Usually, the output characteristics of a photovoltaic (PV) array are significantly affected by nonuniform irradiance which is caused by ambient obstacles, clouds, orientations, tilts, etc. Some local maximum power points (LMPP) in the currentvoltage (IV) curves of a PV array can result in power losses of the array. However, the output power at the global maximum power point (GMPP) is different in different interconnection schemes in a PV array. Therefore, based on the theoretical analysis and mathematical derivation of different topological structures of a PV array, this paper investigated the output characteristics of dual series PV arrays with different interconnections. The proposed mathematical models were also validated by experimental results. Finally, this paper also concluded that in terms of performance, the total cross tied (TCT) interconnection was not always the optimal structure, especially in a dual series PV array. When one of the PV modules was severely mismatched, the TCT worked worse than the series parallel (SP) structure. This research can provide guidance for switching the interconnection to gain the greatest energy yield in a changeable structure PV system.
I. INTRODUCTION
Due to the decreasing supply of traditional fossil fuels and global warming, photovoltaic (PV) power has been widely used as a form of sustainable energy in recent decades. However, because outdoorinstalled PV modules cannot achieve the same performance as those in a laboratory, the operating conditions of a PV array should be taken into careful consideration. As shown in previous studies
[1]

[5]
, the presence of nonuniform irradiance, such as the shadows of surrounding obstacles, clouds, bird droppings, orientation or tilt can result in mismatching losses of a PV array. A PV array made up of PV modules connected in series can be regarded as a PV string. When the shortcircuit current of shaded PV modules is less than the operating current of a PV string, the bypass diodes of the shaded PV modules are conducted and the multiple power peak phenomenon of the IV curve will occur.
Some published papers reported methods for alleviating the influence of a nonuniform irradiance by changing the PV array interconnections
[6]

[8]
. Three main topology structures have been designed for PV arrays: the series parallel (SP), bridge link (BL) and total cross tied (TCT) structures, as shown in
Fig. 1
. In the SP topology, PV strings are not interconnected. However, they are entirely interconnected on each row of junctions in the TCT topology. In the BL topology, PV strings are interconnected alternatively. A common conclusion is that the TCT topology is the most efficient interconnection structure for reducing the loss of generated power under the same conditions
[9]
. In other words, more interconnections can generate more power in a PV array under nonuniform irradiance conditions. In paper
[9]
, in order to improve the output power of a PV array, the reasons why the TCT and BL configurations are superior to the SP configuration were discussed. Interconnected PV strings can mitigate the influence of the shaded PV modules in the seriesconnected branch where the shaded PV modules are located.
Different topology structures of a PV array.
Nevertheless, these papers
[6

7]
analyzed the distinctions of different PV array interconnections only with experimental results, rather than theoretically analyzing the operation process of PV arrays, i.e. obtaining power at the MPP through mathematical derivation.
Therefore, this paper studied the performance of different PV array interconnections under the conditions of nonuniform irradiance. It also presents a theoretical analysis and experimental results. Firstly, a 3×2 PV array with only one shaded PV module was discussed. Then, the performances of the interconnection structures in dual and triple series connected PV systems were investigated. The results indicate that the TCT structure is not always the optimal topology structure for maximizing the output power in a mismatched PV array. The performance of different structures in a PV array is significantly influenced by the mismatch level of the shaded PV module and the shadow location.
II. ANALYSIS OF PV ARRAY TOPOLOGY STRUCTURES
In order to analyze the output performance of each interconnection structure for a PV array with only one shaded PV module, the relationship between the mismatch level and the output power of each structure should be investigated. In this section, the healthy PV modules are assumed to operate under the standard test condition (STC) and the mismatch severity is evaluated based on the level of equivalent irradiance of the shaded PV module. Hence, mathematical equations are deduced to describe the relationship between the level of irradiance of the shaded module and the output power of different PV array structures. The 3×2 PV array, which is formed by 3 rows and 2 columns of PV modules, is discussed first due to its simplicity in terms of deduction and calculation.
