A multi-objective pelican optimization algorithm for dynamic reconfiguration of multi-type rural rooftop PV array

Abstract

Shading and array fault can cause a significant impact on the output power of rural rooftop PV array (RRPVA) and result in power efficiency losses. One of the most popular methods to attenuate the adverse effects of these is reconfiguration in RRPVA. However, the conventional reconfiguration only aims to maximize power output. Hence, this paper proposes a multi-objective pelican optimization algorithm (MOPOA) to improve efficiency and extend the switching life for RRPVA. Comparing the reconfiguration results of the particle swarm algorithm (PSO) and genetic algorithm (GA), the mismatch loss, power loss, performance ratio, and power enhancement percentage of RRPVA under different shading situations are calculated for each of the three algorithms. This paper simulates and analyzes 4×4 symmetric RRPVA and 4×3 asymmetric RRPVA. The results show that MOPOA is 8.4%, 8.5%, 11.2%, 11.5% better than PSO; and 3.8%, 3.5%, 7.6%, 5.6% better than GA in terms of percentage power enhancement (P_en) in 4×4 symmetric RRPVA. In the 4×3 asymmetric RRPVA, the P_en of MOPOA is 5.6%, 9.0%, 10.5%, 9.4% better than PSO, and 4.2%, 2.6%, 3.6%, 2.8% better than GA, respectively. In the case of array fault, the power enhancements were 19.4% and 18.3%, respectively.

Keywords

Dynamic reconfiguration in RRPVA multi-objective pelican optimization algorithm power enhancement multi-type RRPVA

1 Introduction

This section details the current state of research in reconfiguration techniques as well as highlights the contributions made in this paper.

1.1 Present state of the art

With the increase in energy demand, the consumption of coal resources, environmental pollution, and the emergence of sustainability issues, the demand for renewable energy in developing countries is increasing [1, 2]. The advantages of solar cells, such as abundant resources, easy installation, and user-friendly adaptation, make them one of the most popular ways to generate electricity [3, 4]. However, their stochastic nature, dependence on the environment, and nonlinearity make them have many shortcomings in practical applications. In addition, the installation of RRPVA will encounter shadow shading caused by various unavoidable natural factors, such as cloud movement, dust shading, and sometimes shading caused by nearby buildings. Under shadow shading, RRPVA will produce different light signals due to the different light intensities they receive. The difference in light intensity can lead to a multi-peak situation across RRPVA, which results in the stability of RRPVA in terms of power generation [6]. This condition causes the higher current generated by the non-shaded shading module to flow through the shaded module as a load and start consuming current instead of generating a lower current output. It causes severe fluctuations in the power output of RRPVA due to its prolonged presence and also causes hot spots in the shaded modules, which can cause serious physical failures in the shaded modules [7]. As a result, partial shading in the PV array affects the operation and function of the RRPVA, leading to severe losses, failures, and downtime.

1.2 Literature review

The partial shading in the RRPVA is mainly mitigated by using PV array reconfiguration techniques, which are designed to reduce the muti-peak condition and get more fluent PU curves of the RRPVA [8]. Nowadays, there are two reconfigurations: static reconfiguration and dynamic reconfiguration, respectively. Static reconfiguration techniques change the electrical connections and directly modify the physical location of RRPVA. The common static reconfiguration techniques are Soduku [9], Improved Soduku [10], Complementary Sudoku [11], Triple X Soduku [12], Zig-zag [13], Odd Even Configuration (OEC) [14], Lo shu [15], Competence Square (CS) [16], Dominance square (DS) [17], Magic Square (MS) [18] etc. All of the static reconfiguration techniques mentioned above can improve the output efficiency of RRPVA and reduce the multi-peak situation of RRPVA under shading, and it appears to be less reliable for the effects caused by dynamic shading [19, 20].

The reliability issues of static reconfiguration techniques are addressed by dynamic reconfiguration techniques that allow the arrays to operate with higher efficiency during all shading scenarios and effectively eliminate or reduce multiple peak counts from the characteristic curves [21]. Dynamic reconfiguration techniques are mainly mitigated by using advanced swarm intelligence optimization algorithms; such algorithms mainly alter the RRPVA by modifying the electrical connections, enabling the RRPVA to cope effectively with changing shading conditions and increase the output power under shading [22]. Some of the techniques used for dynamic reconfiguration in RRPVA under shading include Genetic Algorithm (GA) [23], Two-step reconfiguration [24], Grey wolf optimization (GWO) [25], Gene Evaluation Algorithm (GEA) [26], Dragonfly algorithm (DA) [27], Harris hawks optimizer (HHO) [28], Particle swarm optimization (PSO) [29], Bald eagle search algorithm (BESA) [30], Divide and Conquer Q-Learning (DCQL) [31] etc. These dynamic reconfiguration techniques can reduce losses under various partial shading conditions and improve output efficiency. However, some things could be improved in the performance of these algorithms for dynamic reconfiguration, and these algorithms only consider a single goal: maximize output power and cannot make further judgments on the reconfiguration situation when facing complex problems.

Before proceeding, this paper provides a brief review of some reconfiguration techniques, elaborating on both static and dynamic reconfiguration methods, and summarizing them in Tables 1 and 2.

Table 1
Reporting methods for PV array’s static reconfiguration

Technique Objective function Advantages Disadvantages

Soduku Maximum output power Simple and easy to implement Only for 9×9 PV arrays

Improved Soduku Maximum output power Simple and easy to implement Campaigns are random and not informative

Complementary Sudoku Maximum output power Improved the shortcomings of Soduku Arrays cannot be expanded

Triple X Soduku Maximum output power Improved the shortcomings of Soduku Only for 9×9 PV arrays

Zig-zag Maximum output power Simple and easy to implement Inability to fully disperse shading

Odd Even Cconfiguration Maximum output power Simple and easy to implement It’s easy to get caught in a dead end.

