Reactive Power Optimization in Distribution Networks of New Power Systems Based on Multi-Objective Particle Swarm Optimization

Abstract

The new power system effectively integrates a large number of distributed renewable energy sources, such as solar photovoltaic, wind energy, small hydropower, and biomass energy. This significantly reduces the reliance on fossil fuels and enhances the sustainability and environmental friendliness of energy supply. Compared to distribution networks in traditional power systems, the new-generation distribution network offers notable advantages in improving energy efficiency, reliability, environmental protection, and system flexibility, but it also faces a series of new challenges. These challenges include potential harmonic issues introduced by the widespread use of power electronic devices (such as inverters for renewable energy generation systems and electric vehicle charging stations) and the voltage fluctuations and flickering caused by the intermittency and uncertainty of renewable energy generation, which may affect the normal operation of electrical equipment. To address these challenges, this study proposes an optimization model aimed at minimizing network losses and voltage deviations, utilizing traditional capacitor adjustments and static var compensators (SVCs) as optimization measures. Furthermore, this study introduces an improved version of the multi-objective particle swarm optimization (MOPSO) algorithm, specifically enhanced to address the unique challenges of reactive power optimization in modern distribution networks. The test results show that this algorithm can effectively generate a large number of Pareto optimal solutions. The application of this algorithm on a 33-node network case study demonstrates its advantages in reactive power optimization. The optimization results highlight the effectiveness and feasibility of the proposed improved algorithm in the application of distribution network reactive power optimization, offering users a uniform and diverse range of reactive compensation solutions.

1. Introduction

With the large-scale integration of renewable energy into the grid, its power generation is greatly affected by climate and environmental factors, significantly increasing the complexity of grid supply and demand balance management. Moreover, the emergence of new loads, such as electric vehicles, has made the grid load curve more volatile, with an increased difference between peak and valley loads, posing new challenges for grid scheduling and operation. Therefore, reactive power optimization of the distribution network becomes a key link to ensure the economy, safety, and stability of the power system. Effective reactive power optimization can significantly improve the power system’s network losses and voltage quality. A reasonable distribution of reactive power is crucial for the safe and stable operation of the power system. Insufficient reactive power compensation can lead to voltage drops, reducing the power system’s static stability limit and, consequently, affecting its stability. Conversely, excessive reactive power might cause an overvoltage, damaging the insulation of electrical equipment and shortening its lifespan [1,2,3,4,5].

The core goal of reactive power optimization is to find the optimal solution to the objective function under the premise of meeting various constraints. As the scale of the power system continues to expand, the distribution network faces increasing supply challenges, and the objectives of reactive power optimization become more diversified [6,7,8,9,10,11]. For instance, ref. [1] proposes a two-stage optimization strategy for solving the voltage-var optimization problem that considers the stochastic penetration of plug-in electric vehicles (PEVs) to unbalanced, three-phase power distribution networks. The authors of Ref. [3] propose a second-order cone power flow (SOC-PF) model combined with chance constraint optimization to formulate the VVO to deal with uncertainties introduced by loads and stochastic distributed energy resources. The authors of ref. [4] propose a data-driven voltage-reactive optimization control strategy that considers the reliability of the photovoltaic inverter. Meanwhile, ref. [12] employs a multi-objective optimization model and applies fuzzy set theory to address the issue of different scales among three objective functions, although its fundamental approach remains a strategy for single-objective optimization. However, the essence of multi-objective optimization lies in attempting to simultaneously optimize multiple objectives, which is often challenging to achieve in practice due to potential conflicts among different objective functions. Therefore, it is necessary to coordinate and compromise among all feasible solutions to obtain a diverse set of solutions (Pareto solutions) [13,14,15,16,17]. To evaluate the diversity of solutions, ref. [10] uses a method of transforming the objective space, applying a parallel grid coordinate system in the new objective space to obtain the distribution entropy. The authors of Ref. [18] adopt a dual-objective optimization model, minimizing active power loss and voltage deviation using a multi-objective harmony algorithm with adaptive parameters to optimize the reactive power output of distributed sources and the switching numbers of capacitors.