 A. Model of a PV Module
A solar cell is usually represented by singlediode or twodiode models. Due to the simplicity and accuracy of the singlediode model
[10]
,
[11]
, it is used in this paper, and its equivalent circuit is shown in
Fig. 2
[12]
.
Equivalent circuit of a solar cell.
The symbols in
Fig.2
are defined as follows:
I
_{ph_cell}
is the cell photocurrent,
I
_{d}
is the current of the parallel diode,
I
_{sh}
is the shunt current,
I
_{cell}
and
V
_{cell}
are the output current and voltage of the cell,
D
is the parallel diode,
R
_{sh_cell}
is the shunt resistance, and
R
_{s_cell}
is the series resistance. The IV equation of the cell in
Fig. 2
is shown as:
Where
I_{o_cell}
is the reverse saturation current of the diode,
q
is the electron charge (1.602×10
^{19}
C),
A_{cell}
is the PN junction ideal factor,
K
is the Boltzmann constant (1.38 × 10
^{23}
J/K), and
T
_{cell}
is the temperature of the cell.
In the one hand, the value of
R
_{sh_cell}
is much greater than the value of
V
_{cell}
+
R
_{s_cell}
I
_{cell}
[13]
. On the other hand, by approximating
I
_{ph_cell}
as
I
_{SC_cell}
, Equation (1) can be simplified by the following equation
[12]
:
Where
I
_{SC_cell}
is the short circuit current of a solar cell.
Equation (2) is valid for a solar cell. PV cells are connected in a series to form a PV model. According to paper
[14]
, the IV equation of a PV module is derived as:
Where
N
_{s}
is the number of series connected solar cells in a PV module.
In addition, Equation (3) can be simplified as:
Where
I
_{SC}
is the shortcircuit current of a PV module,
I
_{o}
is the reverse saturation current of the equivalent diode of a PV module,
A
is the ideal factor,
R
_{s}
is the series resistance,
V
_{module}
and
I
_{module}
are the output voltage and current of a PV module, and
T
is the temperature of a PV module.
According to the model in reference
[13]
, under the STC, the model of a PV module can be obtained from Equation (4).
Where
C
_{1}
and
C
_{2}
are constants under the STC,
V
_{OC}
is the open circuit voltage of the PV module under the STC,
I
_{M}
and
V
_{M}
are the current and voltage at the maximum power point (MPP) under the STC.
Due to the fact that most PV module manufacturers just provide the electrical specifications of PV modules under the STC,
DI
and
DV
are defined as variations of the current and voltage between the STC and the various conditions
[15]
shown in Equations 8 and 9.
Where
T
_{ref}
is the reference temperature of a PV module under the STC,
S
is the inplane irradiance,
S
_{ref}
is the reference irradiance under the STC,
α
and
β
denote the temperature coefficients of
I
_{SC}
and
V
_{OC}
, respectively.
Then, the IV equation of a PV module under various conditions can be deduced as follows:
The following sections will present the performances of PV arrays which are based on Equation (10).
 B. SP Topology Analysis of a 3×2 PV ARRAY
Bypass diodes are usually embedded in PV modules during manufacturing. As shown in
Fig.3
, since the mismatched PV module PVM1 in a 3×2 PV array may be bypassed by a bypass diode, two operating modes of the SP topology exist.
Two operating circuits of SP. (a) Shaded module is by passed and (b) shaded module is not bypassed.
In
Fig. 3
, when PVM1 is bypassed, the operating circuit of the PV array is presented in
Fig. 3
(a). Then, the output power of the PV array can be obtained as follows:
Where
P
is the output power of the PV array,
I
_{1}
is the current of the right branch formed by PVM4, PVM5 and PVM6,
I
_{2}
is the current of the left branch of PVM2 and PVM3, and
V
is the output voltage of the PV array. The forward voltage of the block diode and bypass diode are assumed to be
V
_{d}
.
