 all the
Fig. 2. 2 1 PV array in uniform and mismatching conditions.
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6
Current [A]
4
2
0
0
10
50
0
15
20
Mod. 1,1
Mod. 2,1
Str. 2x1
100
Power [W]
5
Maximum
Power Point
0
5
10
15
Voltage [V]
(a) Uniform conditions
20
Mod. 2,1 active
4
2
Inflection voltage
Mod. 2,1 inactive
0
Power [W]
Current [A]
6
0
5
10
Mod. 1,1
Mod. 2,1
Str. 2x1
60
15
Maximum
Power Points
20
40
20
0
Inflection voltage
0
5
10
15
20
Voltage [V]
(b) Mismatching conditions
Fig. 3. Characteristic curves of 2 1 PV array. Blue lines: module 1,1. Dashed-red lines: module 2,1. Purple lines: 2 1 string. (For interpretation of the
references to color in this figure legend, the reader is referred to the web version of this article.)
current flows through both modules (see Fig. 2(b)) and their voltages are different but higher than zero, as shown in the red
and blue dots in the top of Fig. 3b. Therefore, both modules are active (bypass diodes are turned off) because they are delivering power to the load. When Ist = ISC,2,1 the module 2 becomes inactive since its voltage and power are near to zero. Moreover the bypass diode 2 is turned on, but its current is equal to zero since all the current is flowing through the module 2. For
Ist > ISC,2,1 the difference between Ist and ISC,2,1 flows through the bypass diode 2 as depicted in Fig. 2c, while module 2 current
is its short-circuit current and its voltage is near to zero, therefore its power is negligible. The IV curves of the two modules
and the string are presented in the upper part of Fig. 3b, which illustrate the inflection point that occurs when module 2 becomes active (Ist = ISC,2,1). Such an operating point is also characterized by a discontinuity in the derivative of the current with
respect to the voltage. The voltage at which such an inflection point occurs is named inflection voltage [6]. The power vs. voltage curves of the modules and the string are presented at the bottom of Fig. 3b, where two MPPs are produced by the mismatching conditions and the discontinuity in the power curve of the string at the inflection voltage.
The model considers PV strings with N modules organized in decreasing order of short-circuit currents (ISC,k1,j > ISC,k,j).
That consideration does not introduce calculation errors since the PV modules are connected in series. The inflection voltage
(Vok,j, k 2 [2, n]) is the minimum voltage of the jth string that activates the kth PV module (or deactivates the kth bypass
diode). Such a condition is fulfilled when the current of the string is equal to the short-circuit current of the kth module
(Ist = ISC,k,j). When the string is operating at an inflection voltage, the associated bypass diode is active: the modules having
a current higher than ISC,k,j (from PV1 to PVk1) are active, and the remaining modules (from PVk to PVN) are inactive. Hence,
to calculate an inflection voltage it is necessary to add the voltages of all the active elements in the string with IPV,i,j = Ist = ISC,k,j
as shown in (6). The derivation of such expressions is given in [6].
Vok;j ¼
k1
X
Vxm;k
m¼1
Vxm;k
ISC;m;j ISC;k;j þ Ak;j
1
¼
ln
Bm;j
Am;j
ð6Þ
2.2. String and array current calculation
Fig. 4 shows an example of the inflection points calculated for one string composed by four PV modules with different
irradiance conditions, where three inflection points appear in the curve. Those points, along with the short-circuit current
and open-circuit voltage, can be used to determine the number of active modules (Nac) for a given string voltage. For example, between the short-circuit current and the first inflection point of Fig. 4 only one module in the string is active, this because Ist is lower than ISC,1,1 and higher than the short-circuit current of the other three modules. Similarly, between the first
and second inflections points there are two active modules since ISC,1,1 > ISC,2,1 > Ist > ISC,3,1 > ISC,4,1, and so on. Finally, when
ISC,4,1 > Ist all PV modules are active. Such information is used to construct the system of Nac non-linear equations given in
(7) to calculate the voltages of all active modules for a given voltage of the jth string (Vst). Once the Nac voltages are calculated
the string current can be obtained by using (1) for any active module. The non-linear equation system (7) can be solved by
using the improved Newton–Raphson method presented in [6].
V st ¼
Nac
X
V PV;i;j
i¼1
IPV;1;j ¼ IPV;k;j ; k 2 ½2; Nac ð7Þ
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1 active
module
3 active
modules
2 active
modules
4 active
modules
String current [A]
5
4
3 Short−circuit
point
2
Inflection
points
1
0
Open−circuit
point
0
5
10
15
20
25
30
35
40
String voltage [V]
Fig. 4. Inflection points effect on PV strings.
The PV array model described in this section improves the calculation time of the PV array since the non-linear equation
system to be solved has only one non-linear equation for each active module. Moreover, the calculation of the inflection
points also restricts the search range of the numeric method between the two inflection points at both sides of a given string
voltage. These advantages are useful to perform an evaluation of the IV curves of different possible configurations of the PV
array and to determine which of them provides the highest GMPP.
3. Detection of inflection and open-circuit points from PV string data
Since the bypass diodes operation defines the number of PV modules active for a given string voltage, as in Fig. 4, the
short-circuit, open-circuit and inflection points have to be detected to identify the modules that contribute to the string voltage in each voltage interval between two consecutive points of interest (short-circuit point, inflection points and open-circuit point).
