Performance optimization of electrical equipment in high-altitude photovoltaic power stations based on PSO–MOEAD algorithm

Abstract

The special environment in high-altitude areas poses severe challenges to the performance and lifespan of electrical equipment in photovoltaic power plants. To reduce energy consumption and operation and maintenance costs, a hybrid algorithm based on particle swarm optimization and multi-objective evolutionary decomposition algorithm is proposed in this study. A multi-objective optimization model that comprehensively considers equipment performance, environmental factors, and economy is constructed by integrating voltage compensation mechanisms and thermal parameter optimization schemes. Experimental results show that in standard test functions, the proposed algorithm increases convergence speed by 71% (45.6 s faster than that of the traditional method) and significantly enhances global search capability. In practical high-altitude PV scenarios, it improves voltage stability by 3% (The average voltage increases by 0.01p.u.), reduces power generation costs by 12.3%, and lowers network losses by 15%. Additionally, it effectively optimizes the thermal performance (e.g., 22.8% reduction in peak surface temperature rise of equipment) and reduces the thermal fault maintenance frequency by 64.3%. The results not only provide direct support for the efficient operation of high-altitude photovoltaic power plants, but also open up new ideas for the multi-objective optimization design of other renewable energy systems.

Introduction

With the transformation of the global energy structure, photovoltaic power generation, as a clean energy source, has received extensive attention and application worldwide (Li et al., 2022; Naik, 2024). Especially in high-altitude areas, due to long sunshine hours and high solar radiation intensity, photovoltaic power generation has enormous development potential. However, the unique environmental conditions such as low oxygen, low temperature and strong ultraviolet rays in high-altitude areas put forward higher requirements for electrical equipment in photovoltaic power stations. The performance optimization of electrical equipment in photovoltaic power stations plays a crucial role in promoting the sustainable development of the power industry, advancing power marketization, and enhancing operational efficiency and economic benefits. Meanwhile, energy storage systems can help hydrogen refueling stations purchase electricity at lower costs, providing consumers with low-cost hydrogen and optimizing expected profits and risks. It can also be combined with high-penetration renewable energy. By adjusting the power demand and battery mode, the hydrogen sales volume can be increased, the abandoned photovoltaic power can be reduced, and the system profit can be enhanced (Jordehi et al., 2024a). Renewable energy has prominent advantages, reducing carbon emissions. Combined with storage systems, it can alleviate intermittency, improve energy utilization efficiency, and support the transition to a clean and sustainable energy structure (Jordehi et al., 2024b). Against this backdrop, how to optimize the performance of electrical equipment in photovoltaic power stations, as well as enhance overall efficiency and reliability, is a problem that needs to be addressed.

In high-altitude environments, the performance optimization of electrical equipment in photovoltaic power stations needs to address three key challenges. The first is low pressure. The decrease in air density leads to a decline in heat dissipation efficiency, and inverters and transformers may overheat. The second is strong ultraviolet radiation, which will accelerate the degradation of materials in photovoltaic panels and insulation components. The third factor is extreme temperature fluctuations. The thermal stress caused by low temperatures at night and high solar irradiance during the day affects battery life and power conversion efficiency. Although metaheuristic algorithms do not guarantee global optimality, they are more adaptable in problems such as power energy systems with strong nonlinearity, multiple constraints, and high-dimensional features. Compared to the problem essence that may be lost due to relying on MILP solver linearization, metaheuristic algorithms can better handle the coupling relationship between equipment performance and environmental factors in high-altitude photovoltaic scenarios (Khunkitti et al., 2022; Liu et al., 2023a). Although Particle Swarm Optimization (PSO) algorithm and Multi-Objective Evolutionary Algorithm Based on Decomposition (MOEAD) are traditional algorithms, MOEAD's multi-objective decomposition strategy outperforms single-objective algorithms in handling the performance, economy, and environmental coupling, because PSO has fewer parameters, faster convergence speed, and can adapt to dynamic adjustment requirements (Kessler-Lewis et al., 2024; Wichitkrailat et al., 2024). The limitations of both (such as PSO being prone to local optima and MOEAD having random initial weights) can be compensated for through a hybrid mechanism, which is more stable than some new algorithms in complex scenarios. Therefore, it is chosen as the basis for building a hybrid algorithm.

The PSO algorithm has good adaptability and can handle various types of optimization problems, including nonlinear, multi-modal, and high-dimensional problems, and is widely used to optimize power generation systems. Liu et al. proposed a data cleaning method that combined density clustering to further optimize the power generation prediction accuracy. A prediction model based on PSO-optimized LSTM was constructed. The results showed that the model was superior to existing typical methods and effectively improved the accuracy (Liu et al., 2024). Makhloufi et al. proposed a logarithmic PSO to address the slow convergence speed of PSO algorithm during the search process. Without reducing the search window, the algorithm reduced power oscillations, accelerated convergence speed, and only adjusted one parameter. The effectiveness of this method (Makhloufi & Mekhilef, 2021). To optimize the convergence speed and accuracy in single machine photovoltaic systems, Anbarasi and Kanthalakshmi optimized the PSO algorithm by adding an adaptive inertia weighting factor. The optimized PSO had better tracking speed and efficiency (Anbarasi & Kanthalakshmi, 2023). Moreira et al. optimized the allocation and scale in distribution networks to reduce feeder losses. An improved PSO method based on dynamic momentum was adopted. The economic cost of the distribution system with many decision variables was minimized to the greatest extent possible (Moreira et al., 2022). To optimize the operational efficiency of the complementary power generation system in photovoltaic power plants, Liu et al. established an optimization allocation model for the complementary power generation capacity of photovoltaic power plants using PSO algorithm and BP neural network. The optimization configuration of the complementary capacity was completed. This method reduced the average loss and ensured the effective operation (Liu et al., 2023b).

Multi-objective evolutionary algorithm exerts a meaningful role in optimizing the performance of electrical equipment in photovoltaic power plants. By comprehensively considering multiple objectives, it can effectively improve the economy and stability of photovoltaic power generation systems. To improve the efficiency and accuracy of multi-objective optimization problems, Mohammad Zadeh et al. proposed a hybrid optimization method that integrated multi-objective PSO algorithm and sequential quadratic programming algorithm. This method generated the reference direction through spatial decomposition of the objective function, and introduced a new mutation operator to improve the performance (Mohammad Zadeh & Mohagheghi, 2023). Shi et al. proposed an improved multi-objective optimization algorithm for the optimal design of AC filters in HVDC systems. By adding special mutation operators, the global search ability was improved, and the solution set was optimized in combination with the three-dimensional fuzzy evaluation system. The simulation and dynamic experimental verification of ± 500 kV DC transmission system showed that this method could effectively achieve multi-objective optimal configuration (Shi et al., 2023). To solve the multi-target coverage salesman, Khan et al. proposed the MOEAD-GWO, a two-stage algorithm that integrated improved gray Wolf optimization algorithm and MOEAD framework. The solution quality was improved by K-bit exchange and two-stage update mechanism, and the city clustering and Hamiltonian ring search were jointly optimized for the first time. This algorithm showed advantages in dealing with complex multi-objective problems (Khan et al., 2023). Jordehi et al. proposed a hierarchical stochastic framework for the operation planning of isolated microgrids with energy hubs at hydrogen refueling stations. By designing a hierarchical stochastic framework, hydrogen refueling stations and energy hubs adopted a mixed-integer linear programming model, while microgrids used a mixed-integer quadratic programming model. Taking the 69-bus radial network rich in renewable energy as an example for verification, it showed that the proposed method was highly efficient (Jordehi et al., 2024c). Jordehi et al. proposed a three-level model that integrated hydrogen refueling stations into an interconnected electricity–gas network considering vehicle-to-infrastructure technologies. By processing the nonlinear gas flow model through piecewise linearization, a mixed-integer linear programming model was constructed. The results showed that the total cost of the electricity–gas coupling network was $760.37, enabling it to earn a daily profit of $519 while maintaining gas node pressure and pipeline flow within the predetermined range (Jordehi et al., 2024d). The results of comparison between the proposed method and existing literature are shown in Table 1.