Since the other 5 modules work under a unique irradiance level, their characteristics are the same. Hence, the voltages of modules PVM2 and PVM3 are (
V
+2
V
_{d}
)/2, and the voltages of modules PVM4, PVM5, PVM6 are (
V
+
V
_{d}
)/3. According to Equation (10), the I
_{1}
V equation and I
_{2}
V equation of the two branches in
Fig. 3
(a) are described as:
Where
DV
_{1}
and
DI
_{1}
are the variations of the voltage and current of a healthy PV module between the present condition and the STC, as shown in Equation (8) and (9).
In addition, the derivative of the PV array power with respect to the PV array voltage is zero at the MPP. Thus, the derivative of Equation (11) with respect to V turns to:
Equation (14) is a combination of the equations of dI
_{1}
/dV, dI
_{2}
/dV and Equations (12) and (13). Therefore, a numerical solution for V can be calculated at the MPP. Finally, the output power at the MPP can be obtained.
When the operating current of PV module PVM2 or PVM3 is less than the short circuit current of PVM1, PV module PVM1 will not be bypassed and all 6 PV modules will generate power. The operating mode of the PV array is presented in
Fig. 3
(b).
Under these circumstances, the I
_{1}
V equation of the right branch in
Fig. 3
(b) is the same as Equation (12). For the left branch, the I
_{2}
V equation is given by:
Where
DV
_{1}
and
DI
_{1}
are variations of the voltage and current of an unshaded PV module between the operating condition and the STC based on Equation (8) and (9).
DV
_{2}
and
DI
_{2}
are variations of the voltage and current of a shaded PV module between the operating condition and the STC, as shown in Equations (8) and (9).
Hence, according to Equations (12), (14), (15), dI
_{1}
/dV and dI
_{2}
/dV, the numerical solution of the output power at the MPP can be calculated in the operating mode as shown in
Fig. 3
(b).
 C. TCT Topology Analysis of a 3×2 PV ARRAY
There are also two operating modes of the TCT topology in a 3×2 PV array when a PV module is shaded, as shown in
Fig. 4
.
Two operating circuits of TCT. (a) Shaded module isn’t working and (b) shaded module is working.
When the sum of the short circuit currents of PV modules PVM1 and PVM4 is less than the output current of the PV array, PV module PVM1 and PVM4 will be bypassed. The operating circuit of the PV array is presented in
Fig. 4
(a). Then, the output power of the PV array can be obtained by:
Where
V
_{1}
is the voltage of a single PV module, i.e. PVM2, PVM3, PVM5 or PVM6,
I
is the PV array’s output current.
According to the transformation of Equation (10),
V
_{1}
can be expressed explicitly by
I
:
Similarly, the derivative of the PV array power with respect to the PV array current is zero at the MPP, and the derivative of Equation (16) with respect to I brings:
Equation (18) is composed of dV
_{1}
/dI and Equation (17) leading to a numerical solution of I at the MPP. Therefore, the output power at the MPP can be obtained.
When the sum of the short circuit currents of PV modules PVM1 and PVM4 is greater than the output current of the PV array, all 6 PV modules will work normally.
Fig. 4
(b) reveals the operating circuit of the PV array under these circumstances. Then, P can be obtained by:
Where
V
_{1}
is the voltage of PVM2, PVM3, PVM5 or PVM6,
V
_{2}
is the voltage of PVM1 or PVM4, and
I
is output current of the PV array.
In this mode the IV
_{1}
equation in
Fig. 4
(b) is the same as Equation (16), and the IV
_{2}
equation in
Fig. 4
(b) is described as:
Then, some terms in Equation (19) can be obtained with Equation (17) and (20). Then, it is possible to get the PI formula of the operating circuit:
Due to the derivative of P in Equation (21) with respect to I being zero, the numerical solution of I and the output power can be obtained at the MPP.
 D. BL1 Topology Analysis of a 3×2 PV ARRAY
As shown in
Fig. 5
, there are also two operating modes of the BL1 topology in a 3×2 PV array when a PV module is shaded.
Two operating circuits of BL1. (a) Shaded module is bypassed and (b) shaded module is not bypassed.