The short-circuit point (SCP) is easily detected since it exhibits a PV string voltage equal to zero, as given in (8), where
Vst(i) and Ist(i) are Ndata experimental measurements of the PV string voltage and current, respectively. In contrast, the
open-circuit point (OCP) corresponds to the operating point with PV string current equal to zero and the lowest string voltage, as given in (9).
ISCP ¼ fmaxðIst ðiÞÞ : V st ðiÞ ¼ 0; i ¼ 1 . . . Ndata g
V OCP ¼ fminðV st ðiÞÞ : Ist ðiÞ ¼ 0; i ¼ 1 . . . Ndata g
ð8Þ
ð9Þ
An Inflection Point (INP) is characterized by a large change in the derivative of the current/voltage function, as observed in
Fig. 4. Such a concept is used to detect the INPs from experimental current/voltage measurements of a PV string: when the
magnitude of the derivative (10) at the left side of a given voltage is higher than the magnitude of the derivative at the right
side of it, then such a voltage defines an INP. Moreover, in order to find all INPs in the string it is necessary to scan the entire
IV curve. When a new INP is found, the voltage range to find the next one is constrained from the voltage of the last INP to
VOCP. Such a condition is based on the fact that each PV module exhibits a single maximum power between two INPs, which
permits to constraint the possible voltage solutions and to calculate the maximum number of INPs to be found as given in
(11), where N represents the number of PV modules in the string.
d Ist
Ist ðiÞ Ist ði 1Þ
¼
d V st V st ðiÞ V st ði 1Þ
ð10Þ
NINPmax ¼ N 1
ð11Þ
The algorithm proposed to detect the Interesting Points (IP), which are the SCP, INPs and OCP, were tested considering the
string simulated in Fig. 4: since such a string is composed by four PV modules with mismatching conditions, all the modules
exhibit different IV curves. In such an example, the SCP is the first IP (IP1), the three INPs are the IPs IP2,IP3 and IP4, and the
OCP is the last IP (IP5).
Fig. 5 shows the current/voltage derivative of the IV curve of Fig. 4 and the voltages where the IPs were identified. It is
clear that INPs and OCP voltages where accurately detected, while SCP exhibits zero volts. Then, the currents for the IPs are
extracted from the current/voltage measurements, e.g. Fig. 4.
4. Estimation of PV modules parameters
Based on the INP concept it is possible to obtain part of the current and voltage data sets (IV curves) of all PV modules in
the string and, by using such data sets, to calculate the model parameters of all the modules. To obtain IV curves of all PV
J.D. Bastidas-Rodriguez et al. / Simulation Modelling Practice and Theory 35 (2013) 50–68
String current derivative [A/V]
0.2
57
Inflection points
0
−0.2
−0.4
−0.6
−0.8
−1
Open−circuit
point
−1.2
−1.4
0
5
10
15
20
25
30
35
40
String voltage [V]
Fig. 5. Inflection points detection.
modules it is necessary to analyze the string IV curve from the first to the last IP. Moreover, the number of IPs (NIP) is given by
(12), where NINP is the number of INPs and the other two points correspond to the SCP and OCP. It is important to note that
NINP depends on the number of modules operating in different conditions in the string and its maximum value is shown in
(11), which is obtained when all modules have different operating conditions.
NIP ¼ NINP þ 2
ð12Þ
Between IP1 (SCP) and IP2 (INP1) there could be active one or more PV modules (NamIP,1,2) with the same highest short-circuit
current and the same IV characteristic. The voltages set of that interval is defined as VIP,1,2. To identify the parameters of the
NamIP,1,2 modules the data sets of currents (Iident,1) and voltages (Vident,1) of one module only (as discussed in Section 4.3) are
required, since such NamIP,1,2 modules have the same IV characteristics. To obtain such information the first step is to
determine the set of voltages generated by the NamIP,1,2 modules, defined as VMIP,1,2. Between IP1 and IP2 the condition
VMIP,1,2 = VIP,1,2 holds since there is not active any other module.
The number of modules NamIP,1,2 in this first interval [IP1, IP2] is found by dividing the voltage span generated in such an
interval (max(VMIP,1,2) = VMIP,1,2(Nd,1,2)) by VocSTC and approximating the result to the nearest higher integer, where Nd,1,2 is the
number of data in VMIP,1,2. Finally, the voltage set Vident,1 is obtained by dividing each element of VMIP,1,2 by NamIP,1,2, where
Vident,1 corresponds to the voltage generated by one of the NamIP,1,2 modules. The set Iident,1 is composed by the current points
between IP1 and IP2 since all modules share the same string current.
For the second interval, between IP2 and IP3, there are NamIP,1,2 + NamIP,2,3 active PV modules. Again, it is required to obtain
the voltage set exclusively produced by one of the NamIP,2,3 modules in order to estimate its parameters.
From the data sets Iident,1 and Vident,1 it is possible to estimate the parameters of the NamIP,1,2 modules active in the first
interval (as will be explained in the Section 4.3). Then, such parameters are used to calculate the voltage contribution of
the NamIP,1,2 modules to the string voltage in the second interval VIP,2,3, i.e. between IP2 and IP3. Such data is used to obtain
the set of voltages VMIP,2,3 exclusively produced by the NamIP,2,3 modules that become active in the second interval. Then, similarly than in the first interval, NamIP,2,3 is found by dividing the voltage span generated in the second interval (VMIP,2,3(Nd,2,3))
by VocSTC and approximating the result to the nearest higher integer. Finally, the voltage set Vident,2 is obtained by dividing
each element of VMIP,2,3 by NamIP,2,3, where Vident,2 corresponds to the voltage generated by one of the NamIP,2,3 modules,
and Iident,2 is composed by the currents between IP2 and IP3.