Lo shu Maximum output power Effective in dispersing shading Calculation complexity

Competence Square Maximum output power Reconfiguration works well Not applicable to real-life situations

Dominance Square Maximum output power Simpler calculations Difficult to realize

Magic Square Maximum output power Reduced mismatch losses Method extensions are more complex

circular array data structure [42] Maximum output power Excellent reconfiguration results Can’t cope with fast-moving shading

Henon map [43] Maximum output power Effectively enhances maximum power No reduction in reconfiguration loss

Chaos map [44] Maximum output power Reduced multiple peaks in the PV curve Can’t deal with complex shading

DIAR [45] Maximum output power Enhanced output power No reduction in reconfiguration loss

One-Time Electrical Reconfiguration [46] Maximum output power Reliable operation and easy implementation Inability to cope with variable shading situations

Ancient Chinese magic square [47] Maximum output power Effective in dispersing shading Can’t deal with complex shading

Two-Step Module Placement Approach [48] Maximum output power Reduced reconfiguration loss Cannot always achieve optimal reconfiguration configuration

Technique	Objective function	Advantages	Disadvantages
Soduku	Maximum output power	Simple and easy to implement	Only for 9×9 PV arrays
Improved Soduku	Maximum output power	Simple and easy to implement	Campaigns are random and not informative
Complementary Sudoku	Maximum output power	Improved the shortcomings of Soduku	Arrays cannot be expanded
Triple X Soduku	Maximum output power	Improved the shortcomings of Soduku	Only for 9×9 PV arrays
Zig-zag	Maximum output power	Simple and easy to implement	Inability to fully disperse shading
Odd Even Cconfiguration	Maximum output power	Simple and easy to implement	It’s easy to get caught in a dead end.
Lo shu	Maximum output power	Effective in dispersing shading	Calculation complexity
Competence Square	Maximum output power	Reconfiguration works well	Not applicable to real-life situations
Dominance Square	Maximum output power	Simpler calculations	Difficult to realize
Magic Square	Maximum output power	Reduced mismatch losses	Method extensions are more complex
circular array data structure [42]	Maximum output power	Excellent reconfiguration results	Can’t cope with fast-moving shading
Henon map [43]	Maximum output power	Effectively enhances maximum power	No reduction in reconfiguration loss
Chaos map [44]	Maximum output power	Reduced multiple peaks in the PV curve	Can’t deal with complex shading
DIAR [45]	Maximum output power	Enhanced output power	No reduction in reconfiguration loss
One-Time Electrical Reconfiguration [46]	Maximum output power	Reliable operation and easy implementation	Inability to cope with variable shading situations
Ancient Chinese magic square [47]	Maximum output power	Effective in dispersing shading	Can’t deal with complex shading
Two-Step Module Placement Approach [48]	Maximum output power	Reduced reconfiguration loss	Cannot always achieve optimal reconfiguration configuration

Table 2

Reporting methods for PV array’s dynamic reconfiguration

Technique	Objective function	Advantages	Disadvantages
Genetic Algorithm	Maximum output power	Algorithms are simpler	Inability to fully disperse complex shading
Two-step reconfiguration	Maximum output power	Reconfiguration works well	Algorithms are more complex to calculate
Grey Wolf Optimization	Maximum output power	Effective reduction of multiple peaks in the PV curve	Inability to fully disperse complex shading
Gene Evaluation Algorithm	Maximum output power	Simple and easy to implement	No consideration of line losses
Dragonfly algorithm	Maximum output power	High reliability	More sensors required
Harris hawks optimizer	Maximum output power	Able to stabilize power output	Requires a lot of calculations
Particle swarm optimization	Maximum output power	simple calculation	Easy to fall into local extremes
Bald eagle search algorithm	Maximum output power	Reconfiguration works well	Not applicable to real-life situations
DCQL	Maximum output power	Reconfiguration works excellent	Requires excessive wire connections
Pareto [32]	Maximize power output and minimize the switching number	The operation life of the switching devices can be lengthened	computationally complex
Swarm reinforcement Learning [33]	Maximize power output and power fluctuation	Reduce the optimization difficulty of OMAR, real-time generation schedule	Too much complexity for real life situations
Multi-agent negotiation algorithm [34]	Total profit with hydrogen selling profit, regulation cost	Increase the total profit, achieve an efficient and distributed optimization	Considering the aftermarket leads to excessive computation
Black widow PV array reconfiguration [49]	Maximum output power	Reduced number of switches	No reduction in switching action losses
Adaptive evolutionary jellyfish search algorithm [50]	Maximum output power	Reconfiguration works excellent	Overcalculated

Referring to Tables 1 and 2, it can be seen that most of the reconfiguration techniques mainly focus on the maximum power output of a single objective, which means these reconfiguration techniques do not take into account multiple factors, which can result in excessive switching losses as well as failing to achieve the optimal reconfiguration requirements. The few reconfiguration techniques for muti-objective also do not simultaneously consider minimizing the number of electrical switching actions while achieving optimal reconfiguration criteria. Therefore, to ensure the stability of PV power generation, to solve the multi-peak output power condition during PV power generation, and to reduce unnecessary switching losses as well as to achieve the optimal reconfiguration requirements to increase the power generation efficiency, a multi-objective pelican optimization algorithm (MOPOA) is proposed in this paper to perform dynamic reconfiguration in RRPVA.

1.3 Innovative contribution

The major contributions of the proposed method are emphasized as follows.

In this paper, a novel technique for PV array reconfiguration is proposed. The objective of this technique is to efficiently mitigate shading effects and address contingencies resulting from PV array faults, thereby minimizing power losses due to mismatches in PV arrays.