Table 1. Comparison of the proposed method with existing literature

Research purpose	Method	Result	Shortcomings	References
Power generation prediction optimization	PSO-optimized LSTM with density clustering	Improved prediction accuracy over traditional methods	High computational complexity	Liu et al. (2024)
PSO convergence acceleration	Logarithmic PSO algorithm	Reduced power oscillations, faster convergence	Limited to single-parameter adjustment	Makhloufi and Mekhilef (2021)
Single PV system optimization	PSO with adaptive inertia weighting	Enhanced tracking speed and efficiency	Not applicable to complex networks	Anbarasi and Kanthalakshmi (2023)
Distribution network optimization	Improved PSO with dynamic momentum	Minimized feeder losses and economic costs	Long computation time for large systems	Moreira et al. (2022)
PV complementary capacity configuration	PSO-BP neural network hybrid approach	Reduced average system losses	Requires extensive training data	Liu et al. (2023b)
Multi-objective optimization enhancement	Hybrid MOPSO with spatial decomposition	Improved optimization efficiency	High resource consumption	Mohammad Zadeh and Mohagheghi (2023)
HVDC filter design optimization	Improved MO algorithm with 3D fuzzy evaluation	Effective configuration for ± 500 kV systems	Low generalizability	Shi et al. (2023)
Multi-target coverage optimization	MOEAD-GWO two-stage algorithm	Optimized city clustering and path planning	Not tested in dynamic scenarios	Khan et al. 2023)
Operation planning of isolated microgrids with hydrogen refueling stations and energy hubs	Three-level stochastic framework (MILP for HRS/energy hubs, MIQP for microgrids; Solved by CPLEX/GUROBI; SCENRED for scenario reduction)	Method efficient; battery/wind turbine impact on energy hub evaluated	Applicable only to isolated microgrids	Jordehi et al. (2024c)
HRS integration in interconnected electricity–gas network	Three-level model (piecewise linearization for gas flow; MILP for FCVs/HRSs/power–gas networks solved by CPLEX)	Power–gas network cost $760.37, $17.30 per FCV refuelling	Gas flow linearization leads to precision loss	Jordehi et al. (2024d)
Integration of electricity–gas grid hydrogen refueling stations	PSO–MOEAD hybrid algorithm with voltage compensation and thermal parameter optimization	–	–	This study

In summary, optimizing the performance of electrical equipment in power plants often involves nonlinear and uncertain factors, such as the uncertainty of load forecasting and the volatility of renewable energy. Traditional MOO algorithms lack effective processing mechanisms, which may result in slow convergence speed or inability to find feasible solutions. A new PSO–MOEAD is innovatively proposed to solve the MOO of electrical equipment performance in high-altitude photovoltaic power plants. An optimization model that comprehensively considers equipment performance, environmental factors, and economy is constructed, providing a theoretical basis for the design and operation of photovoltaic power plants.

The innovation of this research lies in: (1) PSO–MOEAD integrates the fast convergence strategy of PSO and the decomposition strategy of MOEAD, introduces the distance particle screening mechanism based on weight vectors, and through dynamic mutation and multi-subgroup collaborative search, overcomes the limitations of traditional algorithms under complex high-altitude constraints through dynamic mutation and multi-subgroup collaborative search, and outperforms existing single parameter tuning or fixed decomposition strategies. (2) A high-altitude exclusive quantitative model is constructed, which integrates the altitude characteristic voltage stability index and temperature radiation correction model to quantify the nonlinear effects of low air pressure and strong ultraviolet rays. A "zoned power generation–centralized grid connection" architecture and multi-node equivalent circuits are designed to adapt to complex network topologies. (3) The convergence speed in the standard test function far exceeds that of similar algorithms. In actual scenarios, it achieves a 12.3% reduction in power generation costs and a 15% reduction in network losses, with the lowest constraint violation rate. The optimization effect on the thermal performance of high-altitude equipment is outstanding, and the overall performance is superior to existing solutions.

The contribution of this paper lies in: (1) the proposed PSO–MOEAD hybrid algorithm combines the fast convergence of PSO and the decomposition strategy of MOEAD, overcoming the shortcomings of traditional algorithms that are prone to getting stuck in local optima or random initial weights. It outperforms existing algorithms in terms of stability and global optimization ability in complex constrained scenarios. (2) A high-altitude exclusive quantitative model is constructed. For the first time, the altitude characteristic voltage stability index and the temperature radiation correction model are integrated to precisely quantify the nonlinear effects of special environments such as low pressure and strong ultraviolet rays, solving the insufficient adaptability of other studies to high-altitude environments. (3) It performs outstandingly in terms of convergence speed, cost reduction, and network loss reduction. Meanwhile, the thermal performance of the equipment has been significantly improved, and its overall performance is superior to existing research, providing a more efficient solution for optimizing high-altitude photovoltaic equipment.

Methods and materials

Firstly, a detailed analysis is conducted on the working environment of photovoltaic power stations in high-altitude areas, and the main factors affecting the performance of electrical equipment are identified. Subsequently, an optimization model based on the PSO–MOEAD algorithm is constructed, including the objective function, constraint conditions, and algorithm parameter settings.

Performance optimization of electrical equipment for photovoltaic power plants in high-altitude areas

High-altitude areas usually have extreme environmental conditions such as low air pressure, low temperature, and high ultraviolet radiation. Relying solely on photovoltaic power stations to supply power to the distribution network not only increases the cost and difficulty of power supply, but also makes it more difficult to restore normal power supply when the distribution network is affected by external environmental factors and malfunctions. Therefore, optimizing the performance of electrical equipment in photovoltaic power plants is particularly necessary. Optimizing the performance of electrical equipment can effectively improve reliability and power generation efficiency, ensure stable operation in harsh environments, and provide solid guarantees for sustainable energy development and power supply in remote areas. The architecture of the photovoltaic power generation system in high-altitude areas is shown in Fig. 1.

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Fig. 1

Architecture of PV power generation system

The system adopts a "zone-based power generation and centralized grid connection" mode to divide the photovoltaic power generation system into multiple power generation units. Each photovoltaic power generation unit is arranged in a square shape. The inverter boosting equipment is centrally arranged, which has higher economic efficiency. The inter output power of a photovoltaic power station can be estimated by Eq. (1):

1

$$P_{\mathit{pv}}=P_{\mathit{stc}}\frac{R}{R_{\mathit{stc}}}\left[1+a_t(T-T_{\mathit{stc}})\right].$$

In Eq. (1), $P_{\mathit{pv}}$ signifies the output power of the power station under actual working conditions. $P_{\mathit{stc}}$ signifies the output power of the power station under standard testing conditions. $a_t$ signifies the temperature coefficient of the power station, reflecting the impact of temperature changes on output power. $R$ represents the actual intensity of solar radiation. $R_{\mathit{stc}}$ is the standard solar radiation intensity, usually set at 1000w/m2. $T$ is the current actual temperature. $T_{\mathit{stc}}$ is the temperature under standard testing conditions, usually set at 25 °C.