When addressing multi-objective optimization problems, finding a solution that optimally satisfies all objectives simultaneously is extremely challenging. This is due to the close connections and mutual constraints among different objectives, which are sometimes even directly contradictory [12,18,19,20]. Traditional multi-objective optimization methods tend to convert them into a comprehensive total objective function through weighting, assigning different weights to each objective and then solving it using single-objective optimization techniques [21]. The main flaw of this conversion to single-objective optimization is that it may overlook the competitive relationships between different objectives, and the allocation of weights for each objective carries a certain degree of subjectivity [22,23,24,25,26,27]. While MOPSO algorithms are well established in various fields, their application in reactive power optimization presents unique challenges. This paper builds on existing MOPSO frameworks, introducing specific improvements that enhance performance in the context of electrical distribution networks. This paper employs an optimization strategy that uses capacitor switching and the adjustment of SVC as the means of optimization, aiming to minimize active network loss and minimize voltage deviation. An improved multi-objective particle swarm optimization algorithm is applied to solve this, calculating the optimal output of control devices every hour within a day, and the accuracy of the optimization results is verified through the IEEE 33-node distribution network system.

2. Optimization Framework with Multiple Objectives for Distribution Networks

This paper delves deeply into the aspects of economic efficiency and operational stability within distribution network systems. It meticulously crafts an optimization model designed to achieve two primary objectives: the reduction of network losses and the minimization of voltage fluctuations. This endeavor is particularly significant in the context of integrating distributed power sources into real-world distribution networks. This study exemplifies this integration by focusing on PQ-type distributed power sources, using them as a case study to develop a comprehensive reactive power optimization model. This model is specifically tailored for distribution networks that incorporate these distributed energy resources, showcasing a forward-thinking approach to enhancing network performance and reliability through strategic optimization.

2.1. Objective Function

The output power of photovoltaic generation mainly depends on factors such as solar radiation intensity, temperature, and characteristics of the PV panels. Generally, the output power of PV generation $P_{PV}$ can be estimated using the following formula:

$$P_{PV}=G\times A\times \eta \times (1-\beta (T-T_{ref}))$$

(1)

where $G$ is the solar radiation intensity (unit: W/m²); $A$ is the area of the photovoltaic panel (unit: m²); $\eta$ is the conversion efficiency of the photovoltaic panel; $\beta$ is the temperature coefficient (unit: %/°C), which represents the efficiency decrease per degree Celsius increase in temperature; $T$ is the actual temperature of the photovoltaic panel (unit: °C); and $T_{ref}$ is the reference temperature, usually taken as 25 °C.

The output power of wind generation mainly depends on the wind speed, the characteristics of the wind turbine (such as blade length, shape, etc.), and air density. The power P_wind of wind generation can be calculated using the following formula:

$$P_{wind}=\frac{1}{2}\rho A{v}^3{C}_p(\lambda ,\beta )$$

(2)

where $\rho$ is the air density (unit: kg/m³); $A$ is the swept area of the wind turbine (unit: m²), typically, where $\lambda$ is the radius of the rotor; $v$ is the wind speed (unit: m/s); and $\beta$ is the power coefficient, which depends on the tip speed ratio. This dimensionless parameter describes the efficiency of the wind turbine, with its theoretical maximum value known as the Betz limit, approximately 0.59.