The algorithm first considers the minimization of the deviation between the row current of RRPVA. The objective function is to minimize the switching losses and maximize the output power of RRPVA under any shading scenarios and fault conditions to construct a three-level multi-objective reconfiguration in the RRPVA model. The switching losses are minimized as much as possible with the maximum output power and the minimum deviation of the row current.

The MOPOA reconfiguration method proposed in this paper is scalable and can adapt to photovoltaic arrays of any size. It is a universal and widely applicable technology. Its effectiveness has been verified through simulation experiments in current research work.

The MOPOA proposed in this paper provides an optimal connectivity matrix for PV array reconfiguration, which establishes the superiority and robustness of the switching scheme by effectively adjusting the weight values of the proposed objective function. Quality reconfiguration results can be obtained using MOPOA in a variety of situations. MOPOA also applies to symmetric and asymmetric PV arrays and can still be used as the PV array expands.

2 Rural rooftop PV array system components and modelling

First of all, introduce the rural rooftop PV array system, whose substance is shown in Fig. 1.

Fig. 1

Physical diagram of RRPVA system.

The process from the PV cell to RRPVA, see Fig. 2(a), (b), (c) and (d), leads to the following current expressions [35, 36]:

Fig. 2

(a) Circuit representation of a PV cell, (b) individual cell, (c) cell forming a PV module (d) module forming a PV array.

$\begin{matrix} I_{PV} = I_{irr} - I_{D} {exp [\frac{q}{nkT} (V_{PV} + I_{PV} R_{s})] - 1} \\ - \frac{V_{PV} + I_{PV} R_{s}}{R_{p}} \end{matrix}$ (1) where I_irr denotes photogenerated current, I_D denotes equivalent diode reverse saturation current, q denotes unit charge, n denotes diode characteristic factor, k denotes Boltzmann’s constant, T denotes temperature, R_p denotes equivalent parallel resistance, R_s denotes equivalent series resistance, V_PV and I_PV are the output voltage and output current of the PU module, respectively. Where, resistance R_p is approximately infinite and resistance R_s is very small negligible [37].

3 Conventional configuration and dynamic reconfiguration techniques for RRPVA

To date, the known conventional configurations of RRPVA include series-parallel (SP), bridge-linked (BL), honeycomb (HC), and total-cross-tied (TCT) architectures, which differ in terms of wiring [38]. The final configuration is then different. This paper illustrates the 4×4 symmetric RRPVA as an example.

Figure 3(a) indicates a SP architecture in which modules are connected in series and parallel connections, and Fig. 3(b) and (c) indicate BL and HC architectures, respectively, in which additional wires are connected to the junction points of the modules in a bridged manner. A schematic architecture for a TCT configuration is indicated in Fig. 3(d), where the wires are connected to the junction points of the modules. This paper selects the TCT architecture.

Fig. 3

4×4 RRPVA conventional configuration.

The total current and voltage of the RRPVA are calculated as follows [39]. $I_{out} = \sum_{j = 1}^{n} I_{ij}, i = 1, 2 \dots \dots m, j = 1, 2 \dots \dots n$ (2) $V_{array} = \sum_{i = 1}^{m} V_{max i}$ (3)

Among them, I_out is the total current generated by the RRPVA, V_array is the total voltage across the terminals of the RRPVA, and V_maxi is the voltage at row i.

4 Multi-objective pelican optimization algorithm

The pelican optimization algorithm (POA) simulates the natural behavior of pelicans during hunting. In POA, the behavior and strategies of pelicans during attack and hunting are simulated to update the candidate solutions. This hunting process is divided into two phases: the exploration phase and the development phase.

The mathematical description of the initialization of the pelican population is as follows [40]: $\begin{matrix} x_{ij} = l_{j} + rand (u_{j} - l_{j}), i = 1, 2 \dots \dots N, \\ j = 1, 2 \dots \dots m \end{matrix}$ (4)

Among them, x_ij is the j _th dimension position of the i _th pelican; N is the population size of pelicans; m is the dimension of the solution problem; rand is a random number in the range of [0,1); u_j and l_j are the upper and lower bounds of the j _th dimension of the solved problem.

In POA, the PV array can be represented by the following population matrix 40]:

$X = {[\begin{matrix} X_{1} \\ ⋮ \\ X_{i} \\ ⋮ \\ X_{N} \end{matrix}]}_{N \times m} = {[\begin{matrix} X_{1, 1} & \dots & X_{1, j} & \dots & X_{1, m} \\ ⋮ & ⋱ & ⋮ & ⋱ & ⋮ \\ X_{i, 1} & \dots & X_{i, j} & \dots & X_{i, m} \\ ⋮ & ⋱ & ⋮ & ⋱ & ⋮ \\ X_{N, 1} & \dots & X_{N, j} & \dots & X_{N, m} \end{matrix}]}_{N \times m}$ (5) where X is the matrix of the PV array, X_i is the i_th position of the PV array.

In POA, the objective function of solving the problem can be used to calculate the objective function value; since it is solved in a single column, the objective function value of the problem can be expressed as a vector of objective function values [40]:

$f = {[\begin{matrix} f_{1} \\ ⋮ \\ f_{i} \\ ⋮ \\ f_{N} \end{matrix}]}_{N \times 1} = {[\begin{matrix} f (X_{1}) \\ ⋮ \\ f (X_{i}) \\ ⋮ \\ f (X_{N}) \end{matrix}]}_{N \times 1}$ (6) where f is the vector of objective functions; f_i is the objective function value of the i _th PV array.