When PVM1 is bypassed, the PV array’s operating circuit is presented in
Fig. 5
(a). The voltage of PVM3 and PVM6 is denoted by V
_{3}
, and the sum of the voltages of PVM4 and PVM5 is V
_{4}
. Then, the output power can be obtained as:
Where
I
is the output current of the PV array.
According to the transformation of Equation (10), the V
_{3}
I equation in
Fig. 5
(a) becomes:
In addition, the IV
_{4}
equation in
Fig. 5
(a) is given by:
Similarly, the derivative of the PV array power with respect to the PV array current is zero at the MPP. The derivative of P in Equation (22) with respect to I brings:
Then terms in Equation (25) can be obtained with dV
_{3}
/dI, dV
_{4}
/dI, Equations (23) and (24), arriving at a numerical solution for V
_{4}
at the MPP. Therefore, the output power at the MPP can be obtained in the operating circuit as shown in
Fig. 5
(a).
When the output current of module PVM2 is less than the short circuit current of PVM1, all 6 PV modules will generate power. The operating circuit of a PV array is presented in
Fig. 5
(b). Then, P can also be expressed as Equation (22), and the V
_{4}
I
_{2}
equation in
Fig. 5
(b) becomes:
Where
I
_{2}
is the current of PVM1 and PVM2. The terms
A
,
B
and
C
can be set as:
Then, Equation (26) can be simplified to:
The solution of I
_{2}
is given by:
In Equation (28), only one solution for I
_{2}
is valid. From reference
[16]
, the following equation can be obtained:
In this equation, I
_{SC2}
and S
_{2}
are the short circuit current and absorbed irradiance of the mismatched PV module. Generally, as the value of
α
in Equation (8) is very small, DI
_{1}
and DI
_{2}
are mainly dominated by absorbed irradiance. Therefore, by neglecting some of the terms in Equation (8), Equations (8) and (9) can be rewritten as:
Where
S
_{1}
is the irradiance of an unshaded PV module.
I
_{SC}
is the short circuit current of a PV module under the STC.
S
_{ref}
is the irradiance of a PV module under the STC.
Through Equations (29), (30) and (31), the following equation can be deduced:
That is:
Therefore, Equation (28) can be rewritten so that:
Eventually, the output current of the PV array can be obtained by:
Where I
_{1}
is the output current of PVM4 and PVM5.
To reduce the number of variables in Equation (22), Equations (23) and (35) are substituted by (22) to get the PV
_{4}
equation. Based on the fact that dP/dV
_{4}
is equal to 0 at the MPP, the output power in the operating circuit of
Fig. 5
(b) can be calculated.
 E. BL2 Topology Analysis of a 3×2 PV ARRAY
There are two operating modes of the BL2 topology in a 3×2 PV array when module PVM1 is shaded, as shown in
Fig. 6
. Obviously, the working modes of the BL2 topology are similar to those of the TCT topology under the same condition for the shading distribution of the specific 3×2 PV array. Therefore, the theoretical analysis and mathematical derivation of the BL2 topology can be omitted.
Two operating modes of BL2. (a) Shaded module isn’t working and (b) shaded module is working.
 F. Simulations of the GMPP Analysis in Different 3×2 PV Array Structures
By analyzing different topology structures for a 3×2 PV array and calculating the GMPP and local maximum power point (LMPP) of each PV array topology, the Mfile of MATLAB codes was programed. In this paper, an EGing50W PV module which is constituted by 36 seriesconnected monocrystalline cells was applied, where 2 groups of 18 cells were paralleled with a bypass diode. The electrical characteristics of the EGing50W under the STC, provided by the manufacturer, are shown in
TABLE I
.
ELECTRICAL CHARACTERISTICS OF EGING50W
ELECTRICAL CHARACTERISTICS OF EGING50W
All of the PV modules are assumed to work under the STC except for PVM1, and all of the PV modules’ temperatures are the same because of the thermal inertia of the PV array. The PV curves of the different PV array topologies will have two peaks when the array works under the above mentioned conditions. The peak at the position of the lower voltage is regarded as the left peak, while the peak at the position of higher voltage is marked as the right peak. According to the mathematical derivations in Section B, C, D and E, the numerical solution of the output power of each PV array topology at the GMPP and LMPP can be calculated by changing the irradiance of PVM1 from 0 to 1000W/m
^{2}
. The corresponding output power at the GMPP and LMPP of each array structure is shown in
Fig. 7
.
