Research on Multi-Objective Reactive Power Optimization of Distribution Grid with Photovoltaics

Abstract

With the introduction of large distributed photovoltaic (PV) power and electric vehicles, the inherent volatility of their output makes it difficult for traditional power grid structures and reactive power optimization methods to meet the needs of the safe operation of distribution grids and economic benefits. Therefore, a multi-objective reactive power optimization method for a distributed grid is proposed under a distributed PV power generation scenario. Aiming at the two objectives of the network loss and voltage fluctuation rate, the improved multi-objective particle swarm optimization algorithm is used to solve the model under the condition that the output of each device does not exceed the constraint and the optimal solution that can reduce the distribution grid loss and improve the voltage stability of the distribution grid is obtained. The simulation was conducted on the IEEE 33-node and 113-node distribution networks to verify the proposed method’s feasibility.

1. Introduction

The large-scale continuous grid connection of new energy power generation equipment is an important development trend of China’s power grid. Under the background of the continuous replacement of a traditional power grid pattern and power structure, the safe and stable operation state of power systems has also undergone profound changes [1,2].

The rapid development of photovoltaic power generation has been unanimously recognized by the public because of its cleanliness, sustainable development, and environmental protection [3]. However, when distributed photovoltaics is connected to the distribution grid, the uncertainty of photovoltaic output will not only increase the system network loss and voltage limit risk, but also lead to the non-convex characteristics of the reactive power optimization model of the distribution network [4]. To ensure power quality and reduce system network loss, it can be adjusted by traditional voltage regulation devices such as capacitor banks and on-load tap changers (OLTCs) or reactive power compensation devices such as photovoltaic grid-connected inverters and static var generators (SVGs). Therefore, the analysis of photovoltaic output characteristics and the research on reactive power optimization scheduling strategy are significant [5].

In analyzing distributed photovoltaic power generation characteristics, Reference [6] established the equivalent circuit model of solar cells by equivalent modeling of solar cells and obtained the working parameters and characteristics of solar cells. Reference [7] used irradiance conversion and the photovoltaic array mathematical model to verify the photovoltaic experiment and received the influence of meteorological factors and temperature on the output of the photovoltaic array through many data simulations. Reference [8] modeled the intensity of light radiation and analyzed the influence of different physical processes on photovoltaic output. Reference [9] improved the stability of a distributed photovoltaic power grid by studying the service life and faults of photovoltaic panels.

The research on reactive power optimization scheduling strategy can be used to study the reactive power optimization model and the optimization algorithm for solving the model. In the establishment of a reactive power optimization model of a distribution network with photovoltaics, study [10] proposes a reactive power/voltage control strategy under a mixed time scale, which comprehensively considers the actions of OLTC and capacitor banks at the day-ahead scale to reduce the number of actions of discrete voltage-regulating equipment, and corrects the reactive power output of photovoltaic inverters at the real-time scale. Regarding photovoltaic inverter control strategy, study [11] proposed a voltage/var control (VVC) optimization model in a reliability-constrained distribution network to minimize grid power loss and optical abandonment power. In establishing the reactive power optimization model of the distribution network, considering the reliability of the photovoltaic power supply, it is necessary to adopt different control strategies for various types of control equipment to further reduce the voltage deviation and network loss [12,13,14]. In optimization algorithm research, particle swarm optimization (PSO) is proposed for the first time in Reference [15]. Because the parameters need to be adjusted when applied to different projects, the practicability is not high, and there is a problem of insufficient global search ability and premature convergence of a local optimum. Reference [16] used a chaotic search to improve the particle swarm optimization algorithm to solve the reactive power optimization scheme of a high-permeability photovoltaic distribution network at a long time scale. Although this method improves the global search ability of the algorithm and weakens the premature problem to a certain extent, the parameters are not adaptive, and the algorithm’s accuracy is also reduced. Reference [17] introduced a state factor to adjust the parameters linearly. Reference [18] proposed a multidimensional collaborative attention optimization algorithm (MCA), which uses a three-branch architecture to infer attention in multiple dimensions, including key adaptive combinations and gating mechanisms. In [19], a coordinated optimization model of active and reactive power in a photovoltaic distribution network based on a cooperative search algorithm (CSA) is proposed. The global search efficiency is improved by combining particle swarm optimization algorithms. However, the calculation process of the algorithm is complex, which loses the simplicity of the particle swarm optimization algorithm, and the parameters are fixed, which cannot meet the parameter requirements of different iteration periods.