In general, the sets of voltages between two consecutive IPs (IPk and IPk + 1) is defined in (13), where k 2 [1, NIP 1] and
the vector VIP contains the voltages of the IPs. Moreover, the set of voltages produced exclusively by the NamIP,k,k+1 active
modules between IPk and IPk + 1 (VMIP,k,k+1) is given by (14), where k 2 [2,NIP 1], Nam,k is the total number of active modules
up to NamIP,k,k+1 (non-inclusive), and Nd,k,k+1 is the number of data in VIP,k,k+1. Nam,k is calculated by using (15) where NamIP,n,n+1
is obtained by (16) with k 2 [1, NIP 1]. The set of voltages used to identify the parameters of one of the modules active between IPk and IPk + 1 is obtained by dividing each element of (14) by (16) as shown in (17). Finally the set of currents used
for the parameters estimation is given in (18), where IIP is a vector with the currents of the inflections points.
V IP;k;kþ1 ¼ fV st ðiÞ : V IP ðkÞ < V st ðiÞ < V IP ðk þ 1Þ; i ¼ 1 . . . N data g
(
)
N am;k X 1
Iscn Ist ðiÞ
V MIP;k;kþ1 ¼ V IP;k;kþ1 ðiÞ ; i ¼ 1 Nd;k;kþ1
ln
Bn
An
n¼1
Nam;k ¼
k1
X
NamIP;n;nþ1
n¼1
NamIP;k;kþ1 ¼ ceil
V MIP;k;kþ1 ðN d;k;kþ1 Þ
V ocSTC
ð13Þ
ð14Þ
ð15Þ
ð16Þ
58
J.D. Bastidas-Rodriguez et al. / Simulation Modelling Practice and Theory 35 (2013) 50–68
V ident;k ¼
V MIP;k;kþ1 ðiÞ
;
NamIP;k;kþ1
i ¼ 1 . . . Nd;k;kþ1
Iident;k ¼ fIst ðiÞ : IIP ðkÞ < Ist ðiÞ < IIP ðk þ 1Þ;
ð17Þ
i ¼ 1 . . . Ndata g
ð18Þ
Flow chart of Fig. 6 illustrates the proposed process to obtain the IPs and the model parameters of each PV module in a
single string. The algorithm is presented in two parts: the left one illustrates the extraction of the IPs, while the right part
presents the iterative process used to obtain the estimation data sets and the model parameters.
From an implementation point of view, it is necessary to detect which physical module is active at any moment during
the voltage scan of the string, this in order to know to which module or modules correspond the estimated parameters between two consecutive IPs. Such a procedure can be easily performed by using binary signals to detect if a given module is
active or inactive (bypassed). Considering a microcontroller-based implementation, a digital input signal can be attached to
each PV module to detect if its voltage is different from zero. This implementation avoids the requirement of Analog-to-Digital converters (ADC) for each PV module; instead, ADCs are required for the current of each string and the array voltage only
(all the strings are connected in parallel), while digital inputs are used to detect the PV modules activation. Moreover, taking
into account that micro-controllers and DSPs provide large amount of digital inputs, such a solution is applicable to medium
of even large PV arrays by interconnecting several micro-controllers or DSPs.
4.1. Model parameters and data sets
The model adopted corresponds to the explicit single-diode approximation given in (1), which has been extensively used
for modeling PV systems and the design of MPP tracking techniques [6,34,35,37]. To simplify the nomenclature, expression
(1) is rewritten as in (19) for a single module.
IPV ¼ ISC A expðB V PV Þ
ð19Þ
Such a model parameters (ISC, A, and B) are valid for a given irradiance and temperature conditions as given in (2)–(5), [6]. But
taking into account that the estimation process is performed online to detect the array configuration that provides higher
power in the present environmental conditions, such a model is accurate enough since it provides a realistic calculation
Fig. 6. Proposed parameters estimation method: flowchart.
J.D. Bastidas-Rodriguez et al. / Simulation Modelling Practice and Theory 35 (2013) 50–68
59
of the GMPP for each possible configuration meanwhile the irradiance and temperature do not change significantly. When
the environmental conditions change, the module parameters must be estimated again.
The parameters of the explicit model must be calculated to accurately reproduce the PV module current/voltage characteristic. But taking into account that (19) is parameterized by three constants, only three points are used to estimate A, B and
ISC. Moreover, to guarantee a satisfactory fitting of the estimated model with the measurements, the short-circuit point of the
module SCPm(Vident,k(1), Iident,k(1)) and the maximum power point of the module MPPm are used. In such a way, the SCPm and
MPPm can be extracted from Vident,k and Iident,k as given in Fig. 7, where SCPm and MPPm are characterized by (0 [V], I0 [A]) and
(VMPP [V], IMPP [A]), respectively.