Phase I: Exploration phase

In the first stage, the pelican approaching prey strategy is modeled so that the POA algorithm can scan the search space and thus exploit the exploration capability of the POA algorithm in different regions of the search space. One point to be emphasized in the POA algorithm is that the location of the prey is randomly generated in the search space. The mathematical modeling of the above concept and the strategy of approximating the prey is as follows[40]:

$x_{i, j}^{P_{1}} = {\begin{matrix} x_{i, j} + rand (P_{j} - S \times x_{i, j}), f_{P} < f_{i} \\ x_{i, j} + rand (x_{i, j} - P_{j}), else \end{matrix}$ (7) where x is the position of the j _th dimension of the i _th PV array based on the first stage update; rand is a random number in the range of [0,1]; S is a random integer of 1 or 2; P_j for the position of the j _th dimension of the prey; f_P is the objective function value of the prey.

In the POA algorithm, the new position of the PV array is accepted if the objective function value is improved at that position. In this type of update, also known as an effective update, the algorithm cannot move to non-optimal regions. This process can be described by the following equation [40]: $x_{i} = {\begin{matrix} x_{i}^{P_{1}}, f_{i}^{P_{1}} < f_{i} \\ x_{i}, else \end{matrix}$ (8)

Among them: $x_{i}^{P_{1}}$ is the new location for the i _th PV array; $f_{i}^{P_{1}}$ is the value of the objective function based on the new position of the i _th PV array after the first stage update.

Phase II: Development phase

In the second stage, the development phase process is modeled, which can make the POA algorithm converge to a better location in the hunting region, which increases the local search capability and development capability of the POA algorithm, mathematically modeled as [40]: $x_{i, j}^{P_{2}} = x_{i, j} + R (1 - \frac{t}{T}) \times (2 \times rand - 1) \times x_{i, j}$ (9) where $x_{i, j}^{P_{2}}$ is the position of the j _th dimension of the i _th PV array based on the second stage update; rand is a random number in the range of [0,1]; R is a random integer of 0 or 2; t is the number of current iterations; T is the maximum number of iterations.

The POA algorithm implements solving single-objective optimization problems. At the same time, for optimizing multi-objective tasks, this paper proposes an alternative version of POA, called the Multi-Objective Pelican Optimization Algorithm (MOPOA), which inherits the main mechanisms of POA. MOPOA is concerned with the problem of taking the best solution in the presence of trade-offs between more conflicting objectives.

The objective of reconfiguration in the RRPVA optimization problem is to maximize the power extracted from the available RRPVA, minimize the deviation between row current levels, and minimize the losses in the switches to achieve smooth PU characteristics and increase the system’s operational reliability. The task can be modeled as a multi-objective optimization problem with three conflicting objective functions. The first objective proposed represents the maximum sum of the power generated in the RRPVA, the second objective is the absolute difference between the top and minimum row current values, and the third objective is the minimization of the switching losses in which each PV array can only exchange its electrical switching states with another PV array in the same column. $F (Obj 1) = sum (P) = \sum_{i = 1}^{n} I_{i} \times V_{i}, n \in Z$ (10) $F (Obj 2) = | I_{max} - I_{min} |$ (11) $F (Obj 3) = \sum_{i = 1}^{n_{row}} \sum_{j = 1}^{n_{col}} sign (| x_{ij}^{new} - x_{ij}^{0} |)$ (12)

For minimizing switching losses, the reconfiguration in RRPVA also needs to consider electrical switching constraints, as follows:

$s . t . {\begin{matrix} x_{ij}^{new} \in {1, 2, \dots \dots, n_{row}}, i = 1, 2, \dots \dots, n_{row}; j = 1, 2, \dots \dots, n_{col} \\ ⋃_{i = 1}^{n_{row}} x_{ij}^{new} = {1, 2, \dots \dots, n_{row}}, j = 1, 2, \dots \dots, n_{col} \end{matrix}$ (13) where I_i and V_i are the current and voltage of RRPVA in row i, respectively. The symbol I is the row current vector, I_max and I_min are the maximum and minimum values of the current in the vector I where sign denotes the sign function, where sign(x) = 0 if x = 0 and sign(x) = 1 if x > 0; $x_{ij}^{new}$ and $x_{ij}^{0}$ denote the new and initial electrical switching states of RRPVA in row i and column j, respectively.

The specific implementation steps of optimization are as follows:

Initialization phase. Building a PV array matrix (determine temperature consistency: 25°C, input light irradiation intensity).

Calculate the sum of the row irradiance intensity (the row irradiance intensity is proportional to the row current) and perform array optimization swapping in each column to minimize the difference between the sum of the row irradiance (i.e., minimize row current deviation).

Create an initial group. Create an initial population, arrange the sequences randomly, and satisfy the switching constraints.

The combination of non-dimension properties of maximizing output power, minimizing row current, and minimizing switching losses are achieved using normalization. Calculate the fitness value for each member.

Perform MOPOA optimization. Randomly select members from the generated population as prey (according to Equation (7)).

According to Equations (10), (11), (12) to complete optimal value selection. Repeat steps 2, 3, and 4 until the array is optimized.

Reconfiguration of the RRPVA. The optimized solution uses the switch matrix to change the electrical connection mode of the PV array to complete dynamic reconfiguration.

The MOPOA pseudocode is shown in Table 3.

Table 3

Pseudocode of MOPOA

Algorithm: Pseudocode of Reconfiguration Based on MOPOA
Inputs: The irradiance levels, V_maxi, and I_i. Population size N, length of archive and number of iterations T.
Outputs: The optimal structure of the switching matrix Initialize the pelican matrix. Compute the initial values of the objective functions 10, 11 and 12 and generate the archive.
Find the non-dominated solutions and initialized the archive with them.
t = 1;
while (termination criteria is not met) do
for (each pelican) do Update the matrix based on the algorithm control equations (4 to 9).
end for
Update pelican’s position
Calculate the objective functions 10, 11 and 12.
Setup Updates.
end while
Select Optimal reconfiguration that corresponded for min I_out values of the second objective function 11.
Select Optimal reconfiguration that corresponded for min switch losses values of the third objective function 12.

