3. Proposed Methodology
3.1. Time Division Method Based on the K-Means++ Clustering Algorithm
To address the issue of frequent reconfigurations in dynamic distribution networks, we employs the K-means++ [
28,
29] clustering algorithm to divide the daily load into periods, effectively reducing the active power loss and voltage offset and minimizing the number of reconfigurations.
Since the load status of each node in the distribution network changes dynamically with time, the load within 24 h is considered a dataset, with each dimension corresponding to the total number of nodes in the system. Using 1 h as a period, the node load of each period is treated as a sample set (O) for cluster analysis, , , representing the apparent power of each node in the distribution network system within that period. During dynamic reconfiguration optimization, reconfiguration is carried out continuously, and interperiod reconfigurations must meet actual needs. Thus, distance segment constraints need to be added to cluster analysis, which is achieved as follows:
Step 1: Determine the collection dataset (O) using the elbow method to determine the optimal range of clustering, i.e., the K value.
Step 2: Randomly select a value from the sample dataset (O) as the initial value center and set the initial value of K to 1.
Step 3: Compute the Euclidean distance from each data object to the center of K. Using the roulette method, select the next cluster center.
Step 4: Repeat Step 3 until K initial value centers are generated.
Step 5: Divide the 24 moments in the sample dataset (O) into the nearest initial value center to form a cluster. Calculate the mean value of each dataset as the new initial value center.
Step 6: Repeat Step 5 until the sum of the squares of the distances from each data point to the corresponding cluster remains unchanged. Then, output the period in each group.
3.2. Dung Beetle Optimizer
The dung beetle optimizer takes inspiration from the biological behavior of dung beetles. The optimal solution is achieved by continuously updating the position updating formulas of four dung beetles, each with its position renewal formula, as follows:
The dung beetles that roll balls are updated and can be expressed as
Encountering obstacles, being unable to move forward, and repositioned dancing can be expressed as
The position-updating formulas for female dung beetles can be expressed as:
The position-updating formulas for small dung beetles can be expressed as:
The position-updating formulas for thief dung beetles can be expressed as:
The values of and are fixed, with set to 0.1 and set to 0.3. The flexure angle () belongs to the interval and is used in the position-updating formulas of the four dung beetles, where and represent the current best position and the global best position, respectively. The oviposition and reproduction areas have upper and lower limits represented by , , , and . The constants and represent independent random vectors of size . The variable is a random number drawn from a normal distribution, while is a random vector with values ranging from 0 to 1. Additionally, a random vector of size that follows a normal distribution is represented by g, and S represents a constant value. The four dung beetle position-updating formulas are constantly updated until the optimal solution is found.
3.3. IMODBO Based on Mixed Strategy
After partitioning the data into time intervals, IMODBO is used to solve the distribution network reconstruction problem for each interval.
3.3.1. Population Initialization Based on a Tent Map
The standard dung beetle optimizer uses random initialization, which cannot guarantee the diversity of the population, affecting the quality of the global optimal solution. Using chaotic mapping to generate individuals randomly, the algorithm can effectively improve the speed and accuracy of the search for the optimal solution. As mentioned in [
30], through many comparative experiments, chaotic tent mapping has better bias adaptation. The iteration equation of chaotic tent mapping is expressed as follows:
Equation (
15) shows that the chaotic tent mapping equation has fewer parameters, making the application convenient and straightforward. The population size is set to 500, the population dimension is to 2, and the parameter
s is set to 0.5. The initial population distribution generated by chaotic tent mapping in the interval [−100, 100] is shown in
Figure 1. The initial population generated by chaotic tent mapping is distributed uniformly in space. Using these chaotic sequences, the initial population
of the dung beetle algorithm can be obtained according to Equation (
16).
In Equation (
16),
and
are the upper and lower limits of the population.
3.3.2. Adaptive Weight Factor
According to the position-updating formula of the standard dung beetle optimizer, the four dung beetles strongly depend on the best position in the position-updating phase. Therefore, an adaptive weight factor is introduced in during the position updating, the formula of which is expressed as follows [
31]:
where
is the maximum value of the adaptive weight,
is the minimum value of the adaptive weight,
represents the maximum number of iterations of the algorithm, and
indicates the current iteration number. After testing, the best algorithm performance is achieved when
and
. As the number of iterations increases, the adaptive weight decreases from 0.9 to 0.4. At the beginning of the iteration, a more considerable adaptive weight improves the exploration ability of the algorithm. As the number of iterations increases, a smaller adaptive weight in the later stage improves the searchability of the algorithm. This prevents falling into a local optimum early, and the convergence in the last stage can gradually approach the optimal value. The position-updating equations are obtained by adding adaptive weighting factors into Equations (10)–(14), which can be expressed as follows:
The dung beetles that roll balls are updated and can be expressed as:
The position-updating formulas for female dung beetles can be expressed as:
The position-updating formulas for small dung beetles are expressed as follows:
The position-updating formulas for thief dung beetles are expressed as follows:
3.3.3. Levy Flight Disturbance Strategy and Variable Spiral Search Strategy
According to the characteristics of the ball-rolling dung beetle algorithm, the ball-rolling dung beetles can expand their search range by “dancing”. However, the search range of the three remaining dung beetles is relatively simple, so the Levy flight perturbation strategy and the variable spiral search strategy should be introduced to increase the global optimal search range. The Levy flight perturbation strategy is introduced, which enables the algorithm to search randomly at different distances, maximizes the diversification of the search domain, and enhances the optimal global searchability. Levy flight not only satisfies the small-scale refined search, ensuring that each target can be searched, but also satisfies the large-scale rough search, avoiding the limitations of local search. The distribution density function of the Levy flight step size change is expressed as [
32]:
where
f is the motion step of Levy flight, which can be expressed by Equation (
24)
where
and
v are random numbers that conform to a normal distribution, and
and
are obtained by Equation (
25):
where the value range of parameter
∂ is
, and generally,
∂ = 1.5.