To comprehensively account for the requirements of network losses and voltage quality in the operation of the power system, this study has chosen the minimization of active network losses and voltage deviations within a 24 h period as the optimization objectives.

$$f_1=\frac{1}{24}{\displaystyle \sum _{t=1}^{24}{U}_{tad}}$$

(3)

where $U_{tad}$ is the voltage deviation per hour, and in the formula ${\displaystyle \sum _{i\in n}\frac{U_i-U_i^{spec}}{\Delta U_i^{\max }}}$, $n$ is the number of nodes, $U_i^{spec}$ is the desired voltage amplitude, and $\Delta U_i^{\max }$ is the maximum voltage deviation.

$$f_2={\displaystyle \sum _{t=1}^{24}{P}_{tloss}}$$

(4)

where $P_{loss}$ is the active loss per hour.

$$P_{loss}={\displaystyle \sum _{i,j\in N}{G}_{ij}}(U_i^2+U_j^2-2U_i{U}_j{\theta }_{ij})$$

(5)

where $N$ is the node label, G_ij is the node $i,\hspace{0.17em}j$ branch conductance, $U_i,\hspace{0.17em}{U}_j$ is the voltage amplitude at the corresponding node, and ${\theta }_{ij}$ is the voltage phase angle difference.

Thus, the objective function is $F=\min (f_1,f_2)$.

2.2. Constraint Conditions

Despite the widespread adoption of devices like static synchronous compensators (STATCOMs) and SVCs in the power system, which can provide continuous reactive power compensation, static capacitors with discrete reactive power compensation capabilities are still commonly used in low-voltage distribution networks due to economic considerations. Therefore, this study has selected the reactive power output of SVCs and the switching operations of shunt capacitors as control variables.

The optimization model should look for the optimal settings of control variables (SVC settings and capacitor switching states) that achieve the minimum objective values while satisfying all system constraints. These settings directly influence the distribution of reactive power, aiming to improve voltage stability and minimize power losses. The SVC’s role, in this case, would be to adjust the reactive power dynamically to maintain voltage levels within desired limits without explicitly detailing the SVC model parameters in the mathematical formulation provided.

Equality constraints are equivalent to flow constraints:

$$\{\begin{array}{c}{P}_i=U_i{\displaystyle \sum _{j=1}^n{U}_j}(G_{ij}\cos {\theta }_{ij}+B_{ij}\sin {\theta }_{ij})\\ Q_i=U_i{\displaystyle \sum _{j=1}^n{U}_j}(G_{ij}\sin {\theta }_{ij}-B_{ij}\cos {\theta }_{ij})\end{array}$$

(6)

Inequality constraint:

$$C_{k\min }<C_k<C_{k\max } k=1,2,\cdots ,N_C$$

(7)

$$Q_{SVCk\min }<Q_{SVCk}<Q_{SVCk\max } k=1,2,\cdots ,N_{SVC}$$

(8)

$$U_{Li\min }\le U_{Li}\le U_{Li\max } i=1,2,\cdots ,N_D$$

(9)

where, $N_C,\hspace{0.17em}{N}_{SVC}$ are the number of compensation capacitor and static reactive power compensator nodes, respectively, N_D the number of load nodes, and U_Li is the i voltage at the nodes.

3. Multi-Objective Particle Swarm Algorithm

3.1. Fundamental Particle Swarm Algorithm

The particle swarm algorithm is a swarm intelligence computation method proposed by Kennedy et al. in 1995 [28]. This approach utilizes two main concepts: pbest, which represents the best previous position of an individual particle, and nbest, reflecting the optimum past positions among its adjacent particles. Guided by these parameters, each particle navigates from its initial location towards a more favorable area. In the t-th iteration, let $x_i(t)$ represent the particle’s position and $v_i(t)$ signify its velocity. The update process adheres to the subsequent formulas.

$$x_i(t)=x_i(t-1)+v_i(t)$$

(10)

$$v_i(t)=c_1{r}_1(x_{pbes{t}_i}-x_i(t))+c_2{r}_2(x_{nbes{t}_i}-x_i(t))+\omega v_i(t-1)$$

(11)

where t is the number of current iterations; t ω is the size of the inertia weight that affects the exploitation and exploration ability of the particle swarm; r₁, r₂ is [0, 1] a random number between; c₁ and c₂ are acceleration coefficients to regulate the proportion of the experience of the individual optimal position and the domain optimal position in the velocity update.