This paper proposes a reactive power optimization strategy for a distribution network based on an improved multi-objective particle swarm optimization algorithm based on the above analysis. Firstly, the characteristics of photovoltaic power generation are analyzed. Secondly, the factors influencing photovoltaic output power are studied. Then, a reactive power optimization model with network loss and voltage offset as the objective is established, and the adaptive parameters and the mutation concept of the genetic algorithm are introduced. It not only improves the adaptability of the algorithm but also improves the global search ability of the optimization algorithm, reduces the possibility of falling into the local optimal problem, makes the strategy obtained by its solution model more accurate, and improves the voltage stability of the distribution network.

2. Reactive Power Optimization Model with PV

2.1. Photovoltaic Generation Principle

The energy conversion in PV power generation is a process where solar energy is converted into direct-current (DC) electric energy and then converted into AC through an inverter and incorporated into the grid or directly passing to a DC load for use because it does not consume primary energy, and does not cause environmental pollution and sustainability problems.

Figure 1 is the PV grid-connected structure, which connects the inverter to the grid. The inverter controller is controlled by the DC/DC output voltage and the grid feedback to control the inverter and realize its reactive power compensation ability. The realization process involves transforming the voltage and current of each phase into the Park transformation and treating their dot product as instantaneous reactive power to separate active and reactive power. The MPPT (maximum power point tracking) control system in Figure 1 comprises a DC/DC conversion circuit combined with PWM and the MPPT algorithm. Among them, the disturbance observation method is used to realize MPPT. Its basic idea is first to disturb the output voltage (or current) of the photovoltaic cell, and then observe the change in the output power of the photovoltaic cell, and continuously change the direction of the disturbance voltage (or) according to the trend of power change, so that the photovoltaic cell finally works at the maximum power point, and no more research is performed here. The DC/DC conversion circuit generally adopts a Boost circuit. The control process involves using the MPPT algorithm to control the duty ratio of the conversion circuit, which is then driven and controlled by the output signal of PWM.

The power generated by photovoltaic power generation [20] fluctuates randomly, and many parameters influence it, including the environment, light radiation intensity, air humidity, actual solar panel area, and solar panel inclination.

The analysis of the solar panel’s inclination needs to be adapted to local conditions. Many methods calculate the incident irradiance on the inclined plane, and the most common model is the Perez model [21].

Among other factors, light intensity and panel temperature are the most critical factors. Higher light intensity is helpful for power generation, but it will also increase the temperature of solar panels. Excessive temperature will reduce power generation efficiency. The solar panel temperature can be calculated by Formula (1).

$$T_{cell}=T_{ambient}+{\displaystyle \frac{G}{G_{NOCT}}}\cdot (T_{NOCT}-T_{a,NOCT})$$

(1)

Among them, T_cell is the temperature of photovoltaic cells, T_ambient is the ambient temperature, G is the current solar radiation intensity, G_NOCT is the solar irradiance at the nominal operating temperature, T_NOCT is the battery temperature at the nominal operating temperature, and T_a,NOCT is the ambient temperature at the nominal operating temperature.

Formula (2) can be used to calculate the output power of the solar panel under the influence of temperature and solar light intensity.

$$P_{PV}=P_{PV}^{STC}\cdot {\displaystyle \frac{G}{G_{STC}}}\cdot \left[1-g\cdot (T_{cell}-T_{STC})\right]$$

(2)

Among them, the output power is of the P_PV solar panel, where $P_{PV}^{STC}$ is the output power under standard operating conditions; G_STC and T_STC are standard illumination and temperature, respectively; and g is the power temperature coefficient.