In distribution network analysis, load fluctuation is a key factor. The distribution of load $P_L$ over a certain period of time is approximated as a normal distribution (Kaur & Chaturvedi, 2024). The research hypothesis is that the mean ${\mu }_p$ represents the average load level, and the standard deviation ${\sigma }_p$ describes the range of load fluctuations. The probability density function $f\left(P_L\right)$ of load $P_L$ is displayed in Eq. (2):

2

$$f\left(P_L\right)=\frac{1}{\sqrt{2\pi {\sigma }_p}}exp\left[-\frac{{(P_L-{\mu }_p)}^2}{2{\sigma }_p^2}\right].$$

Based on this model, the load of distribution network in a specific period can be effectively predicted. In solar power generation, due to its uncertainty and intermittency, the output of distributed power sources will also exhibit uncertainty and intermittency (Varan et al., 2023). Figure 2 shows the equivalent circuit diagram of the photovoltaic grid connected distribution network. This diagram equates the system power supply and photovoltaic system to a series connection of voltage source and power impedance, while the transformer is equivalent to a series connection of impedance and distributed power supply.

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Fig. 2

Radiation distribution network and its equivalent circuit diagram

Before photovoltaic power generation is connected to the grid, the current in the distribution network flows directly from the substation to the users. However, the integration of photovoltaic power generation has made the flow of electricity more complex, not only changing the direction and magnitude of the power flow, but also potentially increasing network losses, which poses a challenge to the stability and reliability. Taking Fig. 2a as an example, before the photovoltaic power generation is connected to the grid, the active loss $P_{\mathit{loss}}$ of this branch can be calculated by Eq. (3) (Dey & Mallik, 2022):

3

$$P_{\mathit{loss}}=R_k\frac{P_i^2+Q_i^2}{U_i}.$$

In Eq. (3), $R_k$ is the line resistance. $P_i$ and $Q_i$ signify the active and reactive power of the node. $U_i$ represents the node voltage. After the photovoltaic power generation is connected, the net power of node $i$ and the loss of line $k$ based on the new power balance and loss calculation are evaluated, as displayed in Eq. (4):

4

$$\left\{\begin{array}{c}{S}_i^{\prime }=P_i^{\prime }+Q_i^{\prime }=(P_i-P_{\mathit{PV}})+j(Q_i-Q_{\mathit{PV}})\\ P_{\mathit{loss}}^{\prime }=\frac{R_k}{U_i^2}\left[{(P_i-P_{\mathit{PV}})}^2+{(Q_i-Q_{\mathit{PV}})}^2\right]\\ \end{array}\right).$$

In Eq. (4), the net power $S_i^{\prime }$ of node $i$ is composed of active power $P_i^{\prime }$ and reactive power $Q_i^{\prime }$. $Q_{\mathit{PV}}$ represents the reactive power injected into node $i$. $P_{\mathit{loss}}^{\prime }$ represents the active loss of line $k$. Based on the equivalent circuit diagram of the distribution network branch (Fig. 2b), the static voltage stability index $U_{sta,k}$ of line $k$ can be determined, as shown in Eq. (5):

5

$$U_{sta,k}=4{(P_j{X}_k-Q_j{R}_k)}^2+Q_j{X}_k+P_j{R}_k.$$

In Eq. (5), $X_k$ represents the reactance value of the $k$th branch. The static voltage stability index $U_{sta,k}$ is a key parameter with a range limited to 0 to 1. This indicator is used to measure the stability. When the indicator of all branches is below 1, it indicates that the system is running stably. On the contrary, if some branches in the system have this index close to or exceeding 1, it may indicate that the stability of the system is threatened. The temperature radiation correction model is used to correct the output power $P_{\text{actual}}$ of photovoltaic modules in high-altitude environments, as shown in Eq. (6) (Dovgilov et al., 2024):

6

$$\left\{\begin{array}{c}{P}_{\text{actual}}=P_{\text{STC}}·\frac{G}{G_{\text{STC}}}·[1-\gamma \left(T_{\text{cell}}-T_{\text{STC}}\right)]·{\eta }_{\text{UV}}\\ {\eta }_{\text{UV}}=1-\alpha ·\left(\frac{{\text{UV}}_{\text{actual}}}{{\text{UV}}_{\text{threshold}}}\right)\\ \end{array}\right).$$

In Eq. (6), $P_{\text{STC}}$ represents the output power under the standard test conditions (irradiance $G_{\text{STC}}$= 1000 W/m2, temperature $T_{\text{STC}}$= 25 ℃). $T_{\text{cell}}$ represents the actual temperature of the solar panel. $\gamma$ is the power temperature coefficient (usually − 0.4%/℃). ${\eta }_{\text{UV}}$ is the attenuation factor of ultraviolet radiation. $\alpha$ is the material degradation coefficient (with a value ranging from 0.05 to 0.1). ${\text{UV}}_{\text{actual}}$ is the actual measured intensity of ultraviolet radiation. ${\text{UV}}_{\text{threshold}}$ refers to the ultraviolet radiation threshold. This model quantifies the nonlinear influence of temperature and ultraviolet radiation on power output and incorporates them as constraint conditions into the optimization objective function.

In power systems, voltage stability is a key issue, especially in environments with high altitudes, low air pressure or large load fluctuations. Voltage sag is usually caused by insufficient reactive power. To maintain voltage stability, reactive power needs to be dynamically injected. Firstly, to make the actual voltage $V_i(t)$ as close as possible to the reference voltage $V_{\text{ref}}$, the voltage error is defined as $e(t)=V_{\text{ref}}-V_i(t)$. Then, based on the classical PI structure in control theory, the error $e(t)$ approaches zero. Therefore, the general form of the control output, that is, the reactive power compensation quantity $\mathrm{\Delta }Q$, is shown in Eq. (7):

7

$$\mathrm{\Delta }Q=K_p·e(t)+K_i·{\int }_0^{t}e(\tau )d\tau .$$

Finally, a distributed static reactive power generator (SVG) or a static synchronous compensator (STATCOM) is adopted. The voltage error $e(t)=V_{\text{ref}}-V_i(t)$ is substituted to obtain Eq. (8):

8

$$\mathrm{\Delta }Q=K_p·(V_{\text{ref}}-V_i)+K_i·\int (V_{\text{ref}}-V_i)dt.$$

In Eq. (8), $K_p=0.5$ and $K_i=0.1$ are control parameters, and $V_{\text{ref}}$ equals 1.0p.u. as the target voltage. Finally, the capacity allocation and switching timing of SVG/STATCOM are optimized through the PSO–MOEAD algorithm to minimize the compensation cost and network loss.

As for photovoltaic power plants, the performance optimization is to minimize the operating costs. This involves fine adjustment of the output power of the generator set to ensure the stability of the entire system while meeting all constraints within the system. To achieve these goals, a performance model of electrical equipment for photovoltaic power plants is constructed, as shown in Eq. (9). This model can be used to allocate and dispatch power resources more scientifically to optimize the overall economic benefits and system performance:

9

$$C=\sum _{i=1}^T\sum _{i=1}^N({\alpha }_i+{\beta }_i{P}_t+{\gamma }_i{P}_t^2).$$

In Eq. (9), the total power generation cost $C$ is determined by the cost coefficients ${\alpha }_i$, ${\beta }_i$, and ${\gamma }_i$ of each generator unit, as well as their actual output $P_t$ during the scheduling week $T$. To measure the stability of system power output, this study also calculates the total deviation of active power, which is the sum of the deviation between the actual active power of each node in the photovoltaic power station and the planned active power. Its expression is shown in Eq. (10):