3.2. Improved Multi-Objective Particle Swarm Algorithm

The particle swarm optimization (PSO) algorithm is widely used in solving multi-objective problems due to its excellent global search ability and fast solving speed. To align with the development trend of multi-objective optimization, combining the basic PSO with appropriate optimization strategies to directly address multi-objective problems has become an inevitable direction for the development of PSO. Consequently, many scholars suggest using the PSO algorithm for multi-objective optimization problems and have extended the traditional PSO to MOPSO. In the PSO method, selecting the current particle’s leader is a key step, and three aspects become particularly critical in a multi-objective optimization environment: the selection of the leader particle, the distribution of solutions, and the convergence speed of the algorithm. In the realm of single-objective optimization, the process of selecting a leader particle is typically straightforward, as the particle boasting the highest fitness value frequently emerges as the prime candidate. This simplicity stems from the clear and singular goal of maximizing or minimizing a specific objective. However, the scenario becomes markedly more complex within the multi-objective optimization framework. Here, the conflicting nature of multiple objectives complicates the leader selection process. Unlike the single-objective approach, multi-objective optimization often prioritizes non-dominated solutions when identifying potential leader candidates, reflecting the need to balance between competing goals. The scholarly pursuit of optimizing solution distribution in both decision and objective spaces remains vigorous and dynamic, with the academic community tirelessly innovating and investigating novel methodologies. Such endeavors are crucial for enhancing the efficacy and applicability of optimization algorithms in diverse practical scenarios [12,18,19]. Furthermore, the issue of convergence speed, particularly within the PSO paradigm, has seen substantial advancements. Through the strategic adoption of various communication topologies, the challenges associated with PSO’s convergence rate have been addressed effectively. Notably, the r3pso method, as discussed in ref. [10], leverages an index-based ring topology, obviating the need for niche parameters. Experimental validations of this method have demonstrated its capacity to form stable niches efficiently, thereby underscoring its effectiveness in optimizing multi-objective problems while maintaining rapid convergence.

3.2.1. File Updates

In this paper, the following improvements are made to the multi-objective optimization algorithm: we first establish a personal best archive (PBA) and a neighborhood best archive (NBA), from which the individual best solution (pbest) and neighborhood best solution (nbest) are respectively selected. A ring topology structure is adopted to facilitate the formation of multiple niches. Moreover, to retain more Pareto frontier solutions, a unique selection mechanism has been introduced. The personal best position is preserved in PBA, while PBA{i} denotes the best position of the i particle found so far.

The term NBA represents the optimal local position, specifically the finest position in the vicinity of the i-th particle, denoted by NBA{i}. In this algorithm, a neighborhood comprises three particles, with each one interacting with the neighbors immediately adjacent on both sides. Diverging from the approach of leveraging the swarm’s global optimum, this method utilizes the local best, or NBA, for each particle. This strategy helps avoid the entire swarm converging to a singular point. Neighborhoods are structured through an index-based ring topology, which prevents direct interaction among particles from different neighborhoods. The NBA confines information flow within the swarm, fostering the emergence of multiple exploration niches and thus broadening the diversity of the search mechanism [29].

3.2.2. Special Crowding Distances

To enhance the diversity of solutions in the decision and objective spaces, ref. [18] employs a special crowding distance, assigning a metric value to each particle within the decision and objective spaces. The special crowding distance is computed in two steps: first, calculating the congestion distance for each particle in the decision space as well as the corresponding congestion distance in the target space, using $C{D}_{i,x}$ and $C{D}_{i,f}$ to represent the congestion distance of the particle $i$ in both spaces; second, calculating the special congestion distance of the non-dominated particles. Based on these distances, non-dominated solutions are ranked in descending order. The foremost position is occupied by the non-dominated solution with the highest special crowding distance. In the minimization problem, when the particle’s contribution $i$ to the first $m$ goal is smallest, let the particle’s contribution $i$ to the first $m$ goal be 1; when the particle’s contribution $m$ to the first goal is largest, let the particle’s contribution $i$ to the first $m$ goal be 0. Take Figure 1 as an example; $x_1$ and $x_2$ can be replaced by $f_1$ and $f_2$, which represent the decision space and goal space, respectively. In the figure, we can see that particle 6 is the maximum value in $f_1$, so the contribution of particle 6 in $C{D}_{6,f}$ is set f₁ to 0. Similarly, particle 6 is the minimum in the second target, so the contribution in $C{D}_{6,f}$ is set f₂ to 1. Therefore CD_{6, f} = (0 + 1) = 1.