In addition, to obtain an accurate fitting of the module open-circuit voltage is required to calculate the value of the module voltage that occurs at 0 A. But such an information is, in general, not detectable from Vident,k and Iident,k since the activation
of the next PV module in the sequence (descendant irradiance) occurs at a current higher than 0 A. Therefore, the closest
measurements available are at the last values of the data sets Vident,k and Iident,k: such a point is named VIend,m and corresponds
to (Vident,k(Nd,k,k+1) [V], Iident,k(Nd,k,k+1) [A]).
Finally, the estimation of the PV model parameters is performed using three points only: SCPm, MPPm and VIend,m. Such a
condition reduces the estimation time in comparison with estimation processes that require large amount of data, this because the number of terms in the function that need to be evaluated in each iteration of the optimization method is proportional to the number of points that the function must be fitted [38].
4.2. Impact of the model parameters error
From the model current/voltage relation (19), the calculation errors in the module current generated by errors on parameters A, B and ISC are given in (20)–(22), respectively, where DIPV represents the error on IPV and DA, DB and DISC represent the
error on the parameters.
@IPV
DA;
@A
@IPV
¼
DB;
@B
@IPV
¼
DISC ;
@ISC
DIPV ¼
DIPV
DIPV
@IPV
¼ expðB V PV Þ
@A
@IPV
¼ A V PV expðB V PV Þ
@B
@IPV
¼1
@ISC
ð20Þ
ð21Þ
ð22Þ
Without loss of generality, such relations will be illustrated using one module of a BP585 panel [36], which nominal parameters are extracted from the electrical information in the manufacturer datasheet (presented in Section 2): A0 = 0.8941 lA,
B0 = 1.4060 V1 and ISC0 = 5.0 A. From the model parameters description (2)–(5) it is noted that the range of variation of ISC is
similar to the range of IPV, and A depends exponentially from BSTC while B depends linearly from BSTC. Therefore, the magnitude of A is strongly smaller in comparison with B, which is illustrated by A0 and B0. From such considerations it is concluded that errors on A (20) have a much smaller impact on IPV than errors on both B and ISC due to the very small
magnitude of DA. In addition, errors caused by B are larger at higher PV voltages, while errors caused by ISC are constant
for PV voltage variations.
Fig. 8 shows the errors introduced in IPV by ±10% errors on the model parameters: as previously predicted, errors on A
have less impact on the module current calculation than errors on both B and ISC. In this way, the MPP prediction of the
PV module is not significantly affected by errors on A, while errors on B generate large errors on the MPP voltage and power
estimation. Similarly, errors on ISC generate large errors on the MPP power prediction with small MPP voltage errors. Such a
I
0
IMPP
MPP
Module Power [W]
Module Current [A]
m
SCP
m
I
IP
VIend,m
V
0
MPP
Module Voltage [V]
Fig. 7. Points used for parameter estimation.
VIP
60
J.D. Bastidas-Rodriguez et al. / Simulation Modelling Practice and Theory 35 (2013) 50–68
behavior is illustrated in the following example: if a variation of ±20% is introduced to A0 the errors in the Impp prediction are
1.38%, while if the same variation is forced in B0 and ISC0 (separately) the errors in the Impp prediction are 17.64% and
±21.38%, respectively.
Therefore, due to the small sensitivity of IPV to changes on A in comparison with changes on B and ISC, larger estimation
errors on A are expected in comparison with the estimation errors on B and ISC. Such a condition is due to the estimation
algorithm can fit the estimation data with a larger range of A in contrast with B and ISC.
4.3. Parameters estimation algorithm
In literature is proposed to extract one or more model parameters of a PV module by fitting a non-linear equation with
experimental data [39]. In this paper the parameters of the PV model are estimated by fitting the three selected points from
each module IV curve (SCPm, MPPm and VIend,m) with the data generated by means of the current/voltage Eq. (19). Moreover,
the Nonlinear Least Square method is adopted to estimate the parameters.
In addition to the three points selected from the IV curve, the method requires some initial values and constraints for the
parameters as well as the current/voltage equation. The initial values for A and B are taken from the manufacturer datasheet
information, while the ISC value is taken from the registered data of the current/voltage curve at zero volts. The search spaces
for the parameter values are defined with upper and lower limits for each parameter. The A and B parameters are constrained
between the 10 % and 200 % of the nominal values to account for the changes caused by the temperature, while ISC is constrained between 98 % and 102 % of the initial value since A parameter is small (micro-Amperes range): for VPV = 0 V, the
relation IPV = ISC A ISC holds.
The estimation procedure is described in Fig. 9. First, the method starts with the initial values of the parameters. Then, it
produces the fitted curve for the current set of parameters. In the third step, the method adjusts the parameters and determines whether the fit improves. The direction and magnitude of the adjustment is made by a trust-region algorithm, which
is based on the trust-region concept described in [40]: the set of parameters are evaluated in the model, and if the squared
error decreases within the trust region, then the region is expanded; otherwise the region is contracted. The trust-region
algorithm was selected since lower and upper parameter boundaries are available. In addition, the convergence criteria
are: a maximum squared error of the optimization function jIPV bI PV j2 6 106 , a maximum number of fit iterations (400 iterations), and a maximum number of model evaluations (600 evaluations).
Finally, the property of requiring only three points per module instead of a large amount of points has been tested: estimating the parameters of four modules using 12 points requires 29 % of the time used to estimate such parameters from
652 points, which is the number of samples taken with steps of 0.1 V for the PV string with VOCP = 65.2 V. Such a condition
put in evidence the reduced amount of calculations avoided by using only three data points, obtaining the same results.