The formula for calculating the switching losses is shown in Equation (14). $S_{L} = \frac{x_{action}}{x_{total}} \times 100$ (14) where S_L is the switching losses, x_action is the number of switch operations, x_total is the total action number of the electrical switch, which is taken as 5000 times in this paper. The specific parameters of the electrical switch are shown in Table 4. (the parameters follow the GB10963.1 and IEC60898-1 standards).

Table 4

Specific parameters of electrical switch: DZ47-63

DZ47-63
Rated current	63A
Rated voltage	220/400 V AC
Impulse withstand voltage	4KV
Mechanical life	About 5000 times
Electrical life	About 2000 times
Environmental temperature	–5°C∼40°C

5 Comparison of reconfiguration techniques in different conditions

Different shading types are set in the simulation experiment platform, and the PSO, GA, and MOPOA are imported for dynamic reconfiguration for 4×4 symmetric RRPVA and 4×3 asymmetric RRPVA, respectively [41]. It is important to note that the performance of the GA, PSO and MOPOA methods are influenced by algorithm parameters, as they follow random initialization and evolve optimal solutions. To achieve optimum performance, three parameters for GA (i.e., population size, probability of crossover, and mutation rate), two parameters for PSO (i.e., social and cognitive constants, c1 and c2, and inertia weight, w) and two parameters for MOPOA (i.e., population size and iterations) are adjusted, and their values are detailed in Table 5. All methods are allowed to run for a maximum of 100 iterations.

Table 5
Parameters for PSO, GA and MOPOA

PSO GA MOPOA

w 0.9 Population size 25 Population size 25

c1 1.2 Mutation 0.3 Maximum number

c2 1.5 Cross over 0.7 of iterations 100

PSO	GA	MOPOA
w	0.9	Population size	25	Population size 25
c1	1.2	Mutation	0.3	Maximum number
c2	1.5	Cross over	0.7	of iterations 100

The shading type settings are shown in Fig. 4, the light intensity settings for RRPVA under different shading conditions are divided into four levels: 1000 W/m², 800 W/m², 600 W/m², 400 W/m². This paper selects the 4×4 RRPVA and 4×3 RRPVA for research target.

Fig. 4

The optimal PV reconfiguration schemes obtained by different algorithms on the 4×4 RRPVA under PSC

5.1 Dynamic reconfiguration of 4×4 symmetric RRPVA

The first simulation experiment is conducted for 4×4 symmetric RRPVA.

As can be seen from the Fig. 4, the dynamic reconfiguration using different algorithms yields completely different results. The row current for the RRPVA after dynamic reconfiguration using different algorithms is calculated, and the row current can be theoretically calculated as follows [39]. $I_{Rw 1} = A_{11} I_{11} + A_{12} I_{12} + A_{13} I_{13} + A_{14} I_{14}$ (15)

The current generated by each module under full irradiation conditions can be specified as I_M.

From Table 6, it can be seen that the row current deviation value is the smallest after reconfiguration using MOPOA. The row current deviation can directly reflect the smoothness of the PV array output to a certain extent. In this case, MOPOA has the most minor row current deviation compared to PSO and GA, and it can be roughly known that the fluctuation of PU output curves of the PV array is minimized after using MOPOA. The switching losses of 4×4 symmetric RRPVA are shown in Table 7, and it can be found that the reconfiguration based on MOPOA has fewer switching times and minimal switching losses.

Table 6

4×4 symmetric RRPVA row current deviation values

	Before reconfiguration	PSO	GA	MOPOA
Trapezoidal shading	1.8I_M	I_M	0.8I_M	0.4I_M
Triangular shading	1.2I_M	1.2I_M	I_M	0.2I_M
Long shading	1.6I_M	1.2I_M	1.2I_M	0.2I_M
Decentralized shading	1.2I_M	I_M	1.2I_M	0.2I_M

Table 7

Switching losses of 4×4 symmetric RRPVA for different conditions

Algorithm Type	Trapezoidal shading	Triangular shading	Long shading	Decentralized shading
PSO	0.2%	0.06%	0.2%	0.18%
GA	0.24%	0.16%	0.22%	0.22%
MOPOA	0.16%	0.1%	0.16%	0.1%

In the case of power multi-peak output, it will make the power generation of RRPVA fluctuate and not stable work, thus leading to the whole RRPVA cannot achieve the maximum power generation, resulting in the waste of resources as well as some economic losses. Therefore, it is necessary for RRPVA to reconfigure to improve its inferior state. Firstly, the PU output curves of 4×4 RRPVA before reconfiguration, trapezoidal shading, triangular shading, long shading, and decentralized shading are illustrated and reconfigured using PSO, GA, and MOPOA, respectively, see Fig. 5.

Fig. 5

PU output curves before/after reconfiguration of 4×4 symmetric RRPVA

The PU output curves of 4×4 symmetric RRPVA before/after dynamic reconfiguration are shown in the Fig. 5, and it can be seen that PU output curves are different under various shading conditions. Figure 5(a) shows trapezoidal shading; Fig. 5(b) shows triangular shading; Fig. 5(c) shows long shading; and Fig. 5(d) shows decentralized shading. As shown in Fig. 5, the PU output curves of 4×4 symmetric RRPVA after dynamic reconfiguration using GA becomes slightly more stable than that after PV reconfiguration using PSO, and its multi-peak situation is somewhat improved. However, it still cannot meet the output requirements of RRPVA. The overall PU output curves of RRPVA were significantly enhanced after dynamic reconfiguration of 4×4 symmetric RRPVA using MOPOA. Under different shading conditions, the PU output curves are smooth and flat. Comparing with before reconfiguration, after PSO reconfiguration, and after GA reconfiguration, the multi-peak situation of the PU output curves of 4×4 symmetric RRPVA are significantly improved at this time, and the speed of reaching the maximum peak is also faster, which meets the stable output requirements of RRPVA.