2.2. Reactive Power Optimization Model

(a) objective function

Considering the safe operation and operation cost of the distribution grid, the model for reactive power optimization of the two objective functions of the minimum grid loss and the voltage fluctuation rate weighted by the penalty function and the average voltage fluctuation rate is constructed as follows:

(1) active grid loss

From the economic point of view, the general goal is to minimize the grid loss:

$$\min f_1=P_L={\displaystyle \sum _{i,j\in {\Omega }^N}{G}_{ij}\left(V_i^2+V_j^2-2V_i{V}_j\cos {\theta }_{ij}\right)}$$

(3)

Among them, P_L is the system grid loss, G_ij is the i th row and the j th column of the real part of the node admittance matrix, V_i and V_j are the voltage magnitudes of node i and node j separately, θ_ij is the difference in voltage phase angles between node i and node j, and Ω^N is the set of all nodes.

(2) voltage fluctuation rate

The node voltage MF’s average rate of fluctuation is

$$MF={\displaystyle \frac{{\displaystyle \sum _{j\in {\Omega }^N}\left|V_j-1\right|}}{N}}$$

(4)

where N represents the number of nodes.

The voltage over-limit penalty function [22] is expressed as

$$\Delta V_j=\left\{\begin{array}{cc}{V}_{j,\min }-V_j\hfill & \hfill V_j<V_{j,\min }\\ 0\hfill & \hfill V_{j,\min }<V_j<V_{j,\max }\\ V_j-V_{j,\max }\hfill & \hfill V_j>V_{j,\max }\end{array}\right.$$

(5)

Among them, V_j,max and V_j,min are the upper and lower boundaries of the voltage amplitude of node j separately.

$$\min f_2={\lambda }_1\cdot MF+{\lambda }_2{\displaystyle \sum _{i\in {\Omega }^V}\alpha \cdot {\left(\frac{\Delta V_i}{V_{i,\max }-V_{i,\min }}\right)}^2}$$

(6)

Among them, Ω^V is the set of over-limit voltage nodes; λ₁ and λ₂ are weight coefficients; and α is the penalty function coefficient of node voltage.

(b) power equation constraints

During the process of VAR optimization, it is essential to calculate the power flow distribution within the grid. The constraint condition for the power flow in the reactive power’s optimization process of the distribution grid with distributed power supply is regarded as its equality constraint. Within this paper, the VAR optimization equipment consists of a reactive power compensator and distributed PV, while also taking into account the influence of distributed energy storage.

$$\left\{\begin{array}{c}{P}_{Grid}+{\displaystyle \sum _{j\in {\Omega }^{PV}}{P}_{PV,j}+P_{DS,j}+P_{EV}={\displaystyle \sum _{j\in {\Omega }^N}{P}_{load,j}+{\displaystyle \sum _{j\in {\Omega }^N}{P}_{i,j}}}}\\ Q_{Grid}+{\displaystyle \sum _{j\in {\Omega }^{PV}}{Q}_{PV,j}+{\displaystyle \sum _{j\in {\Omega }^{SVC}}{Q}_{SVC,j}}={\displaystyle \sum _{j\in {\Omega }^N}{Q}_{load,j}+{\displaystyle \sum _{j\in {\Omega }^N}{Q}_{i,j}}}}\end{array}\right.$$

(7)

In the formula, P_Grid and Q_Grid, respectively, inject active power and reactive power into the superior power grid; P_PV,j, Q_PV,j, P_load,j, Q_load,j, P_i,j, Q_i,j, and P_DS are, respectively, the active power and reactive power of PV injected by node j, active power and reactive power required by the load, and the active and reactive power loss and the power of distributed energy storage. Q_SVC, j is the reactive power of the reactive power device of the node j; Ω^PV and Ω^SVC are the node sets installed with PV and reactive power compensation devices, respectively.

(c) variable constraints

Inequality constraints: Apart from equality constraints, inequality constraints also exist within the power flow calculation process of the distribution grid with distributed power supplies, mainly to constrain the control and state variables involved in reactive power optimization control. In the distribution grid containing distributed generation, according to the operation specification requirements, the grid’s nodal voltage should be changed within the specified range, and the power of each branch is also limited.