4.4. Application example: PV string parameters estimation
5
4
4
A0
3
10% error
−10% error
MPP
2
1
0
0
2
4
6
5
B0
3
10% error
−10% error
MPP
2
1
8
10
0
12
0
2
4
Voltage [V]
6
4
1
8
10
0
12
50
40
40
30
20
30
20
10
10
0
0
6
Voltage [V]
8
10
12
Power [W]
50
4
0
2
4
6
8
10
12
8
10
12
Voltage [V]
40
2
10% error
−10% error
MPP
2
50
0
Isc0
3
Voltage [V]
Power [W]
Power [W]
Current [A]
5
Current [A]
Current [A]
A PV string composed by six PV modules (PV1, PV2, PV3, PV4, PV5 and PV6) was considered to illustrate the proposed
solution. The adopted PV modules are half of a BP585 panel, but to take into account the differences of the temperature
and the tolerances in the PV modules, the Ai and Bi parameters have been generated randomly in the range of ±20 % with
respect to the nominal values A0 and B0 presented in Section 4.2. Moreover, very different short-circuit current parameters
30
20
10
0
2
4
6
8
10
12
0
Voltage [V]
Fig. 8. Effect of parameters errors on the PV module current.
0
2
4
6
Voltage [V]
J.D. Bastidas-Rodriguez et al. / Simulation Modelling Practice and Theory 35 (2013) 50–68
vpv
61
ipv
Photovoltaic
module
+
Least-square error
_
PV module
model
Set of initial
parameters
îpv
Nonlinear least squares
method - Trust region
algorithm
Set of estimated parameters
Input data
Iteration process data
Fig. 9. Parameters estimation procedure from experimental data.
have been used to represent a deep mismatching condition in the PV string. The PV modules parameters are given in the
second column Table 1.
Using the PV modules parameters, the precise current/voltage characteristic of the PV string was generated, measuring
the string voltage and current within the range [0 V, 65.2 V] in steps of 0.1 V, which simulates the online measurements
of the identification system. From such a procedure 652 voltage/current data points were obtained, i.e. Vst and Ist. The PV
string measurements are depicted in Fig. 10 (blue traces), where the string power/voltage characteristic was constructed
from Vst and Ist.
From the measured data, the PV string IPs (SCP, INPs and OCP) were detected using the procedure described in Section 3,
obtaining the results presented in Fig. 10 (black circles), which illustrate the high accuracy of the IP detection algorithm.
Then, the voltage and current data sets related to each PV module, Vident,i and Iident,i, were obtained from the string measurements and IPs following the procedure previously described in this section. Finally, from Vident,i and Iident,i, the estimation
data sets [SCPm, MPPm, VIend,m] were constructed for each PV module.
The PV modules parameters were estimated using the estimation algorithm of Section 4.3 and the values obtained are
presented in the third column of Table 1. Such estimated values are close to the reference values, which illustrates the accuracy of the proposed solution. To provide a quantitative measurement of the estimation accuracy, the Normalized Squared
^ represent the reference and estimated values, respectively. The NSE was
Error (NSE) given in (23) was used, where y and y
calculated for each parameter of all the PV modules, obtaining the fourth column of Table 1. Again, such results put in evidence the satisfactory accuracy of the proposed estimation procedures.
NSE ½% ¼
^ Þ2
ðy y
100
y2
ð23Þ
To test the estimated parameters, the current/voltage characteristic of the complete PV string was reproduced, obtaining
the results presented in red traces of Fig. 10, which illustrates the usefulness of the estimated models. To provide a quantitative measurement of the current curve reproduction, the Normalized Sum Squared Errors NSSE given in (24) was calcu-
Table 1
Reference and estimated PV string parameters.
Parameter
Reference value
Estimated value
NSE (%)
A1 (lA)
B1 (V1)
isc,1 (A)
A2 (lA)
B2 (V1)
isc,2 (A)
A3 (lA)
B3 (V1)
isc,3 (A)
A4 (lA)
B4 (V1)
isc,4 (A)
A5 (lA)
B5 (V1)
isc,5 (A)
A6 (lA)
B6 (V1)
isc,6 (A)
0.8683
1.3238
4.7000
0.9736
1.3501
4.2000
0.9623
1.3978
3.1000
0.9040
1.3302
2.3000
0.9169
1.5028
1.5000
0.9106
1.3202
1.4000
0.8330
1.3148
4.7002
0.8969
1.3584
4.1997
0.9345
1.4009
3.0999
0.8824
1.3324
2.2999
0.9247
1.5011
1.5000
0.8600
1.3259
1.3995
0.1648
0.0046
0.0000
0.6202
0.0037
0.0000
0.0833
0.0004
0.0000
0.0566
0.0002
0.0000
0.0073
0.0001
0.0000
0.3077
0.0018
0.0000
J.D. Bastidas-Rodriguez et al. / Simulation Modelling Practice and Theory 35 (2013) 50–68
String current [A]
62
6
Measurements
Estimated model
4
IP
2
−3
NSSE = 2.874x10 %
0
0
10
20
30
40
50
60
70
50
60
70
String voltage [V]
String power [W]
100
50
−3
NSSE = 9.343x10
%
0
0
10
20
30
40
String voltage [V]
Fig. 10. PV string measurements and estimated models simulation.
lated, where H represents the number of data. The low NSSE value (2.874 103%) provided by the estimated model guarantees a correct reproduction of the current curve; therefore the power curve of the PV string can be accurately reproduced
(NSSE = 9.343 103%) to predict the GMPP.