In this paper, the data of 4×4 symmetric RRPVA will be illustrated in the following aspects, as shown in Table 8. It contains four evaluation metrics for RRPVA, which show the performance comparison between RRPVA before and after reconfiguration in various aspects.

Table 8

Summary of evaluation criteria

Evaluation Criteria	Description of evaluation criteria	Evaluation criteria formula
Mismatch loss	Maximum power deviation under standard operating conditions and shaded shading	P_mistatch = P_{standardcondition} - P_psc
Power loss	Represents the extent of power loss due to PSC	$P_{loss} (%) = \frac{P_{s tan dardcondition} - P_{psc}}{P_{s tan dardcondition}} \times 100 %$
Performance ratio	Reflect the effect of the power loss under PSC	$PR = \frac{P_{psc}}{P_{s tan dardcondition}}$
Power enhancement percentage (P_en)	The ratio of difference between P_re and P_psc under PSC	$P_{enhancement} (%) = \frac{P_{re} - P_{psc}}{P_{re}} \times 100 %$

The maximum output power of 4×4 symmetric RRPVA before reconfiguration under standard operating conditions is 2939 W. The calculation of each evaluation criterion for 4×4 symmetric RRPVA and the calculation results of each evaluation criterion are shown in Table 9.

Table 9

4×4 symmetric RRPVA calculation results of each evaluation criteria

PV array status	P _max	P _mismatch	P _loss	PR	P _en
Before reconfiguration	A:1797W	1142W	38.90%	0.611
	B:2180W	759W	25.80%	0.742
	C:1928W	1011W	34.40%	0.656
	D:1933W	1006W	34.20%	0.658
After PSO reconfiguration	A:2081W	858W	29.10%	0.736	13.60%
	B:2224W	715W	24.30%	0.757	2.00%
	C:2085W	854W	29.10%	0.709	7.50%
	D:2064W	875W	29.80%	0.694	6.30%
After GA reconfiguration	A:2197W	742W	25.20%	0.748	18.20%
	B:2345W	594W	20.20%	0.799	7.00%
	C:2168W	771W	26.20%	0.738	11.10%
	D:2113W	806W	27.40%	0.719	7.80%
After MOPOA reconfiguration	A:2304W	635W	21.60%	0.784	22.00%
	B:2436W	503W	17.10%	0.829	10.50%
	C:2372W	567W	19.20%	0.807	18.70%
	D:2252W	687W	23.40%	0.766	14.10%

Case A, B, C, and D represent the conditions when 4×4 symmetric RRPVA with trapezoidal shading, triangular shading, long shading, and decentralized shading, respectively. From Table 10, it is known that the output power of RRPVA is enhanced using different algorithms, and the power enhancement after reconfiguration using MOPOA is better than GA as well as PSO. Under different shading conditions and using different algorithms, MOPOA has the best dynamic reconfiguration performance. In all the cases mentioned in this section, after reconfiguration using MOPOA, it has the smallest mismatch loss, the smallest P_loss, the largest PR, and the largest P_en.

Table 10

Difference in the power enhancement percentage of different algorithms

Algorithms	Shading type	Difference in P_en
MOPOA-PSO	Trapezoidal shading	8.4%
	Triangular shading	8.5%
	Long shading	11.2%
	Decentralized shading	8.9%
MOPOA-GA	Trapezoidal shading	3.8%
	Triangular shading	3.5%
	Long shading	7.6%
	Decentralized shading	5.6%

Where the power enhancement percentage of 4×4 symmetric RRPVA also difference. As shown in Table 10.

From Table 10, it is clear that the dynamic reconfiguration using MOPOA is more effective in terms of power improvement. In four different shading cases, the power enhancement percentage improves by 8.4%, 8.5%, 11.2%, and 8.9%, respectively, compared to PSO after reconfiguration. Compared to GA in the power enhancement percentage improved by 3.8%, 3.5%, 7.6%, and 5.6%, respectively.

5.2 Dynamic reconfiguration of 4×3 asymmetric RRPVA

For the 4×3 asymmetric RRPVA with shading settings, the RRPVA under its shading case is shown in Fig. 6.

Fig. 6

The optimal PV reconfiguration schemes obtained by different algorithms on the 4×3 RRPVA under PSC

The 4×3 asymmetric RRPVA was analyzed, and the corresponding shadow distribution after dynamic reconfiguration of 4×3 asymmetric RRPVA under different algorithms was obtained after selecting the typical shadows for analysis. The row current deviations of the 4×3 asymmetric RRPVA before/after reconfiguration using different algorithms are calculated using the row current calculation formula, and the obtained results are shown in Table 10. From Table 11, it can be seen that the row current deviation after reconfiguration using MOPOA is the smallest and is 0. It is clear that MOPOA can effectively minimize the row current deviation compared to PSO and GA.

Table 11

4×3 asymmetric RRPVA row current deviation values

	Before reconfiguration	PSO	GA	MOPOA
Full shading	1.2I_M	0.4I_M	0.4I_M	0
Triangular shading	1.2I_M	0.8I_M	0.4I_M	0
Long shading	1.2I_M	0.8I_M	0.4I_M	0
Decentralized shading	0.8I_M	0.8I_M	0.4I_M	0

Table 12 shows that in the case of long shading and decentralized shading, the number of switching based on MOPOA reconfiguration is less, and the switching losses are the smallest. It is worth mentioning that the switching losses are not the lowest in the remaining cases because the reconfiguration first needs to ensure the maximum power output, and then the switching losses are minimized.

