To evaluate the performance of the proposed dynamic reconfiguration method, the specifications of the PV module used in this paper are shown in Table 2, and the PV arrays with 9 × 9 TCT and SP interconnections are evaluated under five different mismatch scenarios, as illustrated in Figs. 7 and 8, respectively.

Table 2 PV module specifications
Fig. 7
figure 7

Four mismatch scenarios (a) Scenario 1 (long and wide shading); (b) Scenario 2 (double parts shading) (c) Scenario 3 (long and wide shading with random modules failure); and (d) Scenario 4 (double parts shading with random failure modules)

Fig. 8
figure 8

Shade dispersion of scenario 5 (moving shading with random modules failure)

Scenario 1 (long and wide shading): the first four PV strings of the array are under partial shading with four different irradiations (900 W/m2, 700 W/m2, 400 W/m2 and 200 W/m2).

Scenario 2 (double parts shading): two parts of the PV array (the bottom left corner and the center position) are under partial shading with different sizes.

Scenario 3 (long and wide shading with random module failures): additional module failures are considered based on Scenario 1. Here, two PV modules labeled ‘46’ and ‘72’ are assumed to have failed with no output power generation.

Scenario 4 (double parts shading with random module failures): based on Scenario 2, two additional PV modules labeled as ‘55’ and ‘97’ have failed.

Scenario 5 (moving shading with random module failures): based on Scenario 4, nine different shading cases constitute top to bottom shading conditions as illustrated for different mismatch cases in Fig. 8.

For each mismatch scenario, the mismatch losses (ML) of each string in the PV array are calculated and compared for two evaluated topologies, i.e., the proposed dynamical reconfiguration based flexible topology and the three-inverter fixed topology. In addition, the maximum power generation of each string and the MPP of the whole array for both fixed and flexible topologies are simulated and compared using Matlab/Simulink (version 2018a).

ML is defined as the difference between maximum power output under uniform irradiance conditions ((MPP_{{{text{uni}}}})) and the global maximum power output under mismatch conditions ((MPP_{{{text{MCS}}}})) [29], determined by (19). In the uniform irradiance condition, the sum of received irradiances is equal to the sum of received irradiances under mismatch conditions.

$$ML = MPP_{uni} – MPP_{MCs}$$

(19)

Simulation experiment for TCT-based PV array

For the TCT-based PV array, the arrangements after carrying out the reconfiguration based on the proposed method for mismatch Scenarios 1 to 4 are presented in Figs. 9a–d, respectively. It can be observed from Fig. 9a that, in Scenario 1, the first inverter is the PV strings that are bypassed. It is shown in Table 3 that the total ML under the mismatch Scenarios 1 to 4, are 7.4 (I_{{text{m}}} V_{{text{m}}}), 6.8 (I_{{text{m}}} V_{{text{m}}}), 8.8 (I_{{text{m}}} V_{{text{m}}}) and 8.2 (I_{{text{m}}} V_{{text{m}}}) for the fixed topology. In contrast, the total ML based on the proposed flexible topology is significantly reduced for all the evaluated scenarios to 1.4 (I_{{text{m}}} V_{{text{m}}}), 1.6 (I_{{text{m}}} V_{{text{m}}}), 3.2 (I_{{text{m}}} V_{{text{m}}}) and 1.2 (I_{{text{m}}} V_{{text{m}}}), respectively. The time for optimization of the proposed reconfiguration solution for all these tested scenarios is less than 0.02 s.

Fig. 9
figure 9

Arrangement after dynamical reconfiguration for TCT-based PV array (a) Scenario 1; (b) Scenario 2; (c) Scenario 3; and (d) Scenario 4

Table 3 The ML of PV array of fixed and flexible topologies for the four scenarios

To further validate the effectiveness of the proposed method, the maximum power generation differences of each string for the two evaluated topologies are shown in Fig. 10. The power difference between different PV strings can be calculated as:

$$Power;difference_{stringi} = P2_{stringi} – P1_{stringi}$$

(20)

where (P2_{stringi}) and (P1_{stringi}) are the maximum power outputs of the ith string for the proposed flexible topology and the three-inverter fixed topology, respectively. Also, the MPP performances for fixed and flexible topologies are evaluated through simulations, and the results are presented in Fig. 11. It can be seen that the maximum power generations of the proposed flexible topology under four mismatch scenarios are 4841.3 W, 4861.9 W, 4634.3 W, and 4766.3 W, respectively, which are 509.0 W, 276.1 W, 459.9 W and 415.5 W higher than those of the fixed topology. Further, the MPPs of the TCT-connected PV array for 9 cases in Scenario 5 are shown in Fig. 12. For all cases, a maximum enhancement of 11.0% and an average enhancement of 4.74% of power generation can be obtained by the proposed solution.