PH
NSSE ½% ¼
2
^
k¼1 ðyðkÞ yðkÞÞ
PH
2
k¼1 ðyðkÞÞ
100
ð24Þ
String power [W]
The predictability of the GMPP makes possible to evaluate the maximum power available in different strings constructed
using the estimated PV modules. To illustrate such a concept, the power curves of two new strings formed by three of the six
previously estimated PV modules have been calculated: the first string was constructed with PV1, PV3 and PV5, while the
second string was formed by PV2, PV4 and PV6. Fig. 11 shows the estimation of the new strings power curves using the estimated model (red traces), which are in agreement with the measurements taken from data generated with the exact string
parameters (blue traces), i.e. second column of Table 1. The low NSSE values obtained: 4.483 102% for the first string (PV1,
PV3 and PV5) and 2.902 104% for the second string (PV2, PV4 and PV6), put in evidence the usefulness of the proposed
estimation procedure in the test of PV strings with different configuration from the one used to estimate the modules
parameters.
Finally, the estimated models were used to reproduce the electrical characteristics of all the PV modules, obtaining the
results reported in Fig. 12, where the satisfactory behavior of the estimated models and low NSSE are observed. Such precision verifies that the estimated models are accurate representations of the reference PV modules.
60
40
20
String of PV1−PV3−PV5:
−2
NSSE = 4.483x10 %
0
0
5
10
15
20
25
30
35
30
35
String power [W]
String voltage [V]
Measurements
Estimated model
50
40
30
20
String of PV2−PV4−PV6:
−4
NSSE = 2.902x10 %
10
0
0
5
10
15
20
25
String voltage [V]
Fig. 11. Estimation of the new strings power curves.
J.D. Bastidas-Rodriguez et al. / Simulation Modelling Practice and Theory 35 (2013) 50–68
63
5. PV array reconfiguration
The estimated model parameters are intended to detect the PV array configuration that provides the highest power. Such
a procedure is performed using the array model presented in Section 2, which requires a reduced number of calculations in
comparison with classical models, providing a fast calculation of the array current [6].
To detect the PV array configuration with highest power, the GMPP of the array configurations must be compared following a search algorithm. Despite multiple optimization algorithms can be adopted, i.e. [15,14,11,16], this paper assesses all
possible SP configurations of the array (exhaustive search) that may provide different GMPP values. Such a exhaustive search
algorithm has been successfully tested in reconfiguration of TCT arrays [15].
The number of possible array configurations with potentially different power in an SP structure is given in (25) where N is
the number of modules in each string and M is the number of strings. In an equivalent TCT structure with N rows of M parallel connected modules the number of feasible possibilities is calculated using (26) [15]. Therefore, for high-voltage requirements, where more PV modules must be connected in series (more rows than strings), the SP structure exhibits a reduced
number of configurations to test. Such a condition makes SP arrays reconfiguration faster for grid connected applications in
contrast with TCT structures. In any case, the solution proposed in this paper can be extended to arrays in TCT configuration
with some modifications in the model and the sensors connected to the array.
NRSP ¼
NRTCT
ðN MÞ!
ð25Þ
ðN!ÞM M!
ðN MÞ!
¼
ðM!ÞN N!
ð26Þ
To illustrate the SP array possible configurations, an array composed by two strings of three PV modules each (3 2) is
considered, it exhibiting 10 possible configurations that provide different power. Such possible configurations are shown in
Table 2, where the modules connection in the strings are given as columns: in example, the first configuration has the first
string composed by PV1, PV2 and PV3, while the second string is composed by PV4, PV5 and PV6.
5.1. Reconfiguration system
To automatically reconfigure the PV array several switches (switches matrix) are required to change the strings connections. The number of switches required by SP connection following the structure presented in Fig. 13 is given by (27), while
the number of switches requiered by the TCT structure presented in [15] is given by (28).
NSwSP ¼ 2 N ðM 1Þ þ ðN 1Þ ðM 2Þ ðM 1Þ ðN M N þ 1Þ ðN M N þ 2Þ
þ
þ ðN 2Þ ðN M N þ 1Þ
2
2
NSwTCT ¼ N ðN þ 1Þ 2 þ 2 M ðN M NÞ
ð27Þ
ð28Þ
4
3
2
1
0
NSSE = 7.105x10−2%
0
5
5
4
3
2
1
0
10
NSSE = 3.779x10−4%
0
Module voltage [V]
5
Module current [A]
5
Module current [A]
Module current [A]
Fig. 13 shows a switches matrix to reconfigure the adopted 3 2 PV array, which requires 20 single-position switches. In
the connection circuit, the upper switches (S2U, S3U, S4U, S5U and S6U) are composed by 2, 3 or 4 single-position switches,
while the lower switches (S3D, S4D and S5D) are single-position switches. Therefore, upper switches are always connected
5
4
3
2
1
NSSE = 3.779x10−4%
0
0
10
Module voltage [V]
5
10
Module voltage [V]
NSSE = 2.910x10−6%
4
3
2
1
0
0
5
10
Module voltage [V]
5
NSSE = 1.995x10−4%
4
3
2
1
0
0
5
10
Module voltage [V]
Module current [A]
5
Module current [A]
Module current [A]
PV Module characteristic
Estimated model
5
NSSE = 5.490x10−5%
4
3
2
1
0
0
5
10
Module voltage [V]
Fig. 12. PV modules characteristics and estimated models.