Table 12

Switching losses for different cases of 4×3 asymmetric RRPVA

Algorithm Type	Full shading	Triangular shading	Long shading	Decentralized shading
PSO	0.10%	0.10%	0.04%	0.06%
GA	0.06%	0.08%	0.04%	0.06%
MOPOA	0.12%	0.12%	0.04%	0.04%

Next, the PU output curves of 4×3 asymmetric RRPVA before reconfiguration, full shading, triangular shading, long shading and decentralized shading are illustrated and reconfigured using PSO, GA and MOPOA, respectively, as shown in Fig. 10. Figure 10(a) shows full shading with severe output power loss of RRPVA; Figure 10(b) shows triangular shading; Fig. 10(c) shows long shading; and Fig. 10(d) shows decentralized shading.

From Fig. 7, it can be seen that under different shading conditions, at this time, after using MOPOA dynamically, the PU output curves of 4×3 asymmetric RRPVA are smooth and flat compared with the PU output curves before reconfiguration, PSO, and GA reconfiguration, and all of them only have a single peak.

Fig. 7

PU output curves before/after reconfiguration of 4×3 asymmetric RRPVA

The PU output curves of 4×3 asymmetric RRPVA before/after dynamic reconfiguration are shown in the Fig. 7. Figure 7(a) shows full shading; Fig. 7(b) shows triangular shading; Fig. 7(c) shows long shading; and Fig. 7(d) shows decentralized shading. As can be seen in Fig. 7, the PU output curves are the smoothest after the reconfiguration using MOPOA, which result in a more stable RRPVA output and also have the highest output power.

The evaluation criteria of the 4×3 asymmetric RRPVA were calculated and the results are shown in Table 12. A, B, C, and D represent the cases when the 4×3 asymmetric RRPVA is in shading mask after PSO reconfiguration, after GA reconfiguration, and after MOPOA reconfiguration, respectively. From Table 13, it can be learned that after dynamic reconfiguration of 4×3 asymmetric RRPVA, the power is improved, and the PR is increased, representing that the RRPVA power output situation becomes better than before reconfiguration, in which the dynamic reconfiguration performance of MOPOA is the best. At the same time, it has the smallest mismatch loss, the smallest P_loss, the largest PR, and the largestP_en.

Table 13

Calculation results of each evaluation criterion for 4×3 asymmetric RRPVA

PV Array Status	P_max	P_mismatch	P_loss	PR	P_en
Array without shading	2204W
Full shading	A:1015W	1189W	53.9%	0.461
	B:1228W	976W	44.3%	0.557	17.3%
	C:1249W	955W	43.3%	0.567	18.7%
	D:1316W	888W	40.3%	0.597	22.9%
Triangular shading	A:1446W	758W	34.4%	0.656
	B:1586W	618W	28.0%	0.720	8.8%
	C:1705W	499W	22.6%	0.774	15.2%
	D:1760W	444W	20.1%	0.799	17.8%
Long shading	A:1417W	787W	35.7%	0.643
	B:1557W	647W	29.4%	0.707	9.0%
	C:1684W	520W	23.6%	0.764	15.9%
	D:1760W	444W	20.1%	0.799	19.5%
Decentralized shading	A:1557W	647W	29.4%	0.706
	B:1591W	613W	27.8%	0.722	2.1%
	C:1705W	499W	22.6%	0.774	8.7%
	D:1760W	444W	20.1%	0.799	11.5%

For more intuitive comparison, the power enhancement percentage of 4×3 asymmetric RRPVA is illustrated for the difference with Table 14.

Table 14

Difference in the power enhancement percentage of different algorithms

Comparison of different algorithms	Shading type	Difference in P_en
MOPOA-PSO	Full shading	5.6%
	Triangular shading	9.0%
	Long shading	10.5%
	Decentralized shading	9.4%
MOPOA-GA	Full shading	4.2%
	Triangular shading	2.6%
	Long shading	3.6%
	Decentralized shading	2.8%

From Table 14, it is clear that the dynamic reconfiguration using MOPOA is more effective in terms of power improvement. For the four different shading cases, the power enhancement percentage improves by 5.6%, 9.0%, 10.5%, and 9.4%, respectively, compared to PSO after reconfiguration. Compared to GA in the power enhancement percentage improved by 4.2%, 2.6%, 3.6%, and 2.8%, respectively.

5.3 Dynamic Reconfiguration in RRPVA under fault condition

In actual life, it is not only shading that affects the power generation of RRPVA but also the possibility of RRPVA failure, so need to reconfigure to ensure that the impact caused by RRPVA fault is minimized. The PV array in black indicates the PV array fault, 4×4 symmetric RRPVA fault case before and after reconfiguration condition is shown in Fig. 8.

Fig. 8

4×4 symmetric RRPVA reconfiguration in fault condition

When the switching loss is 0.1%, the faulty PV array is reconfigured with uniform distribution in each row after reconfiguring 4×4 symmetric RRPVA.

$I_{Rw 1} = I_{Rw 2} = 4 (\frac{1000}{1000}) I_{M} {= 4 I}_{M}$ (16) $I_{Rw 3} {= 3 I}_{M} + \frac{0}{1000} I_{M} {= 3 I}_{M}$ (17) $I_{Rw 4} = 3 (\frac{0}{1000}) I_{M} {+ I}_{M} {= I}_{M}$ (18)

The row current deviation at this point is 3I_M. The calculated row current deviation after reconfiguration is 0 and the results are shown in Equation (19).