Maximum power of each PV array topology structure versus the irradiance of shaded PV module.
On the one hand, with an increase in the irradiance of PVM1, the power of each PV array topology at the MPP in the right peak increases as shown in
Fig. 7
. On the other hand, the power of each PV array topology at the MPP in the left peak is constant, due to the shaded PV module PVM1 being bypassed by the bypass diode.
In
Fig. 7
, the corresponding output power at the GMPP of each PV array topology is obtained as shown in
Fig. 8
.
Maximum power of each PV array topology structure versus the irradiance of shaded PV module.
According to
Fig. 7
and
Fig. 8
, the GMPP of each PV array topology is likely to appear at the right peak when PVM1 is slightly shaded. In addition, compared with the other topology structures, the TCT can generate greater power at the GMPP. On the other hand, when PVM1 is seriously shaded, the GMPP appears at the left peak. The SP can generate a greater power at the GMPP.
As shown in
Fig. 8
, the percentage of the power deviation of each PV array topology with respect to the SP at the GMPP is shown in
Fig. 9
.
The power deviation percentage at GMPP versus the irradiance of shaded PV module.
When the GMPP appears at the left peak for each PV array structure, the greatest power of the topology SP occurs in the three topology structures. However, when the GMPP appears in the right peak of each PV array topology, the power of the TCT topology is superior. When the equivalent irradiance of the mismatched PV module is approximately 360W/m
^{2}
, the TCT interconnection achieves greater superiority than the SP interconnection.
III. EXPERIMENTAL VALIDATION OF DIFFERENT INTERCONNECTIONS OF A 3×2 PV ARRAY
 A. Implementation of Outdoor Experiments
To verify the proposed model for the generated power of different PV array topology structures, a flexible experimental platform was built on the rooftop of a laboratory building on campus, as shown in
Fig. 10
. In addition, the structure of the PV array was altered by switching the structures in
Fig. 3
,
4
,
5
and
6
. The specifications of the experimental PV modules are shown in
TABLE I
.
Flexible experimental platform.
Three PV modules were mounted on a frame. A solar power meter (model: TES1333) was used to measure the instant irradiance and a digital multimeter (model: Fluke17B) was used to measure the back sheet temperature of the PV modules. A selfdesigned programmable electronic load was used to measure the IV curve after altering the structure of the PV array. The resolution of the measured voltage and current are 10 mV in a 090 V range and 1 mA in a 010 A range, respectively
[17]
. Finally, the IV curves of different topology structures and their corresponding ambient parameters, i.e. irradiance and PV module temperature, are obtained simultaneously by this platform.
 B. Study Cases of PV Array Interconnections under the Conditions of a Nonuniform Irradiance
Based on the platform mentioned above, the rationality of the mathematical derivation and theoretical analysis of the 3×2 PV array in Section II can be validated. In addition, the performance of different PV array topology structures under the conditions of a nonuniform irradiance can also be compared.
The conditions of a nonuniform irradiance can be simulated by using several pieces of translucent plastic to decrease the inplane irradiance of one PV module, i.e. PV module PVM1 in
Fig. 3
,
4
,
5
and
6
. Moreover, the temperature of the PV modules can be considered to be the same due to the thermal inertia of the PV array.
Fig. 11
shows the measured PV characteristics of the investigated 3×2 PV array with different topology structures. The irradiance level of the shaded PV module is 320W/m
^{2}
and that of the other PV modules is 780W/m
^{2}
. The back sheet temperature of the PV modules approximates 42℃. As shown in
Fig. 11
, the right peak on the PV curve of the different interconnected PV arrays is the GMPP. Compared with the other interconnection structures, the TCT and BL2 interconnections both generated greater output power. These experimental results agree with the conclusions summarized in
[6]
,
[8]
,
[9]
, i.e. the TCT interconnection is the most efficient structure and can generate much more power than the SP structure when PV array mismatch occurs.