The variable helical search strategy allows for the development of various position-updating search paths, balancing the global and local searches of the algorithm. In the position-updating process, the parameter
z is designed as an adaptive variable to solve the monotony of the spiral parameter in the search method in case of falling into a locally optimal solution. It is used to dynamically adjust the spiral shape of the three dung beetle searches, thereby broadening the ability of three dung beetles to explore unknown areas and improving the algorithm’s search efficiency and global search performance. The formula for the variable helical search strategy is as follows [
33]:
where
Q is the spiral factor, and
h is the random number of (−1, 1), where
, and
.
In summary, the position updates of the four improved dung beetles are expressed as follows:
The position-updating formulas for the dung beetles that roll balls are updated and can be expressed as:
The position-updating formulas for female dung beetles can be expressed as:
The position-updating formulas for small dung beetles are expressed as follows:
The position-updating formulas for thief dung beetles are expressed as follows:
3.4. Pareto Dominance Theory
The traditional dung beetle optimizer selection strategy is suitable for single-objective optimization problems but not for multi-objective distribution network optimization and reconstruction problems. Therefore, in this paper, we use the Pareto dominance theory to evaluate the solutions generated by the distribution network optimization and reconstruction problem. Individuals with high degrees of optimization and extensive coverage ranges are selected to form the population for the next iteration. According to the Pareto theory, the objective function of this iteration is ranked by non-dominant levels. Every time a fitness calculation is performed, it is compared with the solution set. The solution with the highest non-dominant level is stored in the external archive as the current optimal solution, and solutions with lower non-dominant levels are removed. This process is repeated until the fitness calculation is completed.
Step 1 involves reading the distribution network data and determining the loop and the branch switches of each loop participating in the coding.
Step 2 sets the initial parameters, including the number of populations, the maximum number of iterations (T), the population dimension (dim), and the upper and lower limits of each dimension component of IMODBO based on the number of branch switches of each loop participating in the encoding.
Step 3 generates the population using the chaotic tent map according to Equation (
16).
Step 4 calculates the fitness of each individual in the population, performs Pareto screening, selects the solution with the highest non-domination level, and stores it in an external file.
Step 5 divides the population into four types of dung beetles and updates the positions of the four dung beetles according to Equations (27)–(31).
Step 6 recalculates the fitness of the updated positions, compares them with the external file using greedy selection, and updates the external file.
Step 7 checks whether the current iteration number of the algorithm reaches T. If not, the algorithm proceeds to Step 8; otherwise, it returns to Step 4.
Step 8 outputs the optimal solution from the external file.
In summary, the steps of the improved multi-objective dung beetle algorithm for solving distribution network optimization and reconstruction problems are shown in the
Figure 2.
5. Conclusions
This study proposes an optimization method based on the improved multi-objective dung beetle optimizer (IMODBO) to address the dynamic reconfiguration optimization problem in distribution networks with distributed power sources. The IMODBO method, in conjunction with the K-means++ clustering algorithm, is proposed and validated for this problem. Furthermore, multiple scenarios are designed in standard distribution network systems to verify the effectiveness of the proposed method. The main contributions of this study are as follows:
Through testing on both single-objective and multi-objective test functions, the IMODBO algorithm outperforms other optimization algorithms such as DBO, ISSA, NGO, WOA, and GWO in terms of convergence rate, convergence accuracy, and overall performance in single-objective optimization. In multi-objective optimization, the IMODBO algorithm exhibits better performance than MOALO, NSGA-II, MODA, MOSMA, MOPSO, and MOWOA in terms of convergence, solution set distribution, and comprehensive performance.
An optimization scheme is proposed for dynamic reconfiguration in distribution networks with distributed power sources, utilizing the IMODBO algorithm and K-means++ clustering algorithm. The objective is to reduce the active power loss, stabilize node voltages, and minimize switch operations.
In the IEEE-33 nodes and PG69 nodes test systems, the proposed method achieves the minimum active power loss and voltage deviation among multiple scenarios. In a single-period comparison experiment, the Pareto solutions obtained by the IMODBO algorithm dominate over the solutions obtained by NSGA-II, MOALO, and MOPSO algorithms. The proposed method also exhibits good performance in terms of multi-objective performance metrics. The Pareto solutions obtained by the IMODBO algorithm have the minimum IGD and maximum HV values in both test systems. This indicates that the proposed method can simultaneously seek solutions that are close to the optimal Pareto front and provide diverse options for decision makers.
This study has significant implications for achieving sustainable energy goals. The proposed IMODBO method demonstrates superior optimization performance and provides an effective solution to reduce energy loss and improve power quality in distribution networks. However, there are limitations to this study. For instance, it does not consider the randomness of photovoltaic power generation and wind power generation. Future research should take into account more factors to make the results more applicable to real-world projects. Additionally, testing in larger and more complex systems with a wider range of distributed power sources can be conducted to further increase the penetration rate of renewable energy and achieve carbon reduction and neutrality goals.