64
J.D. Bastidas-Rodriguez et al. / Simulation Modelling Practice and Theory 35 (2013) 50–68
Table 2
Possible configurations of a 3 2 PV array.
1
2
3
4
5
6
7
8
9
10
14
kk
25
kk
36
13
kk
25
kk
46
13
kk
24
kk
56
13
kk
24
kk
65
12
kk
35
kk
46
12
kk
34
kk
56
12
kk
34
kk
65
12
kk
43
kk
56
12
kk
43
kk
65
12
kk
53
kk
64
Ist2
Ist1
+
1
4
1 2
S2U
4
1 2 3 4
S5U
S4U
S3U
S6U
Va
PV1
PV2
PV3
PV4
S4D
S3D
1
PV5
PV6
-
S5D
1
1
Fig. 13. Switches matrix scheme for the 3 2 PV array reconfiguration.
to one terminal, i.e. one single-position switch connected at time (1, 2, 3 or 4), while the lower switches could be connected
(1) or disconnected (0) only. The positions of the switches (switches states) to obtain the configurations of Table 2 are described in Table 3.
In addition, the switches matrix of Fig. 13 considers current sensors for each string (Ist1 and Ist2). Such sensors are used to
obtain measurements of the IV characteristics of each string of the array from a single scan of the array voltage. Such IV characteristics are required to estimate the PV modules parameters following the procedures described in Sections 3 and 4.
5.2. Experimental measurements
To validate the proposed approach, an application example based on experimental measurements from ERDM85 PV panels [41] is presented. The main datasheet information of such modules are: ISTC = 5.13 A, VocSTC = 21.78 V, Impp = 4.8 A,
Vmpp = 17.95 V, aI = 0.020 %/°K, aV = 0.34 %/°K and Ns = 36. The experimental system depicted in Fig. 14 was used to collect
measurements from PV modules at different operation conditions. Such real modules characteristics were used to experimentally test the proposed identification procedure. The experiments were carried out during a clearly sky day in the
south-west region of Colombia. During the experiments the approximated values of the ambient temperature and irradiance
were 24 °C and 580 W/m2, respectively.
The experimental measurements were used to construct the current/voltage characteristics of the 3 2 SP array, for the
10 configurations of Table 2. Such a procedure was performed by implementing the switches matrix of Fig. 13 in Matlab,
controlling the switches states as described in Table 3.
Without loss of generality, this example considers the switches matrix operating in the first configuration of Table 2,
named Config. 1, in which the current/voltage measurements of the two strings were generated from the experimental modules characteristics. Such strings measurements were used to estimate the PV modules parameters following the procedures
Table 3
Switches matrix states for the 3 2 PV array configurations.
Configuration
[S2U]
[S3U, S3D]
[S4U, S4D]
[S5U, S5D]
[S6U]
1
2
3
4
5
6
7
8
9
10
[1]
[1]
[1]
[1]
[4]
[4]
[4]
[4]
[4]
[4]
[2,
[4,
[4,
[4,
[1,
[1,
[1,
[2,
[2,
[2,
[4,
[2,
[3,
[3,
[3,
[2,
[2,
[1,
[1,
[3,
[4,
[3,
[2,
[4,
[2,
[3,
[4,
[4,
[3,
[1,
[4]
[4]
[3]
[1]
[4]
[3]
[2]
[2]
[3]
[4]
1]
0]
0]
0]
0]
0]
0]
0]
0]
0]
0]
1]
0]
0]
1]
0]
0]
0]
0]
1]
0]
0]
1]
1]
0]
1]
1]
1]
1]
0]
65
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Artificial
Shade
Connection
Table
DC Load
Fig. 14. Experimental test platform.
First string power [W]
described in the flow chart of Fig. 6. Fig. 15 shows both experimental and estimated power/voltage characteristics of the first
and second strings, where the estimated model provides satisfactory results (NSSE = 0.1078 % and NSSE = 0.0951 %,
respectively).
The estimated model parameters were used to reproduce the experimental PV modules characteristics, obtaining the
satisfactory results reported in Fig. 16, where the three IV curves in the upper and the three IV curves in the bottom
of the figure correspond to the first and the second strings, respectively. Such calculations, characterized by low NSSE values, put in evidence the accurate prediction of the real modules electrical behavior. But it must be pointed out that, in the
proposed procedure, the estimation error is propagated across the estimated modules in the same string, since the estimated model of the first module (in a string) is used to generate the estimation data for the second module, then the errors on the first module parameters increase the estimation error on the second module. In general, the estimation errors
increase with the number of modules previously estimated for the same string. But, since the proposed procedure estimates the PV modules in descendent order of irradiance, the modules parameters with higher errors have less contribution to the PV array power estimation.
The estimated parameters were used to test the 10 possible configurations of the 3 2 PV array, obtaining the results
presented in Table 4, where the experimental and estimated GMPP are reported. From such results it is demonstrated that
the estimated models accurately predict the array configuration that provides the highest power production. Moreover, Table 4 also reports the NSSE for each configuration, which put in evidence the satisfactory results of the proposed solution.