Variable constraints [23] can be classified into constraints on the upper and lower limits of voltage and current, reactive power compensator constraints, OLTC gear constraints, and PV reactive power compensation constraints.

(1) voltage and current constraints

$$\left\{\begin{array}{c}{I}_{\min }\le I_j\le I_{\max }\hfill \\ V_{\min }\le V_j\le V_{\max }\hfill \end{array}\right.$$

(8)

As shown in the formula, I_max and I_min are the upper and lower boundaries of the current individually; V_max and V_min are the upper and lower boundaries of the voltage individually. I_j and V_j are the current and voltage of node j separately.

(2) PV constraints

$$\left\{\begin{array}{c}{P}_{PV,\min }\le P_{PV,j}\le P_{PV,\max }\\ Q_{PV,\min }\le Q_{PV,j}\le Q_{PV,\max }\end{array}\right.$$

(9)

Among them, P_PV,max and P_PV,min are the upper and lower limits of PV active power output, respectively; Q_PV,max, Q_PV,min are the upper and lower boundaries of the PV reactive power output, respectively.

(3) reactive power compensator constraints

$$Q_{SVC,j\min }\le Q_{SVC,j}\le Q_{SVC,j\max }$$

(10)

In the formula, Q_SVC,jmax and Q_SVC,jmin are the upper and lower limits of the reactive power compensator output, respectively.

(4) OLTC gear constraint

$$K_{\min }\le K\le K_{\max }$$

(11)

Among them, K is the OLTC gear; K_min and K_max are the OLTC gear upper and lower limits, respectively.

(5) energy storage constraints

$$\begin{array}{l}SO{C}_{DS}^{\min }\le SO{C}_{DS}\le SO{C}_{DS}^{\max }\\ P_{DS}^{\min }\le P_{DS}\le P_{DS}^{\max }\end{array}$$

(12)

where $SO{C}_{DS}^{\max }$, $P_{DS}^{\min }$, and $P_{DS}^{\max }$ are the charging state of the energy storage device and the minimum and maximum values of a single charge and discharge, respectively.

(6) Electric vehicle constraints

$$\begin{array}{l}{\eta }_1\cdot P_{ev}\left(t\right)\le P_{EV}\left(t\right)\le {\eta }_2\cdot P_{ev}\left(t\right)\\ \begin{array}{cc}& \end{array}{\displaystyle \sum _{t=1}^{24}{P}_{ev}\left(t\right)=}{\displaystyle \sum _{t=1}^{24}{P}_{EV}\left(t\right)}\end{array}$$

(13)

Among them, η₁ and η₂ are the upper and lower limit coefficients, respectively; P_ev is the pre-dispatch load power; and P_EV is the post-dispatch load power.

3. Improved MOPSO Algorithm

The reactive power optimization of the distribution grid with PV is a multi-objective optimization problem involving multiple factors. The PSO algorithm is well suited for complex nonlinear problems. Initially, it distributes the population within the specified space and then gradually moves towards the optimal direction based on the minimum fitness target. By setting the number of iterations, the optimal Pareto solution set can be achieved [24]. Meanwhile, since the multi-objective algorithm needs to achieve the convergence, diversity, and homogeneity of the optimal Pareto solution set, it is essential to set the relevant parameters accurately.

3.1. Basic Principle of PSO Algorithm

The particle swarm optimization algorithm [25,26,27,28] is an algorithm that conducts optimization within the global scope based on swarm intelligence. The traditional particle swarm optimization algorithm was applied to solve the problem of optimizing single-objective functions, and later, it developed into dealing with multi-objective issues due to the increase in objective functions. In the selected search reference space, the positions of particles in the population will keep changing as the iteration proceeds until the iteration ends and the optimal solution is found. The steps of the single-objective particle swarm optimization algorithm are as follows: First, initialize the population size, positions, velocities, individual best (pbest), and global best (gbest), assuming that they are all feasible solutions initially. With each iteration, calculate the fitness of the corresponding particles, compare them with the current best to obtain the individual best and global best of the corresponding particles, and record the optimal positions and the corresponding fitness for the preparation of the following comparison. Then, update the velocity for the next iteration according to the pbest, gbest, and the current velocity, and update the position for the next iteration through the updated velocity to obtain a new fitness. Proceed in this way until the number of iterations is reached.