In step 2, the particular crowding distance of particles in general is S_i taken as in Equation (12) and calculated as in Equation (13), when the crowding distance of decision-space or target-space particles is greater than the average crowding distance. S_i includes a maximum or minimum selection step, which includes the crowding metrics from both decision and target spaces. Thus, this method can promote diversity in both spaces simultaneously.

$$S_i=\max (C{D}_{i,x},C{D}_{i,f})$$

(12)

$$S_i=\min (C{D}_{i,x},C{D}_{i,f})$$

(13)

3.3. Algorithm Implementation Steps

This algorithm is a multi-objective optimization variant of the PSO method, often referred to as MO_Ring PSO_SCD. The code of the algorithm is shown in Figure 2. The steps of the algorithm are as follows:

• Initialization
  • Initialize a population of particles P(0) where each particle represents a potential solution.
  • Evaluate the particle population (calculate the fitness value for each particle).
  • Initialize two archives, the PBA and the NBA, with each particle’s initial position being its personal and neighborhood best.

2.

Main Loop

Loop until the maximum number of generations, Max-Generations, is reached.

3.

NBA Update

For each particle i in the population construct a temporary neighborhood best array (temp_NBA) based on the adjacency of particle i, including the left and right neighbors. From the temporary neighborhood best array, select the non-dominated particles to be the new neighborhood best (NBA[i]).

4.

Output

At the end of the algorithm, output the non-dominated particles from the NBA.

3.4. Algorithm Performance Verification

MMF4 and MMF8 are test functions for multimodal multi-objective optimization problems (MMOPs) that evaluate the performance of multi-objective optimization algorithms. They are characterized by having multiple global optima (Pareto optimal sets) as well as numerous local optima in the solution space. An ideal algorithm in multi-objective optimization should not only find these global optima but also maintain a diverse set of solutions, covering the entire Pareto front as much as possible [30].

MMF4:

$$f_1=\sin \left|x_1\right|$$

(14)

$$f_2=\{\begin{array}{ll}1-x_1^2+2{(x_2-\sin (7\pi \left|x_1\right|))}^2\hfill & 0\le x_2<1\hfill \\ 1-x_1^2+2{(x_2-1-\sin (7\pi \left|x_1\right|))}^2\hfill & 1\le x_2<2\hfill \end{array}$$

(15)

$$x_1\in [-1,1],\hspace{0.17em}\hspace{0.17em}\hspace{0.17em} x_2\in [0,2]. x_2=\{\begin{array}{ll}\sin (7\pi |x_1|)\hfill & 0\le x_1<1\hfill \\ \sin (7\pi |x_1|)+1\hfill & 1<x_2\le 2\hfill \end{array}$$

(16)

MM8:

$$f_1=\sin \left|x_1\right|$$

(17)

$$f_2=\{\begin{array}{ll}\sqrt{1-{(\sin |x_1|)}^2}+2{(x_2-\sin |x_1|-|x_1|)}^2\hfill & 0\le x_2\le 4\hfill \\ \sqrt{1-{(\sin |x_1|)}^2}+2{(x_2-4-\sin |x_1|-|x_1|)}^2\hfill & 4<x_2\le 9\hfill \end{array}$$

(18)

$$x_1\in [-\pi ,\pi ],\hspace{0.17em}\hspace{0.17em}\hspace{0.17em} x_2\in [0,9]. x_2=\{\begin{array}{ll}\sin \left|x_1\right|+\left|x_1\right|\hfill & 0\le x_1\le 4\hfill \\ \sin \left|x_1\right|+\left|x_1\right|+4\hfill & 4<x_1\le 9\hfill \end{array}$$