Experimental PV curves when GMPP appears at the right peak.
Another experiment was implemented to evaluate the different topology structures of the 3×2 PV array when the GMPP appears in the left peak of the PV curve. The irradiance of the shaded PV module is 51W/m
^{2}
and that of the other PV modules is 903W/m
^{2}
. Although such a low irradiance may rarely happen in reality, some problems can be found in the experiments. The average back sheet temperature of each PV module is approximately 62℃. As shown in
Fig. 12
, when the mismatched PV module is severely shaded, the GMPP of the PV curves will appear in the left peak regardless of the structure of the PV array. In fact, in a dual series connected PV array, the SP structure can generate much more power than the TCT or BL2 structure when the mismatched PV module is severely shaded. This result contradicts the conventional conclusions from other studies. This is mainly because the bypass diode of the shaded PV module conducts, which makes the PV module in the same row in a dual series connected TCT structure become bypassed. However, it does not influence the operation of the other PV modules in a SP interconnected PV array.
Experimental PV curves when GMPP appears at the left peak.
Fig. 11
and
Fig. 12
reveal two peaks of the PV array’s output power under the conditions of a nonuniform irradiance. Some conclusion can be generalized as follows:

1) Two peaks exist in a 3×2 PV array when a PV module is shaded. The GMPP may appear at the left peak or the right peak depending on the mismatch severity of the shaded PV module.

2) According to the results of the experiment, when the mismatched module is severely shaded, the superiority order of the output power of each topology structures is: SP>BL1>TCT=BL2.

3) From the results shown inFig. 8, when the GMPP appears at the left peak, the SP interconnection has better performance than the TCT interconnection.
IV. SIMULATION ANALYSIS OF M×N PV ARRAYS
 A. The extension and Analysis of N×2 PV Arrays
The output power at the GMPP or LMPP of different interconnections in a 3×2 PV array has been analyzed by deriving the relationship between the irradiance of the shaded PV module and the output power of an array. Therefore, the output power at the GMPP of a PV array that contains a different number of seriesconnected PV modules should be studied, such as a 5×2 or 7×2 PV array. Since the performance of the BL interconnection is between the SP and TCT for a fixed PV array, the SP and TCT topology structures are mainly discussed in this section.
Based on the derived model of the differently connected PV array in section II, the output characteristics of the SP or TCT topology structures of a 5×2 or 7×2 PV array can be obtained, as shown in
Fig. 13
. In this simulation, all of the PV modules are assumed to be under the STC except for the shaded PV module. The maximum power for each PV array topology structure of a 5×2 or 7×2 PV array can be calculated when the absorbed irradiance of PVM1 is increased from 0 to 1000W/m
^{2}
. Therefore, the corresponding output power at the GMPP of each topology structure versus the irradiance of the mismatched PV module can be plotted in
Fig. 13
.
Maximum power of each topology in different PV array versus the irradiance of shaded PV module.
In
Fig. 13
, the output power of the SP structure is greater than that of the TCT interconnection when only one PV module is severely shaded in 3×2, 5×2 and 7×2 PV arrays. On the other hand, the output power of the TCT structure is greater than that of the SP structure when the PV module is slightly shaded.
In addition, only a critical point exists on the curves of each PV array structure, which has been marked in
Fig. 13
. If the equivalent irradiance of the shaded PV module can be estimated, this model can assist in switching the structure between the SP and TCT interconnections in a changeableinterconnected PV system.
Furthermore, the critical point in
Fig. 13
moves towards a higher irradiance level with an increasing number of series connected PV modules in a dual series PV array. Thus, in such large scale dual series connected PV systems, the SP interconnection is recommended due to its simplicity and lower cost compared with the TCT interconnection.