It is assumed that there is a population in the N-dimensional space with several m particles, and the number of iterations is n times. The position vector of the ith particle is X_i = (X_i1, X_i2, …, X_in), and its velocity vector is V_i = (V_i1, V_i2, …, V_in). It is assumed that the optimal individual position of the jth iteration is P_j = (P_j1, P_j2, …, P_jm), which is called the individual optimal. The best position in history is P_jy, the best among the individuals so far and the current global optimal. In the jth iteration, at the next moment, the speed and position update formula for the ith example is

$$V_i\left(t+1\right)=\omega V_i\left(t\right)+c_1{r}_1(P_{ji}-X_{ij})+c_2{r}_2\left(P_{jy}-X_{ij}\right)$$

(14)

$$X_i(t+1)=X_i(t)+V_i\left(t+1\right)$$

(15)

Among them, ω represents the inertia coefficient, while c₁ and c₂ are the learning factors. r₁ and r₂ are random numbers within the range of [0, 1], and t is the current iteration number.

3.2. Improved PSO

The convergence of PSO is affected by its weighting parameter and learning parameter. The fixed parameter values cannot meet the requirements of its full convergence. To prevent the occurrence of premature problems, it should pay attention to the global search in the early stage and pay attention to optimization convergence in the later stage. Therefore, this paper improves its three parameters as follows:

$$\omega (t)={\omega }_{start}-({\omega }_{start}-{\omega }_{end})({\displaystyle \frac{t}{T_{max}}}{)}^2$$

(16)

$$c_1^t=c_1^{start}+\left(c_1^{end}-c_1^{start}\right){\displaystyle \frac{t}{T_{\max }}}$$

(17)

$$c_2^t=c_2^{start}+\left(c_2^{end}-c_2^{start}\right)\times {\displaystyle \frac{t}{T_{\max }}}$$

(18)

In the formula, ω_start, ω_end, $c_1^{start}$, $c_1^{end}$, $c_2^{start}$, and $c_2^{end}$ are the starting and ending values of the weight factor, the learning factors of individual particles, and the learning factors of leaders, respectively.

Meanwhile, to further prevent the premature problem of local convergence, the mutation algorithm of the genetic algorithm is added to upgrade the global search ability. The specific improvements are

$$\begin{array}{l}{X}_{new1}(t)=\theta X_a(t)+\left(1-\theta \right)X_b(t)\\ X_{new2}(t)=\theta X_b(t)+\left(1-\theta \right)X_a(t)\end{array}$$

(19)

In the formula, X_new1(t) and X_new2(t) are the positions of the mutated particles, X_a(t) and X_b(t) are the positions of two random particles in the current population, and θ is the mutation parameter.

3.3. Principle of Improved MOPSO Algorithm

MOPSO, like other algorithms, produces a collection of non-inferior solutions at one iteration. It is optimized by mutual learning among individuals. The key is how to control the dimension of external archives. The difference between it and the single-objective PSO algorithm is that it only satisfies that the new pbest selected after the pbest of the previous iteration does not dominate each iteration, and the single-objective is better; secondly, in order to cover the entire reference surface as much as possible, its gbest is selected according to the final external archive using the adaptive grid method. The particles with the least crowding degree are chosen. Taking this paper as an example, the variable data and external archives are initialized. The 24 h PV reactive power output, reactive power compensator output, and OLTC ratio are taken as variables. After the first iteration, the new grid loss, voltage fluctuation rate, and voltage amplitude average are obtained, corresponding to the three fitness targets in the algorithm. Then, each particle is compared with the new fitness according to the initial individual optimal-corresponding fitness. The new solution is retained if the fitness of the new three targets is better than that of the initial three. The original solution is maintained if the new three fitness targets are better than or worse than the initial three fitness targets. The corresponding individual optimal (pbest) solution set is obtained. On this basis, it is selected with the initialized external archive. Firstly, a part of the solution is removed according to the relationship between dominance and dominance. Then, the solution other than the memory size of the external archive is removed according to the adaptive grid method, and the overall optimal solution gbest is selected. In continuous development, it can be proved that this is a point worthy of study.