Fig. 10
figure 10

Maximum power difference of each string between the two evaluated topologies and the total maximum power difference

Fig. 11
figure 11

MPP performance comparison under the four mismatch scenarios

Fig. 12
figure 12

MPPs of the TCT connected PV array for the 9 cases in Scenario 5

Simulation experiment for SP-based PV array

The proposed method for an SP-based PV array is further evaluated, and the four different mismatch scenarios illustrated in Fig. 7 are also used. The detailed arrangements of PV modules in the PV array after array reconfiguration are illustrated in Fig. 13. Also, the ML of individual strings for the fixed and flexible topologies are calculated and compared in Table 4. In this table, some information is simplified in expression, e.g., the horizontal lines divide the string numbers into three parts, indicating that the PV strings in each part are connected to the same inverter. It can be seen that the total ML of the three-inverter fixed topology for Scenarios 1 to 4 are 8.9 (I_{{text{m}}} V_{{text{m}}}), 18.3 (I_{{text{m}}} V_{{text{m}}}), 12.1 (I_{{text{m}}} V_{{text{m}}}) and 18.1 (I_{{text{m}}} V_{{text{m}}}), respectively. In comparison, the proposed method provides significant improvement in mismatch conditions with the ML reduced to 6.6 (I_{{text{m}}} V_{{text{m}}}), 13.4 (I_{{text{m}}} V_{{text{m}}}), 6.8 (I_{{text{m}}} V_{{text{m}}}) and 13.0 (I_{{text{m}}} V_{{text{m}}}), respectively.

Fig. 13
figure 13

Arrangement after dynamical reconfiguration for SP-based PV array (a) Scenario 1; (b) Scenario 2; (c) Scenario 3; and (d) Scenario 4

Table 4 The ML of PV array of the fixed and flexible topologies for the four scenarios

To further validate the effectiveness of the proposed dynamic array reconfiguration method, the maximum power difference of individual strings in the PV array is calculated and evaluated for both fixed and flexible topologies, as presented in Fig. 14. Also, the MPP of the whole PV array is simulated and examined, and the results are presented in Fig. 15. It shows that the maximum power generation values of the proposed dynamic reconfiguration based flexible topology under the four mismatch scenarios are 4403.1 W, 3879.4 W, 4274.0 W and 3739.9 W, respectively. The results are 99.5 W, 324.7 W, 221.5 W and 370.4 W higher than those of the fixed topology, with the same mismatch scenarios. The time for optimization of the proposed reconfiguration solution for all the tested scenarios is under 3 s.

Fig. 14
figure 14

Maximum power difference of each string between the two evaluated topologies and the total maximum power difference

Fig. 15
figure 15

MPP of the fixed and flexible topologies for SP-based PV array under the four mismatch scenarios

In addition, the MPPs of the SP-based PV array for the 9 cases in Scenario 5 are shown in Fig. 16. It can be seen that the maximum power obtained in the proposed flexible topology is improved by 5.55% on average compared to the fixed topology for all the tested cases.

Fig. 16
figure 16

MPPs of the SP-based PV array for the 9 cases in Scenario 5

Cost–benefit analysis for the proposed dynamic PV reconfiguration method

The required hardware of the proposed dynamic PV reconfiguration method mainly includes the PV monitoring instruments, switching matrix and driving circuits [10]. The prices of the chosen components are shown in Table 5. Note that the selected measuring instruments, i.e., voltage and current sensors, are useful for PV panels with a voltage range of between 0 and 25 V and current range between 0 and 20 A [30]. In the switching matrix, each switch consists of an electromechanical relay and one or several semiconductor devices, e.g., MOSFETs. One MOSFET has to be connected to each relay with a driver required for controlling the MOSFET [10].

Table 5 Price of the required hardware of the dynamic PV reconfiguration method

The number of required components which should be used in the 9 × 9 and 9 × 20 TCT and SP interconnected PV arrays with three inverters are shown in Table 6, as well as the total installation cost. As mentioned above, the size of the switching matrix is only determined by the numbers of the PV strings and inverters. Therefore, for the 9 × 20 PV array, the number of relays, MOSFETs and drivers required will not increase.

Table 6 Components considered for each topology

As mentioned in the introduction, the mismatch losses problem will occur because of module non-uniform aging, module failure and shading. Except for the shading caused by surrounding buildings, the other factors are random and therefore the increment obtained by the dynamic reconfiguration method is hard to estimate. Indeed, the increment of power generation depends on the degree of non-uniform aging, the position of the failed modules, the size and location of shadings as well as the topology of the PV array. For this reason, in this work, for a PV array of 25 years’ life, working time of five hours per day, and an average power reduction of 35% is considered, and the same value of power increment has been considered for each topology and fixed to 5% for simplicity [10]. The economic analysis of the reconfiguration method is calculated as $83 per megawatt hour (MWh), which is the weighted average wholesale price for solar PV-generated electricity in the USA in 2019 [31].

To further assess the cost–benefit for module types with different capacities, an analysis of module capacity of 83 kW, 150 kW and 250 kW is carried out. Considering a PV array lifetime of 25 years, for the 9 × 9 and 9 × 20 TCT and SP interconnected PV arrays, the economic estimates of the proposed reconfiguration method are shown in Tables 7 and 8, respectively. It can be seen that the benefit of the PV array increases with the increase of the module capacity and the number of PV modules in a string. In addition, the deployment of the proposed reconfiguration method can enhance the condition monitoring and analysis of the operating PV modules, and improve the maintenance of the PV systems [32].

Table 7 The economic estimates of the proposed reconfiguration method for TCT-based PV array
Table 8 The economic estimates of the proposed reconfiguration method for SP-based PV array

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