 B. The Simulation Analysis of More General Conditions
Considering more general conditions, 5×2 and 5×3 PV arrays are simulated when more than one module or more than one series is shaded. In this paper, in order to study the output power under more general mismatch conditions, the equivalent irradiances of the shaded PV modules are different. One of the irradiances of the two shaded PV modules is 0 to 1000 W/m
^{2}
, and the other one is 0 to 800 W/m
^{2}
. Firstly, a 5×2 PV array in different topology structures is simulated to analyze the relationship between shadow severity and output power.
Fig. 15
indicates the change of the array’s maximum power with the equivalent irradiance of the shaded PV module. The position of the shadow is shown in
Fig. 14
. Similarly, in
Fig. 16
and
Fig. 18
, two PV modules are set to be shaded and the output powers are also analyzed with respect to the shadow severity. The results are shown in
Fig. 17
and
Fig. 19
. Apparently, in a 5×2 PV array, the TCT can generate more power than the other structures when a slight shadow occurs. This is the same as the phenomena in a 3×2 PV array. In addition, a 5×3 PV array is also investigated.
Fig. 20
,
Fig. 22
and
Fig. 24
show the topology structures in a 5×3 PV array when one and two modules are shaded.
Fig. 21
,
Fig. 23
and
Fig. 25
are the results of the corresponding simulations. According to
Fig. 23
, the TCT topology can always generate greater power when two shaded modules are in the same series. This is due to the fact that the shaded PV module has not been bypassed by the bypass diodes, and the GMPP only exists at the right peak of the PV curve.
One shaded module in different topology structures in 5×2 PV array.
Maximum power of each topology in 5×2 PV array versus the irradiance of one shaded PV module.
Two shaded module in different topology structures in 5×2 PV array.
Maximum power of each topology in 5×2 PV array versus the irradiance of two shaded PV modules in the same series.
Two shaded modules in different series in 5×2 PV array.
Maximum power of each topology of two shaded PV modules in different series in 5×2 PV array.
One shaded module in different topology structures in 5×3 PV array.
Maximum power of each topology in 5×3 PV array versus the irradiance of one shaded PV module.
Two shaded module in different topology structures in 5×3 PV array.
Maximum power of each topology in 5×3 PV array versus the irradiance of two shaded PV modules in the same series.
Two shaded modules in different series in 5×3 PV array.
Maximum power of each topology of two shaded PV modules in different series in 5×3 PV array.
By observing the critical points of each topology structure in
Fig. 15
and
Fig. 21
, these critical points move towards the lower irradiance level with increments of the PV strings. When a mismatched PV module is severely shaded and bypassed, the healthy PV module is also bypassed since the operation current of the array is greater than its shortcircuit current in the TCT topology structure of a 5×2 PV array. However, in a 5×3 PV array, the operation current can be shunted through other healthy PV modules in the same row as the shaded module. Therefore, they cannot be bypassed and can generate a certain amount of power.
Comparing
Fig. 17
with
Fig. 19
, it can be seen that the output power of the SP topology structure in
Fig. 19
is much greater than that in
Fig. 17
. If two shaded PV modules exist in the same PV string, the operation voltage of the whole array is significantly reduced due to the mismatched string. As a result, the operation point of the other healthy PV modules will be different from the individual MPP. For the whole array, this phenomenon can result in a heavier power loss. However, if two shaded PV module are in different series, as shown
Fig. 18
, this will cause less power loss than the situation in
Fig. 16
. Similarly, in
Fig. 23
, the GMPP only appears at the right peak of the PV curves, and the GMPP still appears at the left peak in
Fig. 25
.
In general, in an M×N (M≥3, N≥2) PV array, the TCT has the highest efficiency under most conditions. The output power of the SP topology depends on the structure of the PV array and the severity of the shadow. The power generated by the BL is between the SP and the TCT.