10

$$min{f}_2={\sum }_{i=1}^n\left|P_{i,actual}-P_{i,plan}\right|.$$

In Eq. (10), $P_{i,actual}$ represents the actual active power of the node $i$. $P_{i,plan}$ represents the planned active power of the node. $n$ represents the total number of nodes. The output of the generator set must be between the preset upper and lower limits $P_i^{max}$ and $P_i^{min}$ to ensure that they operate within a safe and economical range. The upper and lower limits of the output of the generator set are constrained, as shown in Eq. (11):

11

$$P_i^{min}\le P_t\le P_i^{max}i\in N,t\in T.$$

In practical applications, to ensure the safe and reliable operation of the generator, a climbing rate that is too fast or too slow is not conducive to ensuring the safe and reliable operation of the generator. Therefore, the climbing rate needs to meet Eq. (12) (Bai et al., 2025):

12

$$\left\{\begin{array}{c}{P}_t-P_{i(t-1)}\le R_{u,i}i\in N,t\in T\\ P_{i(t-1)}-P_t\le R_{d,i}i\in N,t\in T\\ \sum _{i=1}^N{P}_t-P_{f,t}=0t\in T.\\ \end{array}\right)$$

In Eq. (12), $R_{u,i}$ and $R_{d,i}$ signify the upper and lower limits of the climbing rate, which limit the rate of change in output. $P_{f,t}$ signifies the total load at $t$. These measures collectively maintain the stability and safety of photovoltaic power plants. To meet the power flow constraints of the system, the Newton–Raphson method is adopted for power flow calculation. First, the grid-connected nodes of the photovoltaic power station are set as PQ nodes (specifying active power $P$ and reactive power $Q$). The main substation nodes are set as balance nodes (specifying node voltage $V$ and phase angle $\theta$). The nonlinear equation system is constructed based on Kirchhoff's law, as shown in Eq. (13):

13

$$\left\{\begin{array}{c}{P}_i=V_i\sum _{j=1}^N{V}_j(G_{\mathit{ij}}cos{\theta }_{\mathit{ij}}+B_{\mathit{ij}}sin{\theta }_{\mathit{ij}})\\ Q_i=V_i\sum _{j=1}^N{V}_j(G_{\mathit{ij}}sin{\theta }_{\mathit{ij}}-B_{\mathit{ij}}cos{\theta }_{\mathit{ij}})\\ \end{array}\right).$$

In Eq. (13), $G_{\mathit{ij}}$ and $B_{\mathit{ij}}$ are elements of the node admittance matrix, and ${\theta }_{\mathit{ij}}={\theta }_i-{\theta }_j$. The power flow constraint is integrated into the objective function through the penalty function method. If the bus voltage or line power exceeds the limit, the penalty term $F_{\text{penalty}}$ shown in Eq. (14) is added:

14

$$\left\{\begin{array}{c}{F}_{\text{penalty}}={\lambda }_V\sum {(V_i-V_{\text{lim}})}^2+{\lambda }_S\sum {(S_{\mathit{ij}}-S_{\mathit{ij}}^{\text{max}})}^2\\ 0.95\text{p}.\text{u}.\le V_i\le 1.05\text{p}.\text{u}.\\ S_{\mathit{ij}}\le S_{\mathit{ij}}^{max}\\ \sum P_{\text{gen}}=\sum P_{\text{load}}+P_{\text{loss}}\\ \end{array}\right).$$

In Eq. (14), $S_{\mathit{ij}}$ is the apparent power of the line. $P_{\text{gen}}$, $P_{\text{load}}$, and $P_{\text{loss}}$ represent the generating power, load power, and loss power, respectively. ${\lambda }_V$ and ${\lambda }_S$ are the penalty factors, set as 100 and 50, respectively.

Performance optimization of electrical equipment based on PSO–MOEAD algorithm

The previous section delves into the constituent units and system architecture of high-altitude distribution networks, and analyzes and quantifies the intermittent effects of photovoltaic power generation in high-altitude distribution networks. To achieve the optimal performance configuration of electrical equipment for photovoltaic power plants in high-altitude areas, the PSO–MOEAD algorithm is proposed. This algorithm cleverly decomposes complex MOO problems into several relatively simple single-objective optimization sub-problems, and simultaneously processes these sub-problems through aggregation strategies, thereby collaboratively optimizing multiple sub-problems. This method not only improves the solving efficiency, but also effectively finds the optimal solution, providing strong support for the stable operation and performance improvement of photovoltaic power generation in high-altitude areas.

In the application of traditional MOEAD, the randomness of the population initialization stage may result in a longer distance between the weight vector and the corresponding particles, as shown in the relationship between particle $x_1$ and its weight vector ${\lambda }_1$ in Fig. 3a. This increase in distance may prolong the iteration required for the algorithm to find the optimal solution. The PSO–MOEAD algorithm adopts a distance-based strategy during the population initialization phase. Specifically, it selects particles based on their distance from the weight vector, prioritizing those particles that are closer in distance.

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Fig. 3

Multi-objective weight summation

In the MOO, the process of generating weight vectors is crucial for determining the optimization direction. By uniformly generating these vectors, a set of evenly distributed reference points can be formed in the target space, with each point pointing to a unique optimization objective (Wang et al., 2022). This method ensures that the reference points can extensively cover the Pareto front, enabling the optimization process to comprehensively explore the solution space. Due to the uniform distribution of weight vectors, reference points based on neighborhoods will also be more evenly distributed on the Pareto front. Compared with reference points that rely on the entire population, neighborhood reference points are closer to the optimal solution of the current sub-problem, providing more accurate guidance for the population evolution (Dhandapani et al., 2023; Hao et al., 2022). This neighborhood oriented method helps the algorithm to more finely adjust the population to approximate the optimal solution.

In Fig. 3b, the reference point $h_i$ of each neighborhood is a vector $h_i={(h_i^1,...,h_i^m)}^T,i=1,2,...,n$ composed of its minimum values in each target dimension. $i$ represents the number of neighborhood numbers, $m$ signifies the number of objective functions, and $n$ signifies the size of the population. Correspondingly, the reference point $h_{}^{\ast }$ of the entire population is composed of the minimum objective function values in all dimensions, denoted as $h_{}^{\ast }=(h_1^{\ast },...,h_j^{\ast })_{}^T,j=1,2,...,m$. These reference points can be achieved through Eq. (15) (Yasuda et al., 2025):

15

$$\left\{\begin{array}{c}{h}_i^j=min\{f_j(\text{x})|x\in {\psi }_i^T\},j\in m,f_j(x)\in {\tau }_i^T\\ h_j^{\ast }=min\{f_j(x)|x\in \psi \},j\in m,f_j(x)\in \tau \\ \end{array}\right).$$

In Eq. (15), ${\psi }_i^T$ represents the decision space of particles in the neighborhood $L_i$. ${\tau }_i^T$ is the result of $f_j(\text{x})$ mapping the decision space ${\psi }_i^T$ to the target space. $\psi$ and $\tau$ represent the decision space and objective space of the entire population, respectively. $f_j(\text{x})$ signifies the $j$th objective function value. The PSO–MOEAD algorithm dynamically adjusts the reference points of particles through the reference points $h_i\left(i\in n\right)$ of each neighborhood and the reference points $h_{}^{\ast }$ of the entire population. If the reference point of a neighborhood is far away from the reference point, then the reference point of the entire population will be given greater weight. Conversely, it also holds true. This method optimizes the search behavior of PSO by balancing the influence of neighborhood and population reference points.