Finally, to illustrate the estimation accuracy, Fig. 17 shows the comparison of the experimental and estimated electrical characteristics of the array configurations that produces the highest (Config. 2) and the lowest (Config. 9) GMPP.
Experimental data
Estimated model
80
60
40
20
0
String of: PV1−PV2−PV3
0
10
20
30
40
50
60
50
60
Second string power [W]
PV array voltage [V]
80
60
40
20
0
String of: PV4−PV5−PV6
0
10
20
30
40
PV array voltage [V]
Fig. 15. Experimental and estimated strings power curves.
J.D. Bastidas-Rodriguez et al. / Simulation Modelling Practice and Theory 35 (2013) 50–68
2
1
PV1
NSSE = 0.0577 %
0
0
10
2
1
PV2
NSSE = 0.0897 %
0
20
3
0
Module voltage [V]
2
1
PV4
NSSE = 0.0501 %
0
10
1
PV3
NSSE = 2.2183 %
0
20
0
2
1
PV5
NSSE = 0.0952 %
0
20
0
Module voltage [V]
10
20
Module voltage [V]
Experimental data
Estimated models
3
Module current [A]
Module current [A]
10
2
Module voltage [V]
3
0
Module current [A]
3
Module current [A]
Module current [A]
3
10
20
3
Module current [A]
66
2
1
PV6
NSSE = 0.4333 %
0
Module voltage [V]
0
10
20
Module voltage [V]
Fig. 16. Experimental and estimated PV modules characteristics.
Experimental GMPP (W)
Estimated GMPP (W)
NSSE (%)
1
2
3
4
5
6
7
8
9
10
133.02
154.00
134.33
131.40
133.61
131.61
134.11
135.05
130.70
132.83
133.96
155.57
136.17
132.26
134.29
132.40
135.87
136.91
131.79
133.84
0.1266
0.0581
0.0931
0.1191
0.1235
0.1163
0.0950
0.0884
0.1087
0.1226
PV array Current [A]
Config.
6
4
2
0
0
10
Experimental data
Estimated model
20
30
40
PVGMPP
array voltage [V]
Estimated GMPP
50
60
PV array Power [W]
PV array Power [W]
PV array Current [A]
Table 4
Experimental and estimated GMPP.
200
150
100
50
0
NSSE = 0.0581 %
0
10
20
30
40
50
60
6
4
2
0
0
10
Experimental data
Estimated model
20
30
40
PV GMPP
array voltage [V]
Estimated GMPP
50
60
50
60
200
150
100
50
0
NSSE = 0.1087 %
0
10
20
30
40
PV array voltage [V]
PV array voltage [V]
(a) Config.2 (highest power)
(b) Config.9 (lowest power)
Fig. 17. Experimental and estimated power curves of the best and worst configurations.
J.D. Bastidas-Rodriguez et al. / Simulation Modelling Practice and Theory 35 (2013) 50–68
67
6. Conclusions
A method to determine the SP configuration that provides the highest GMPP, in uniform or mismatching conditions, from
the measurements of the array voltage and the currents in each string, is proposed. The first step of the method is to perform
a voltage scanning of the array and measure the currents in each string to obtain the IV curve of each string. From such data
the SCP, INPs and OCP points are detected to extract the part of the data that correspond to each module. Then, the parameters of the modules are calculate with a suitable estimation method. When all parameters are calculated, the model presented in [6] is used to find the GMPP of each configuration to determine the best one for a given mismatching profile
and weather conditions.
The model proposed in [6] was selected to evaluate all the feasible configurations due to its high calculation speed. Nonetheless, it is important to note that such a model assumes ideal bypass diodes. However, when the bypass diodes are active
(PV modules are bypassed) the output voltages of the modules are negative depending on the current that flows through the
bypass diodes. Although such negative voltages are small, they produce small shifts between the experimental measurements and the data predicted by the model as analyzed in [10]. Nevertheless, such shifts may not affect the shape of the
curves and the estimation of the best configuration should not be affected.
The solution proposed in this paper could be improved by introducing a more detailed model of the bypass diodes in order to reproduce the small voltage shifts in the IV curves produced by such diodes. Nevertheless it is important to highlight
that the parameter calculation of the bypass diodes is not a trivial task since the measurement of the diodes current are required, but such nodes are not accessible in all the commercial PV panels. Moreover, since the exhaustive search algorithm
tests all the possible configurations, a more optimized search method is required to apply the proposed solution in middlesize and large-size PV arrays. Finally, it is important to note that the proposed method can be extended to PV arrays in TCT
configurations. To achieve that goal it is required a model to predict the electrical behavior of PV arrays in TCT configuration
as the one presented in [42]. In addition it would be necessary to change the number and locations of the current and voltage
sensors in the array since all the modules in each row share the same voltage, but the current of each module may be
different.
Acknowledgments
The authors acknowledge the scholarships APCC-ND-66-197 and 095-2005 to the Colombian Departamento Administrativo de Ciencia, Tecnología e Innovación (COLCIENCIAS). This work was also supported by the Universidad Nacional de
Colombia under the projects VECTORIAL-MPPT of GAUNAL research group and PORTAFOLIO DE PROYECTOS DE INNOVACIÓN
2008-2011 (Código Quipu 20701008044) of Facultad de Minas - Sede Medellín.
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