(19)

Each particle exchanges information with three others within its neighborhood, selecting leaders through the use of dominance relationships and a special crowding distance. The integrated application of this method effectively provides a solution for multimodal multi-objective problems. To assess the algorithm’s performance, it is tested using the multimodal multi-objective optimization problems proposed in reference. The algorithm parameters were set with a maximum number of iterations of 2000, a population size of 200, an external file capacity of 20, and a variance probability as in Equation (17), c1 = 1, c2 = 2, ω decreasing linearly from 1 to 0.4, and running 20 iterations. The Pareto-optimal set (PS) consists of solutions within the decision space that no other solutions outperform. Similarly, the Pareto-optimal front (PF) represents a collection of vectors in the objective space that align with these non-dominated solutions. The solution distribution of the function MMF4 is shown in Figure 3. Figure 3a shows the comparison of the true solution with the solution obtained by this algorithm, and Figure 3b represents the comparison of the Pareto front. Similarly, Figure 4 shows the comparison results for MMF8. It can be seen from the figure that the obtained solutions do not differ much from the ideal Pareto frontier, and the distributions and uniformity are quite convincing.

4. Simulation Analysis

4.1. Control Variables

To reduce the algorithm’s time complexity, this study treats the entire day’s reactive power optimization issue as a whole. This approach aims to simplify the dynamic changes over time into variations in the values of control variables, but it also results in a 24-fold increase in the computational space. This means that the output of the static var compensator and the switching operations of shunt capacitors within a day collectively constitute a single entity within the algorithm, as shown in Figure 5.

4.2. Example of Reactive Power Optimization

In this paper, the IEEE33 node network is used as an example to verify the advantages of the improved algorithm for reactive power optimization applications. The reference value of system power is 10 MVA, the voltage reference value is 12.66 kV, and the upper and lower voltage limits are 1.07 and 0.93, respectively; in consideration of the fluctuation of the 24 h load, the daily load variation is shown in Figure 6. Predictions of wind power generation and solar photovoltaic generation for a typical day in a certain region are shown in Figure 7, displaying their output over a 24 h period. Wind and solar power are integrated into the grid in the form of distributed energy resources. Due to insufficient or non-existent grid coverage in remote areas, distributed energy systems often need to be designed based on local natural resources (such as sunlight, wind, water resources, etc.) and user demand. Thus, in this paper, DGs are added to nodes 18, 21, and 33 near the end of the arithmetic network. Compared to traditional large-scale power stations, distributed energy systems are generally smaller, allowing them to be built close to the point of consumption, reducing transmission losses and increasing energy efficiency. This small scale also makes distributed energy more flexible, enabling adjustments and expansions according to specific regional demands. The maximum number of iterations is set to 100 in the program, the number of populations is 20, two groups of SVCs are connected to nodes 12 and 24, and the number of PBAs is five, because there are three particles in each neighborhood, and the size of NBA is three times the size of PBA; the parallel capacitor is connected to node 30 with a capacity $0.5\mathrm{M} \mathrm{var}\times 10$. To validate the effectiveness of the method proposed in this paper, the reactive power optimization model described above was implemented on the YALMIP platform using a mixed-integer optimization framework. The Gurobi 9.0.3 solver was employed to solve the optimization problem.