Step 1: Initialize the population number, individual optimal, global optimal, external files, variables (PV, reactive power compensator 24 h reactive power output, and transformer ratio), and the fitness, speed, and position of each particle.

Step 2: Based on the initial velocity and position and the corresponding parameters, each particle’s velocity and position are updated, and the new fitness related to it is calculated.

Step 3: Each particle compares the new fitness with the original fitness to obtain the dominant relationship between the two. If the former dominates the latter or does not dominate the other, the original solution is retained; if the latter dominates it before and after, it is updated to a new individual optimal (pbest).

Step 4: After the particles receive the individual optimal, the individual optimal set is first screened by the dominance relationship, and then the individual optimal set is screened after the dominance relationship screening, and the solution is set in the external file.

Step 5: According to the adaptive grid method and the capacity set by the external file, the truncation operation is performed, the redundant solution is screened to obtain the Pareto set of solutions, and the global optimal solution is obtained from the external file according to the adaptive grid method. If the iteration count does not meet the requirements, step 2 is returned to, and the operation is repeated.

The algorithm flowchart is presented in Figure 2.

4. Example Analysis

4.1. IEEE33 Node Example

This paper uses MATLAB to conduct simulations on reactive power optimization. As shown in Figure 3, the IEEE 33-node system is used for simulation to verify the effect of the algorithm. The IEEE 33-node system has 33 nodes, 32 branches, three photovoltaic generators, one adjustable transformer, a group of electric vehicles, and 2 nodes with shunt compensation capacitors. The base power is 100 MVA, the population size is 100, the repository size is 100, the number of dimension grids is 7, the number of iterations is 150, ω is linearly transformed between 0.5 and 0.001, c₁ = 0.1, c₂ = 0.2, and the mutation rates are all 0.5. The power factor of the photovoltaics is set to 0.8, with the lower limit being 0. The per-unit value of the reactive power compensator has an upper limit of 0.08 and a lower limit of 0. The upper limit of the transformer turn ratio is 1.1, and the lower limit is 0.9. Respectively, η₁ and η₂ are 0.8 and 1.2.

Figure 3 is the IEEE33 node topology diagram, with nodes 13, 22, and 30 being photovoltaic nodes and node 7 being an electric vehicle node.

Figure 4 compares the Pareto frontiers under four different algorithms featured in this paper. These algorithms are the improved multi-objective particle swarm optimization algorithm, the enhanced particle swarm optimization algorithm referenced in [17], the MCA algorithm, and the basic particle swarm optimization algorithm. The figure shows that the Pareto frontier of the algorithm proposed in this paper is closer to the coordinate axes, indicating a superior performance. This translates to lower network loss and voltage deviation. Figure 5, Figure 6 and Figure 7 depict the active and reactive power output from the photovoltaic system, the output from the reactive power compensator, and the ratio of the on-load voltage regulator, respectively.

Figure 8 shows the comparison of load distribution in the distribution network before and after load transfer through electric vehicles after algorithm regulation. Figure 9 shows the change in charge and discharge power and state of charge of electric vehicles within 24 h during the regulation process.

Figure 10 is a comparison diagram of network losses over 24 h. It can be seen from the figure that the network losses after optimization have decreased significantly, and the improved algorithm can achieve even lower network losses. The decrease in network losses is more noticeable after node 10.

It can be concluded from Figure 11 that after dispatching the distributed energy resources in combination with the on-load tap changer (OLTC), the overall voltage has been significantly improved and is closer to the reference voltage. Meanwhile, the parts initially higher than the reference voltage remain unchanged, and only those lower than the reference voltage have been enhanced and improved, demonstrating the optimization’s advantages. This also verifies that the algorithm proposed in this paper can improve the volatility of distributed power generation connected to the grid, ensure the voltage stability of the system, and increase the voltage simultaneously, achieving the corresponding effects.