In addition, to address the uneven distribution of Pareto front solutions that may arise from relying solely on population reference points, the PSO–MOEAD algorithm employs a Tchebycheff aggregation method based on neighborhood reference points. This method helps improve the distribution of solutions, thereby enhancing the performance in MOO problems. The specific mathematical expression is shown in Eq. (16):

16

$$\left\{\begin{array}{c}g(x|\lambda ,r)=max\{{\lambda }_i|f_i(x)-r_i|\},i=1,2,...,n\\ r_i=\frac{(D_i-D_{min})h_{}^{\ast }}{D_{max}-D_{min}}+\frac{(D_{max}-D_i)h_i}{D_{max}-D_{min}},i=1,2,...,n\\ \end{array}\right).$$

In Eq. (16), the Euclidean distance between the reference point $h_i$ of the $i$th particle in its neighborhood and the reference point $h_{}^{\ast }$ of the entire population is $D_i=\left|h_i-h_{}^{\ast }\right|$. $D_{min}$ and $D_{max}$ represent the minimum and maximum distances between $h_i$ and $h_{}^{\ast }$, respectively. To evaluate the constraint violation of infeasible solutions in the population, the concept of constraint violation degree is applied, and the average constraint violation of particles is included as a penalty term in the objective function. When the constraint violation degree is zero, it means that all solutions in the population are feasible. The constraint violation of the $i$th particle is described by Eq. (17):

17

$$W_i^j=\left\{\begin{array}{c}max\{0,g_j(x)\},j=1,...,q,\hfill \\ max\{0,|h_j(x)-\mu |\},j=q+1,...,p\hfill \end{array}\right),i\in n.$$

In Eq. (17), $g_j(x)$ and $h_j(x)$ signify the $j$th inequality and equality constraint conditions, where $g_j(x)\le 0$ and $h_j(x)=0$. $\mu$ is the allowable error range for equality constraints. $p$ signifies the total number of constraints. $q$ signifies the inequality constraint. $p-q$ signifies the equality constraint. If $g_j(x)$ is a positive number or $|h_j(x)-\mu |$ is not equal to 0, it indicates that the solution is infeasible and there are unfulfilled constraints. When the $W_i^j$ is zero, it indicates that the particle is a feasible solution.

Aiming at the possible data loss of some nodes (such as local sensor failure or communication delay) in high-altitude photovoltaic power stations, the distributed Kalman filtering and spatio-temporal interpolation model are used to estimate the state of nodes that are not directly observed (such as remote inverters) through the voltage and power data of adjacent nodes and the grid topology relationship based on Eq. (18):

18

$$\widehat{V}_{_i^{}}^{}=\sum _{j\in \mathcal{N}(i)}{w}_{\mathit{ij}}(V_j+\mathrm{\Delta }{P}_{\mathit{ij}}{R}_{\mathit{ij}}).$$

In Eq. (18), $\mathcal{N}(i)$ is the neighbor set of the node, $w_{\mathit{ij}}$ is the weight coefficient, and $R_{\mathit{ij}}$ is the line resistance. To avoid PSO–MOEAD falling into local optimum, the dynamic mutation strategy and the collaborative search of multiple subgroups are introduced. When the population diversity is lower than the threshold, Gaussian variation is performed on 20% of the particles according to Eq. (19):

19

$$\left\{\begin{array}{c}{x}_i^{\text{new}}=x_i+\mathcal{N}(0,{\sigma }^2)\\ \sigma =0.1·(x_{\text{max}}-x_{\text{min}})\\ \end{array}\right).$$

In Eq. (19), $x_i$ represents the position or solution of the current particle. $x_i^{\text{new}}$ represents the position or solution of the new particle after the mutation operation. $\sigma$ is the variation step size. The population is divided into three subgroups, and different weight vector update strategies (global reference, neighborhood reference, and random exploration) are adopted, respectively, and the optimal solution is shared through the elite transfer mechanism.

In the PSO–MOEAD algorithm, the size of the external archive set is equal to the number of weight vectors, and the archive set is initially empty. As the algorithm progresses, particles in the population will undergo fitness evaluation and identify non-dominated solutions through non-dominated sorting. Taking dual objective optimization as an example, in the screening process shown in Fig. 4, all six particles are identified as non-dominated solutions. There is no mutual dominance relationship between them, that is, they are all strong non-dominated solutions. The dominant regions of these particles do not overlap with each other. For example, the dominant region of particle $x_1$ may be region A, while the dominant region of another particle $x_4$ may be region B. After the new particle is generated, the algorithm will evaluate each member in the external archive set to determine whether the new particle can dominate these members. If the new particle is able to dominate certain members in the archive set, these dominated members are removed, and the new particle is added to the archive set. On the contrary, if a new particle cannot dominate any members in the archive set and is itself dominated by some members in the archive set, the new particle will not be added to the archive set.

[See PDF for image]

Fig. 4

Dominance relationships among individuals in multi-objective evolutionary

The pseudocode of PSO–MOEAD is shown in Fig. 5. It clearly defines the complete process of the algorithm, from initializing weight vectors and populations, decomposing multi-objective problems, updating particles based on PSO strategy, introducing dynamic mutation and constraint processing, to updating external archives and reference points. By standardizing the operational logic of each link, the multi-objective optimization problem of high-altitude photovoltaic equipment has been solved. Finally, the Pareto optimal solution set is output.

[See PDF for image]

Fig. 5

PSO–MOEAD pseudocode

The PSO–MOEAD algorithm flow is shown in Fig. 6. Firstly, the parameters are initialized to generate a uniform weight vector, and initial particles are selected based on distance to construct a population and neighborhood. The multi-objective problem is decomposed into sub-problems through MOEAD. Within each sub-problem neighborhood, the particle velocity and position are updated according to the PSO strategy. The optimal solutions of individuals and neighborhoods are combined for optimization. When the population diversity is insufficient, Gaussian mutation is triggered, and the feasibility of handling constraint violations is calculated simultaneously. The external archive is dynamically updates, retaining non-dominated solutions and deleting dominated solutions, and controlling the scale through congestion. The neighborhood and global reference points are dynamically adjusted, iterated to the maximum number of times or to achieve the target rate of change, and then output the Pareto front in the archive.

[See PDF for image]

Fig. 6

Flowchart of PSO–MOEAD

Results

Based on the performance optimization model of electrical equipment in high-altitude photovoltaic power plants proposed in the study, Schaffer function, Rastrigin function, ZDT1 and ZDT2 are applied to assess the PSO–-MOEAD. The PSO–MOEAD is compared with other advanced MOO algorithms.

Standard testing function simulation

The control parameter settings of PSO–MOEAD comprehensively consider the balance between algorithm performance and adaptability to high-altitude photovoltaic scenarios. The specific parameters and settings are shown in Table 2. The population size is set to 100, which can not only avoid premature convergence caused by a too small population, but also prevent the increased computational burden due to an overly large scale, which is suitable for the search requirements of complex network topologies. The neighborhood size is set to 20 to ensure the coordination between local search and global exploration. The learning factors are c1 = c2 = 2.0, taking into account the influence of local optima and global information on particle update. The inertia weight is linearly reduced from 0.4 to 0.1. In the early stage, the global exploration ability is enhanced to cover a wider solution space. In the later stage, the convergence accuracy is improved to approach the optimal solution. The mutation probability is set to 0.1, and the adaptive Gaussian perturbation is triggered only when the population diversity is lower than the threshold to avoid the interference of invalid mutations on the convergence speed. The maximum number of iterations is 100, combined with a fitness change rate threshold of 10–4, to ensure optimization accuracy while controlling computational costs. The above parameter settings are verified in subsequent experiments by standard test functions (such as Schaffer, Rastrigin, and ZDT series).