$I_{Rw 1} {= I}_{Rw 2} {= I}_{Rw 3} {= I}_{Rw 4} {= 3 I}_{M} + \frac{0}{1000} I_{M} {= 3 I}_{M}$ (19)

Obviously, the row currents are the same at this point, and there is no row current deviation after reconfiguration using MOPOA.

Reconfiguration in 4×3 asymmetric RRPVA under fault condition using MOPOA, when the switching losses are 0.1%, and the reconfiguration forward current deviation calculation and results are shown in Equations (20)–(22). $I_{Rw 1} {= I}_{Rw 2} = 3 (\frac{1000}{1000}) I_{M} {= 3 I}_{M}$ (20) $I_{Rw 3} {= 2 I}_{M} + \frac{0}{1000} I_{M} {= 2 I}_{M}$ (21) $I_{Rw 4} = 3 (\frac{0}{1000}) I_{M} = 0$ (22)

The row current deviation at this point is 3I_M. The calculated row current deviation after reconfiguration is 0 and the results are shown in Equation (23). $I_{Rw 1} {= I}_{Rw 2} {= I}_{Rw 3} {= I}_{Rw 4} {= 2 I}_{M} + \frac{0}{1000} I_{M} {= 2 I}_{M}$ (23)

The corresponding model was built using the simulation software formation platform for a series of simulation experiments to obtain the PU output curves before/after reconfiguration under the fault condition of RRPVA, as shown in Fig. 10.

Fig. 9

4×3 asymmetric RRPVA reconfiguration in fault condition

Fig. 10

PU output curves before/after reconfiguration in case of failure of RRPVA

Figure 10(a) shows the PU output curves before/after reconfiguration of 4×4 symmetric RRPVA under array fault, and Fig. 10(b) shows the PU output curves before/after reconfiguration of 4×3 asymmetric RRPVA under array fault. From the output curves, it can be seen that after reconfiguration, the PU output curves’ peaks are enhanced, the power becomes better, and the power has been enhanced by 19.4% and 18.3%, respectively, which can effectively reduce the damage caused by the failure ofRRPVA.

The data descriptions for the RRPVA for the fault case are shown in Table 15, where Case A represents the RRPVA before reconfiguration and Case B represents the RRPVA after reconfiguration using MOPOA. It can be seen that MOPOA has good reconfiguration performance not only for the PV array in shading situations but also in the case of array fault. In the 4×4 symmetric RRPVA, P_mismatch decreased by 425 W, P_loss decreased by 14.4%, and PR increased by 0.145; in the 4×3 asymmetric RRPVA, P_mismatch decreased by 267 W, P_loss decreased by 12.3%, and PR increased by 0.111.

Table 15

Description of data for RRPVA under fault conditions

PV array	P_max	P_mismatch	P_loss	PR	P_en
4×4 symmetric	A:1770W	1169W	39.7%	0.602
RRPVA	B:2195W	744W	25.3%	0.747	19.4%
4×3 asymmetric	A:1193W	1011W	45.9%	0.541
RRPVA	B:1460W	744W	33.6%	0.652	18.3%

6 Conclusion

This paper presents an in-depth analysis of the output characteristics of RRPVA with multiple types and under different shading, and proposes an MOPOA-based reconfiguration strategy for multiple types of RRPVA.

After dynamic reconfiguration using MOPOA, the power enhancement percentage of 4×4 symmetric RRPVA is 22.0%, 10.5%, 18.7%, and 14.1% for trapezoidal shading, triangular shading, long shading, and decentralized shading, respectively. 4×3 asymmetric RRPVA is 22.9%, 17.8%, 19.5%, and 11.5% for full shading, triangular shading, long shading, and decentralized shading, respectively. It works better than both PSO and GA. In the presence of faulty PV arrays, the reconfigured 4×4 symmetric RRPVA showed a power enhancement of 19.4% compared to the pre-reconfiguration power, and the reconfigured 4×3 asymmetric RRPVA showed a power enhancement of 18.3% compared to the pre-reconfiguration power. The use of MOPOA ensures the minimization of row current deviation and the maximization of output power while minimizing switching losses and extending the life of electrical switches. MOPOA has the smallest mismatch loss, the smallest P_loss, the largest PR, and the largest P_en.

By comparing the simulation results, it is found that the reconfiguration of RRPVA optimizes the output characteristic curves, it reduces the output characteristic multi-peak phenomenon, and solves the mismatch problem of RRPVA under the influence of multiple factors, which lays the foundation for the subsequent research of maximum power point tracking. Additionally, we think that there are still some shortcomings in our study. Practical validation of the proposed approach will be conducted in future works to ensure the practical feasibility of the MOPOA we have proposed.

7 Foundation

This work was supported by the National Natural Science Foundation of China (61572416), Hunan province Natural science Zhuzhou United foundation (2022JJ50132), Key Laboratory Open Project Fund of Disaster Prevention and Mitigation for Power Grid Transmission and Transformation Equipment.

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A multi-objective pelican optimization algorithm for dynamic reconfiguration of multi-type rural rooftop PV array

Abstract

Keywords

1 Introduction

1.1 Present state of the art

1.2 Literature review

2 Rural rooftop PV array system components and modelling

Table 5 Parameters for PSO, GA and MOPOA PSO GA MOPOA w 0.9 Population size 25 Population size 25 c1 1.2 Mutation 0.3 Maximum number c2 1.5 Cross over 0.7 of iterations 100

7 Foundation

References

Table 5
Parameters for PSO, GA and MOPOA

PSO GA MOPOA

w 0.9 Population size 25 Population size 25

c1 1.2 Mutation 0.3 Maximum number

c2 1.5 Cross over 0.7 of iterations 100