Table 1. IEEE33 Objective function comparison.

Scene	Network Loss/Pu	Voltage Deviation/Pu
Not optimized	0.5091	304.6793
PSO	0.3215	139.957
MCA	0.3209	138.79
Other improved PSO	0.3189	135.011
The improved PSO in this paper	0.3177	128.6216

shows that, compared to the unoptimized scenario, the network loss achieved through the four different algorithms decreased by 36.8%, 37.6%, 37.3%, and 36.97%, respectively. Additionally, the voltage deviation obtained by these algorithms was reduced by 54.06%, 57.79%, 55.58%, and 54.44%, respectively. This indicates that the improved optimization algorithms exhibit higher accuracy.

4.2. The 113-Node Distribution Grade Example

To verify the expansibility and superiority of the algorithm, a simulation was conducted using a power grid diagram consisting of 113 nodes, as shown in Figure 12. The system includes 113 nodes, 112 branches, ten photovoltaic generators, and three substations connected to three lower-level power grid systems. Additionally, there are two nodes for reactive power compensation capacitors. The reference power is 100 MVA, with a population size of 100 and a storage size of 100. The dimension grid number is set to 7, and the number of iterations is 100. The weight (ω) is linearly adjusted between 0.5 and 0.001, with coefficients c₁ = 0.1 and c₂ = 0.2. The mutation rate is established at 0.5, and the photovoltaic power factor is set to 0.8. The lower limit for the variables is 0, while the upper limits for the per-unit value of the reactive power compensator and the transformer ratio are 0.08 and 1.1, respectively, with a lower limit of 0.9. Respectively, η₁ and η₂ are 0.8 and 1.2.

In the analysis, both the original particle swarm optimization algorithm and several improved versions were employed to address the problem optimally. The results of these analyses are illustrated in the following figures: Figure 13 shows the comparison of network losses, while Figure 14 presents the comparison of voltage amplitudes.

The data presented in Figure 13 indicate that implementing reactive power optimization has effectively reduced network loss, particularly between points 10 and 17. The network loss optimization results are similar when comparing the five algorithms. Figure 14 shows that all five optimization algorithms yield significant improvements. The overall voltage levels have been markedly enhanced, aligning more closely with the reference voltage, leading to significantly improved voltage stability. In addition, the improved multi-objective particle swarm optimization algorithm introduced in this paper shows comparable performance to the other algorithms for the first 58 nodes. However, its effectiveness is noticeably more significant at the terminal node, demonstrating its advantages in enhancing voltage stability.

Table 2. IEEE113 Objective function comparison.

Scene	Network Loss/Pu	Voltage Deviation/Pu
Not optimized	1.235	1495.4652
PSO	0.7423	743.8013
MCA	0.7297	735.1938
Other improved PSO	0.7161	725.3565
The improved PSO in this paper	0.6933	723.7394

reveals that, in contrast to the unoptimized scenario, the network loss achieved by the four different algorithms decreased by 39.89%, 43.86%, 42.01%, and 40.91%, respectively. Furthermore, the voltage deviation achieved by these algorithms decreased by 50.26%, 51.61%, 51.47%, and 51.49%. This highlights that the improved optimization algorithm maintains superior accuracy in the 113-node network.

5. Conclusions

Based on the reactive power compensation capability of distributed photovoltaic power sources, this paper proposes a multi-objective reactive power optimization method for distribution networks in distributed photovoltaic power generation to solve the reactive power optimization problem of distribution networks containing distributed power sources. This algorithm combines the improved multi-objective particle swarm optimization algorithm and the genetic algorithm to construct a reactive power optimization model for distribution networks with photovoltaic power. Through simulation comparison, it can be known that the particle swarm algorithm improved by the genetic algorithm can obtain a more advanced Pareto solution set and has better convergence accuracy. This method not only improves the convergence accuracy of the algorithm but also enhances the diversity of finding the Pareto solution set as it takes the lowest crowding degree as the criterion. It is proved that by utilizing the reactive power compensation capability of distributed power sources in this paper, network losses can be effectively reduced, and voltage quality can be improved.