Table 2. _{Overview of simulation environment and algorithm parameters}

	Parameters/configuration	Value/instructions
Software tools	Algorithm development platform	MATLAB R2022b
	Power grid simulation tool	DIgSILENT PowerFactory 2023
	Data analysis tools	Python 3.9 (NumPy, Pandas, Matplotlib)
Hardware configuration	CPU	Intel Xeon Gold 6248R
	GPU acceleration	NVIDIA A6000
Solver	Nonlinear optimization solver	MATLAB fmincon
	Mixed integer programming solver	Gurobi 10.0
Sampling quantity	The number of samples in Monte Carlo simulation	1000
Algorithm parameters	Population size	100
	Neighborhood size	20
	Learning factor	2
	Inertia weight	0.4 → 0.1
	Mutation probability	0.1
Convergence condition	Maximum number of iterations	100
	Threshold of fitness change rate	10−4

The various parameters in the performance optimization model for photovoltaic power stations in high-altitude areas are shown in Table 3. Numbers 1–5 are five-unit models, and numbers 1–10 are ten-unit models. The various parameters are mainly calibrated based on the long-term operation logs, unit performance test reports, and maintenance records of a certain photovoltaic power station's historical operation data. The climbing rate constraint is set in combination with the physical characteristics of the equipment and the dispatching requirements of the power grid. To protect data privacy, a hierarchical desensitization technique is adopted in the research. Firstly, the sensitive fields in the original data are generalized. Secondly, a differential privacy mechanism is adopted. During the iterative process of the optimization algorithm, controllable noise is injected into the objective function to ensure that the output results cannot be inferred back to specific devices or power stations, thereby enhancing the privacy protection capability while ensuring the model accuracy.

Table 3. Generator characteristic parameters and power upper and lower limits

Serial number	$P_i^{max}$(Mw)	$P_i^{min}$(Mw)	${\alpha }_i$($/h)	${\beta }_i$($/Mwh)	${\gamma }_i$(Mw2/h)	$R_{u,i}$(Mw/h)	$R_{d,i}$(Mw/h)
1	465	140	786.3584	37.3624	0.1498	90	90
2	465	125	458.3524	44.3281	0.1046	90	90
3	335	65	1047.9651	43.6218	0.0277	90	90
4	295	55	1248.3614	37.2519	0.0360	60	60
5	235	65	1659.2147	34.6302	0.0209	60	60
6	155	50	1353.6815	37.9240	0.0181	60	60
7	125	15	1459.3249	38.9227	0.0119	40	40
8	115	40	1454.6582	37.6399	0.0119	40	40
9	75	15	1450.9321	38.7283	0.1088	40	40
10	50	5	1469.2166	41.0327	0.1301	40	40

The Schaffer function and Rastrigin function are benchmark functions for testing the performance of optimization algorithms. By testing optimization algorithms on these benchmark functions, the algorithm performance on different types of problems can be evaluated, such as global search ability, local search ability, convergence speed, etc. All the algorithms involved in the comparison are evaluated for performance under the same number of function evaluations. In the experiment, the maximum number of iterations is uniformly set, and the number of function evaluations corresponding to each iteration remains consistent for all algorithms. This ensures that the performance comparison of different algorithms is based on fair benchmarks and avoids interference with the results due to differences in the number of functional evaluations.

As shown in Fig. 7, on the Schaffer function test, the PSO–MOEAD algorithm quickly approached the global optimal value within the first 20 iterations, while the MOEAD-AWA and MOPSO algorithms became stable after 40 iterations. This is because PSO–MOEAD combines the global search capability of PSO algorithm with the decomposition strategy of MOEAD, and dynamically adjusts the search direction by weight vector, avoiding the problem that traditional algorithms are prone to fall into local optimization in complex multi-modal functions.

[See PDF for image]

Fig. 7

Schaffer function test results

Figure 8 depicts the performance of four different algorithms on the Rastrigin function. MOEAD-AWA and MOPSO reached their optimal solutions after approximately 10 iterations in the early stages. The optimal value of the Rastrigin function should be 0. In fact, the MOEAD-AWA algorithm obtained the optimal value of 1.7914, while the MOPSO algorithm obtained 0.8433, indicating that these two algorithms may only have found local optimal solutions. The global optimal solution of PSO–MOEAD algorithm was 0, and its search speed was also the fastest, with the entire optimization process taking only 1.238 s. This result highlights the superiority of the PSO–MOEAD algorithm on global search capability and convergence speed. Compared to the other three algorithms, it demonstrates higher efficiency and accuracy in solving complex problems with multiple local optima, such as the Rastrigin function. PSO–MOEAD has a global optimal value of 0 on the Rastrigin function, while other algorithms only find local optima. This is because PSO–MOEAD enhances diversity through the Tchebycheff polymerization method of neighborhood reference points, avoiding premature convergence.

[See PDF for image]

Fig. 8

Test results of Rastrigin function

The PSO–MOEAD algorithm and MOBPSO algorithm were evaluated through three key performance indicators, convergence, diversity, and uniformity. These indicators were obtained through the two multi-objective testing functions, ZDT1 and ZDT2, as displayed in Table 4. Both on the ZDT1 and ZDT2 test functions, the PSO–MOEAD algorithm outperformed the MOBPSO algorithm on running time, convergence, uniformity, and diversity. This means that the PSO–MOEAD algorithm can not only achieve faster solutions when solving MOO problems, but also provide higher quality solutions. In addition, the PSO–MOEAD algorithm has demonstrated stronger ability in avoiding premature convergence, thereby improving the global search capability.

Table 4. Test values of convergence, uniformity, and diversity of ZDT1 and ZDT2

Index		ZDT1		ZDT2
		MOBPSO	PSO–MOEAD	MOBPSO	PSO–MOEAD
Running time (s)		8.512	8.026	15.101	8.247
Convergence	Best	2.29E−04	2.44E−05	3.19E−04	4.85E−04
	Mean	1.30E-03	1.30E−04	1.21E−03	1.45E−03
	Var	4.11E−06	4.09E−07	1.40E−06	1.27E−06
Uniformity		6.53E−04	6.51E−04	3.67E−04	3.49E−04
Diversity		0.70437	0.71285	0.71469	0.68572

Example verification

Figure 9 compares the computational efficiency of the PSO–MOEAD algorithm and the MOPSO algorithm on the five-unit model and the ten-unit model in the high-altitude PV case. The results showed that in both models, the calculation time of PSO–MOEAD algorithm was significantly higher than that of MOPSO algorithm, and the p-value was less than 0.01, indicating that the difference was statistically significant. Specifically, for the five-unit model, the median computation time of MOPSO algorithm was about 30 s, while that of PSO–MOEAD algorithm was about 60 s. For the ten-unit model, the MOPSO algorithm had a median computation time of about 100 s, while the PSO–MOEAD algorithm had a median computation time of about 150 s. This shows that the MOPSO algorithm is significantly superior to the PSO–MOEAD algorithm in computational efficiency, especially when dealing with more complex ten-unit models. Therefore, in application scenarios that require fast calculation results, MOPSO algorithm may be a better choice.