V. CONCLUSION
Considering only one shaded PV module in a 3×2 PV array, the advantages and disadvantages of different PV array topology structures were investigated in this paper. These topology structures include the SP, BL and TCT interconnections. Based on the mathematical derivation, the output power at the GMPP or LMPP of each PV array topology structure can be obtained by increasing the irradiance of the shaded PV module from 0 to 1000 W/m
^{2}
. Then, an experimental platform was designed to validate the mathematical derivation and theoretical analysis of different topology structures in a 3×2 PV array. In addition, the SP and TCT topology structures were also studied in a 5×2 or 7×2 PV array model. Finally, the shadowing of more than one module or series were simulated. Some findings can be observed from this paper:
1. The topology structure of the SP works better than that of the TCT or BL when a mismatched PV module is severely shaded in a N×2 PV array. The topology structure of the TCT performs better than that of the SP or BL when it is slightly shaded.
2. With an increment in the number of seriesconnected PV modules in a N×2 PV array, the SP interconnection is more likely to perform better than that of the TCT when only one PV module is shaded.
3. According to an analysis of the output characteristics of different topology structures in 5×2 and 5×3 PV arrays, if two or more shadows are located on different series, the critical points of the SP and TCT are approximate between 500700W/m
^{2}
. If the shadows are in the same series, the left peak becomes the only global MPP when the modules are severely shaded, and the TCT performs significantly better than the SP topology structure.
Further research will be focused on analyzing the behavior of differently structured PV arrays when two or more PV modules are shaded at the same or at different shadow levels. These circumstances are more complex than the currently built PV plants.
Acknowledgements
This work was supported by the Natural Science Foundation of Jiangsu Province (No. BK20131134) and State Key Laboratory of PV Science and Technology Open Foundation (No. 201400035879) ，also supported by Changzhou Key Laboratory of Photovoltaic System Integration and Production Equipment Technology, Jiangsu, China.
BIO
Kun Ding received his MS degree in Materials Processing Engineering and his Ph.D. degree in Electrical Engineering from Hohai University, Jiangsu, China, in 2000 and 2009, respectively. He is presently working as an Associate Professor in the College of Mechanical and Electrical Engineering, Hohai University. His current research interests include power conversion and control for renewable energy interfaces, especially photovoltaics (PV), PV array configurations, PV system designs, and dc–dc converters.
Li Feng was born in Suining, Sichuan, China. She is presently working towards her M.S. degree in the College of Mechanical and Electrical Engineering, Hohai University, Changzhou, China. Her current research interests include fault diagnosis methods and performance evaluations of distributed photovoltaic systems.
SiYu Qin was born in Nanning, Guangxi, China, in 1990. He received his M.S. degree in 2016 from the College of Mechanical and Electrical Engineering, Hohai University, Changzhou, China. His current research interests include the development of PV module test equipment and the analysis of the outdoor performance of PV modules.
Jing Mao received her B.S. degree from the Yangzhou University, Yangzhou, China, in 2008. In 2008 she joined Trina Solar Limited (TSL), and she is presently a Senior Engineer of the State of Key Laboratory of Photovoltaic Science at Changzhou Trina Solar Energy CO., LTD., Changzhou, China. Her current research interests include reliability assessment, failure analysis and generation analysis of PV modules and systems.
JingWei Zhang was born in Nanjing, Jiangsu, China, in 1989. He is presently working towards his Ph.D. degree in the College of Energy and Electrical Engineering, Hohai University, Nanjing, China. His current research interests include the R&D of PV inverters and PV systems, power electronics and drive systems.
Xiang Wang is presently working towards his M.S. degree in the College of Mechanical and Electrical Engineering, Hohai University, Changzhou, China. His current research interests include DSPbased control and gridconnected converters for the photovoltaic (PV) modules in distributed PV systems.
Tao Peng was born in Yancheng, Jiangsu, China, in 1988. He is presently working towards his M.S. degree in the College of Mechanical and Electrical Engineering, Hohai University, Changzhou, China. His current research interests include DSPbased control and gridconnected converters for the photovoltaic (PV) modules in distributed PV systems.
QuanXin Zhai is presently working towards his M.S. degree in the College of Mechanical and Electrical Engineering, Hohai University, Changzhou, China. His current research interests include DSPbased control and gridconnected converters for the photovoltaic (PV) modules in distributed PV systems.
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