The traditional multi-objective particle swarm optimization algorithm is referred to as Algorithm 1, and the improved multi-objective particle swarm optimization algorithm proposed in this paper is referred to as Algorithm 2. A comparison of the daily average voltage at each node before and after optimization is shown in Figure 8; Algorithm 2 demonstrates superiority in maintaining higher voltage levels across most nodes. At node 20, Algorithm 2 improved the voltage by about 3.3%, whereas Algorithm 1 achieved an increase of approximately 2.5%. At node 25, Algorithm 2 showed a voltage improvement of around 2.7% compared to Algorithm 1’s improvement of about 1.6%. The changes in network losses before and after optimization are shown in Figure 9; from these curves, we can observe that the loss curves exhibit a clear diurnal pattern, with higher losses at certain times. This could be due to the fluctuating demand for electricity over time or the variability in supply from renewable sources such as solar and wind power. For most of the time period, Algorithm 2 resulted in lower network losses compared to both the pre-optimization levels and those achieved with Algorithm 1, suggesting that Algorithm 2 is more effective in reducing network losses. Thus, the improved multi-objective particle swarm optimization algorithm performs better in terms of increasing voltage levels and reducing network losses compared to the traditional multi-objective particle swarm optimization algorithm and the state before optimization.

This study aimed to minimize overall network losses and ensure voltage stability within a day by considering the constraint of discrete equipment operation counts and coordinating the timing of equipment actions to achieve optimization. Based on the optimization outcomes at different time intervals, effective reduction of system network losses was achieved through coordinated control of equipment actions, resulting in a significant decrease in the total active power loss to 1.378 MW within a day. Under the constraint of equipment operation counts, adjustments to the capacitor bank are limited within a certain range. In such cases, further refinement of reactive power flow control is achieved by adjusting the output of the static var compensator (SVC), thereby reducing system reactive power losses. By coordinating the timing of equipment actions to achieve optimization objectives, the system can dynamically respond to variations in network conditions and load demands over different time intervals. This dynamic adjustment capability enhances the system’s resilience by improving its ability to adapt to external disturbances and changes.

Figure 10 shows the Pareto optimal frontier obtained by MOPSO, AMOPSO, and the improvement of these three algorithms proposed in this paper. The Pareto frontier of the MOPSO algorithm and AMOPSO algorithm contains the limitation of the candidate solution distribution. The competitive relationship between network loss and voltage deviation can be clearly seen from the figure, and there is ambivalence. When the network loss is very small, the voltage deviation cannot be small at the same time, which may not satisfy the voltage quality requirement in the actual distribution network operation. However, when the voltage deviation is small, the target value of the corresponding solution for network loss cannot be small at the same time, which cannot meet the requirement of economy. If the voltage deviation at distribution nodes is taken as the primary optimization objective, optimization plan A can be chosen; if the emphasis is on minimizing active power network losses, solution C can be selected; if there is no particular preference for either objective, optimization plan B can be chosen.

5. Conclusions

With the increasing requirements of users on power quality, the objective function of reactive power optimization in distribution networks becomes more and more diverse. In order to enhance the diversity of solutions and the uniformity of distribution, this paper proposes an improved multi-objective particle swarm algorithm to form a stable small habitat using ring topology induction, and uses the special congestion distance as a metric to apply a test function to verify the performance of the algorithm. The minimization of network loss and voltage offset are established as the optimization objectives in the arithmetic analysis, and the improved algorithm is used to solve the problem in order to better match the actual distribution network, taking into account the access of distributed power sources and daily load fluctuations. The following conclusions were obtained:

• The performance of the algorithm was verified using the MMF multimodal multi-objective optimization family of test functions, where the solutions are uniformly distributed over the objective space and do not differ significantly from the ideal Pareto front.
• In the IEEE33 node network, this algorithm clearly showed that there is a contradictory relationship between the two objectives of network loss and voltage deviation. The improved algorithm can be effectively applied in reactive power optimization, and an effective reactive power optimization scheme can be obtained. After reactive power compensation, the system voltage was significantly supported, and the amounts of network loss and voltage quality were significantly improved.
• The uniformly distributed reactive power optimization scheme obtained after solving can also be used by users to choose flexibly according to their needs, which is more practical in the application of reactive power optimization.

However, only two performance indicators, distribution network loss and voltage offset, were considered, and there are other factors in the distribution network, such as the static voltage stability margin of the system and the limit of the number of reactive power compensation equipment throwing, etc. The establishment of a more realistic mathematical model in subsequent research needs to be strengthened.