[See PDF for image]

Fig. 9

Comparison of computational efficiency of PSO–MOEAD in a high-altitude PV case (30 independent runs)

Table 5 compares the comprehensive performance of PSO–MOEAD, MOPSO, reference (Vigneshwar et al., 2024) and reference (Zhang et al., 2024), and evaluates their performance through multiple indicators. In terms of convergence speed, PSO–MOEAD was the fastest (45.5 s), followed by the method proposed in reference (Zhang et al., 2024) (78.2 s), and the MOPSO and the reference (Vigneshwar et al., 2024) were slower. In terms of voltage stability, the method proposed in reference (Vigneshwar et al., 2024) was the best (0.80%), and PSO–MOEAD was the worst (3.00%). In terms of cost reduction rate, PSO–MOEAD was the highest (12.30%), and the method mentioned in reference (Vigneshwar et al., 2024) was the lowest (5.60%). In terms of hypervolume, PSO–MOEAD algorithm had the highest (0.821), and other algorithms were lower than that of PSO–MOEAD algorithm. In terms of violation rate, PSO–MOEAD had the lowest (0.40%), while the method mentioned in reference (Vigneshwar et al., 2024) had the highest (3.70%). Overall, PSO–MOEAD has the best performance on convergence speed, cost reduction rate, hypervolume and constraint violation rate, while the method proposed in reference (Vigneshwar et al., 2024) has the best voltage stability. Therefore, PSO–MOEAD has the best comprehensive performance among these four algorithms.

Table 5. Comparative analysis of comprehensive performance of multiple algorithms

Index	PSO–MOEAD	MOPSO	Ref. Vigneshwar et al. (2024)	Ref. Zhang et al. (2024)
Rate of convergence	45.6 s	89.4 s (+ 96%)	132.7 s (+ 191%)	78.2 s (+ 71%)
Voltage stability	3.00%	1.20%	0.80%	1.50%
Cost reduction rate	12.30%	7.80%	5.60%	9.10%
Hypervolume	0.821	0.763 (-7.1%)	0.712 (-13.3%)	0.749 (-8.8%)
Constraint violation rate	0.40%	2.10%	3.70%	1.80%

Figure 10 shows the Pareto optimal solutions of the MOPSO and PSO–MOEAD algorithms under the five-unit and ten-unit models. Figure 10a shows the pre-Pareto comparison of the five-unit model. The Pareto pre-curves obtained by the three algorithms on the five-unit model indicate that the total deviation decreases as the power generation cost increases. The performance of the two algorithms is similar. However, within a range of power generation costs, the total active power deviation obtained by the PSO–MOEAD algorithm is slightly lower than that of the MOPSO algorithm and the MOEAD algorithm. Figure 10b shows the pre-Pareto comparison of the ten-unit model. In most power generation cost ranges, PSO–MOEAD achieved a lower total active power deviation than that of the MOPSO and the MOEAD. This indicates that in more complex models, PSO–MOEAD may have better performance.

[See PDF for image]

Fig. 10

Pareto optimal solution graph

Figure 11 shows the indicator change of the distribution network before and after applying PSO–MOEAD. From Fig. 11a, the voltage value before optimization fluctuated greatly and was below 0.95 in some periods. After PSO–MOEAD optimization, the average voltage value increased by about 0.01p.u., which reduced the adverse effects of voltage fluctuations and improved the efficiency and reliability of power transmission. This indicates that the optimized distribution network performs better in maintaining voltage stability. The Load Voltage Stability Index (LSPI) is an indicator used to measure the voltage stability of power systems, with lower values indicating better voltage stability. From Fig. 11b, the optimized LSPI was generally lower than that of before optimization, indicating a significant improvement in voltage stability.

[See PDF for image]

Fig. 11

Comparison of distribution network performance indicators

Figure 12 shows the changes in energy indicators during the operation of electrical equipment performance in photovoltaic power plants before and after optimization. From Fig. 12a, the optimized distribution network loss was lower than that of before optimization in most time periods, indicating that the optimization measures effectively reduced the loss of the distribution network. Especially around 15:00 and 22:00, the optimized loss was significantly reduced, which may be related to the optimization algorithm adjusting the operating parameters of the power grid. From Fig. 12b, the proposed method could effectively regulate the voltage when it deviated from the standard level, ensuring that it remained within an appropriate fluctuation range. The improvement of units 4 to 6 was particularly significant. The voltage of unit 6 increased by 0.0187, with an increase of 3%. Based on the proposed method, photovoltaic power plants can convert and distribute electricity more efficiently, while ensuring the stability and reliability.

[See PDF for image]

Fig. 12

Comparison of operation indicators of distribution network

The uncertainty components and the optimization results of thermal performance are shown in Table 6. After optimization, the surface temperature rise of the equipment was significantly reduced (with a peak reduction of 22.8%), and the heat dissipation efficiency was increased by 16.9%. Combined with the improved voltage stability index in Fig. 11b, it indicated that thermal parameter optimization indirectly improved the stability of the power grid by reducing equipment heat loss. Furthermore, the average annual maintenance frequency of thermal faults decreased by 64.3%, verifying the effectiveness of the optimized thermal stress distribution. To further verify the robustness of the model, this study also analyzed the uncertainty of key parameters. Specifically, for parameters such as solar radiation intensity, temperature coefficient and load fluctuation, 1000 groups of random samples are generated, respectively, within their typical distribution ranges. The PSO–MOEAD algorithm is used to optimize and solve each group of samples. The distribution characteristics of voltage stability, power generation cost, and network loss are statistically analyzed, and their standard uncertainty and extended uncertainty are calculated. It can be known from the table that irradiance fluctuation and temperature coefficient error have the most significant influence on voltage stability.

Table 6. Uncertainty components and thermal performance optimization results

	Parameters	Before optimization	After optimization	Rate of change
Optimization results of thermal parameters	Surface temperature rise of the equipment (℃)	32.5	25.1	-0.228
	Thermal resistance of insulating materials (K·m2/W)	0.85	0.68	-0.2
	Heat dissipation efficiency	72.4	84.6	0.169
	Uniformity index of thermal stress distribution	0.53	0.78	0.472
	Average annual maintenance frequency (times/year)	4.2	1.5	-0.643

		Sources of uncertainty	Standard uncertainty (%)	Expanded uncertainty (%)
Uncertainty analysis	Irradiance	Instrument calibration error	2.03	4.06
	Temperature coefficient	Accuracy of temperature probe	1.82	3.64
	Load fluctuation	Standard deviation of normal distribution	1.3	2.6
	Synthetic uncertainty	/	3.06	6.12

Table 7 shows the performance differences of the algorithms for optimizing power generation costs and network losses. In terms of power generation costs, PSO–MOEAD performed the best in both cost and efficiency with an optimal value of 78.3584 CNY/h, 45 iterations, and a time consumption of 45.6 s. Although the optimal value of MOPSO was 84.2011 CNY/h with a limited reduction, it still outperformed some reference algorithms. In terms of network loss, PSO–MOEAD achieved a maximum reduction of 15.0% with a per-unit value of 0.041, and demonstrated high efficiency with 38 iterations and a time consumption of 42.1 s. The MOPSO decreased by 9.8%, and both the optimization effect and efficiency were weaker than those of PSO–MOEAD.

Table 7. Comparison of optimization results of single-objective function

Optimization objective	Algorithm	Optimal value	Rate of decline from baseline (%)	Number of iterations	Computation time (s)
Power generation cost	PSO–MOEAD	78.3584 CNY/h	− 7.8	45	45.6
	MOPSO	84.2011 CNY/h	− 0.9	89	89.4
	Ref. Vigneshwar et al. (2024)	83.1274 CNY/h	− 2.2	120	132.7
	Ref. Zhang et al. (2024)	81.5923 CNY/h	− 4.0	75	78.2
Network loss	PSO–MOEAD	0.041 p.u	− 15.0	38	42.1
	MOPSO	0.046 p.u	− 9.8	72	85.3
	Ref. Vigneshwar et al. (2024)	0.048 p.u	− 6.3	110	128.5
	Ref. Zhang et al. (2024)	0.044 p.u	− 12.1	68	74.9

Discussion and conclusion

The PSO–MOEAD hybrid algorithm is proposed to optimize the performance of high-altitude photovoltaic power plant. From the standard test function and the actual case verification, the algorithm shows significant advantages. The experimental results showed that on the standard test function, the convergence speed of this algorithm was 71% higher than that of the traditional method (45.6 s faster), and it only found the global optimal solution (at 0) in the Rastrigin function. The global optimization ability is significantly better than that of the comparison algorithm. In actual high-altitude scenarios, the algorithm achieved a 3% improvement in voltage stability (an average voltage increase of 0.01p.u.), a 12.3% reduction in power generation costs, and a 15% decrease in network losses. The thermal performance of the equipment was optimized. The peak surface temperature rise was reduced by 22.8%, the heat dissipation efficiency was increased by 16.9%, the annual maintenance frequency of thermal faults was reduced by 64.3%, and the constraint violation rate was as low as 0.4% (the lowest among the comparison algorithms). In summary, PSO–MOEAD balances convergence speed, diversity, and stability by integrating dynamic weight screening and multi-node equivalent circuit models, providing an effective solution for optimizing high-altitude photovoltaic equipment. However, PSO–MOEAD improves the quality of the solution set and increases the computational burden through dynamic neighborhood reference points, external archive maintenance, and multi-objective aggregation strategies. To address the high computational burden on algorithms, in the future, lightweight improvement strategies (such as adaptive population size adjustment and parallel computing acceleration) can be explored to enhance real-time performance in large-scale photovoltaic power stations.

Acknowledgements

Not applicable.

Author contributions

Zhijun Xiao makes all the contributions related to this research.

Funding

Not applicable.

Data availability

All data generated or analyzed during this study are included in this article.

Declarations

Ethics approval and consent to participate

Not applicable.

Consent for publication

Not applicable.

Human participants

Not applicable.

Competing interests

The authors declare no competing interests.

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References

Anbarasi, MP; Kanthalakshmi, S. Power maximization in standalone photovoltaic system: An adaptive PSO approach. Soft Computing; 2023; 27, 12 pp. 8223-8232.

Bai, W; Xiaoping Liu, P; Wang, H. Fixed-time fault-tolerant control for power system with multiple actuator constraints. Asian Journal of Control; 2025; 27, 2 pp. 828-839.4880741

Dey, S; Mallik, A. Switching network loss minimization through multivariable modulation in a multiactive bridge converter. IEEE Transactions on Industrial Electronics; 2022; 70, 11 pp. 10833-10847.

Dhandapani, L; Shinde, SB; Wadhwa, L. A deep learning-based approach to optimize power systems with hybrid renewable energy sources. Electric Power Components and Systems; 2023; 51, 16 pp. 1740-1755.

Dovgilov, NL; Morozova, SP; Alekseev, SV; Dunaev, AY; Gavrilov, VR; Dmitriev, IY et al. Precision black body model at a temperature range of 200–450 K: Metrological assurance for optical-electronic equipment for remote sensing of the Earth in the infrared range of the spectrum. Measurement Techniques; 2024; 66, 11 pp. 872-878.

Hao, WK; Li, YP; Wang, JS; Zhu, Q. Solving economic load dispatch problem of power system based on differential evolution algorithm with different mutation strategies. IAENG International Journal of Computer Science; 2022; 49, pp. 156-165.

Jordehi, AR; Mansouri, SA; Tostado-Véliz, M; Hossain, MJ; Nasir, M; Jurado, F. Optimal placement of hydrogen fuel stations in power systems with high photovoltaic penetration and responsive electric demands in presence of local hydrogen markets. International Journal of Hydrogen Energy; 2024; 50, pp. 62-76.

Jordehi, AR; Mansouri, SA; Tostado-Véliz, M; Safaraliev, M; Hakimi, SM; Nasir, M. A tri-level stochastic model for operational planning of microgrids with hydrogen refuelling station-integrated energy hubs. International Journal of Hydrogen Energy; 2024; 96, pp. 1131-1145.

Jordehi, AR; Mansouri, SA; Tostado-Véliz, M; Sirjani, R; Safaraliev, M; Nasir, M. A three-level model for integration of hydrogen refuelling stations in interconnected power-gas networks considering vehicle-to-infrastructure (V2I) technology. Energy; 2024; 308, 132937.

Jordehi, AR; Tostado-Véliz, M; Mansouri, SA; Ahmarinejad, A; Ahmadi, A; Safaraliev, M et al. A two-stage stochastic framework for hydrogen pricing in green hydrogen stations including high penetration of hydrogen storage systems. Journal of Energy Storage; 2024; 100, 113567.

Kaur, P; Chaturvedi, KT. Economic power dispatching from distributed generations: Review of optimization techniques. Energy Engineering; 2024; 121, 3 pp. 557-579.

Kessler-Lewis, E; Polly, SJ; Sacchitella, E; Hubbard, SM; Hoheisel, R. Demonstration of a monolithically integrated hybrid device for simultaneous power generation and data modulation. IEEE Journal of Photovoltaics; 2024; 14, 2 pp. 272-280.

Khan, I; Basuli, K; Maiti, MK. Multi-objective covering salesman problem: A decomposition approach using grey wolf optimization. Knowledge and Information Systems; 2023; 65, 1 pp. 281-339.

Khunkitti, S; Siritaratiwat, A; Premrudeepreechacharn, S. A many-objective marine predators algorithm for solving many-objective optimal power flow problem. Applied Sciences; 2022; 12, 22 11829.

Li, D; Wang, L; Li, L; Guo, W; Wu, Q; Lerch, A. A large-scale multiobjective particle swarm optimizer with enhanced balance of convergence and diversity. IEEE Transactions on Cybernetics; 2022; 54, 3 pp. 1596-1607.

Liu, Q; Xu, G; Li, Z; Gao, Y; Yang, Y; Li, Y et al. A rapid prediction model of photovoltaic power generation for autonomous long-duration aerostat. IET Renewable Power Generation; 2023; 17, 6 pp. 1597-1608.

Liu, Y; Zhang, Z; Ma, L; Jia, Y; Yin, W; Liu, Z. Short-term prediction of photovoltaic power based on DBSCAN-SVM data cleaning and PSO-LSTM model. Energy Engineering; 2024; 121, 10 pp. 3019-3035.

Liu, Z; Liu, H; Zhang, D. PSO-BP-based optimal allocation model for complementary generation capacity of the photovoltaic power station. Energy Engineering; 2023; 120, 7 pp. 1717-1727.

Makhloufi, S; Mekhilef, S. Logarithmic PSO-based global/local maximum power point tracker for partially shaded photovoltaic systems. IEEE Journal of Emerging and Selected Topics in Power Electronics; 2021; 10, 1 pp. 375-386.

Mohammad Zadeh, P; Mohagheghi, M. Enhanced decomposition-based hybrid evolutionary and gradient-based algorithm for many-objective optimization. Applied Intelligence; 2023; 53, 24 pp. 30497-30522.

Moreira, A; Molina, Y; Aquino, R; Ñaupari, Z. Allocation and sizing of photovoltaic systems to reduce power losses and economic aspects using a new PSO approach. IEEE Latin America Transactions; 2022; 20, 6 pp. 977-985.

Naik, A. Multi-objective social group optimization for machining process. Evolutionary Intelligence; 2024; 17, 3 pp. 1655-1676.

Shi, F; Wang, H; Lu, T; Huang, J; Wang, C. Multi-objective optimal design for AC filters in high voltage direct current system based on improved multi-objective evolutionary algorithm based on decomposition. High Voltage; 2023; 8, 3 pp. 492-503.

Varan, M; Erduman, A; Menevşeoğlu, F. A grey wolf optimization algorithm-based optimal reactive power dispatch with wind-integrated power systems. Energies; 2023; 16, 13 pp. 1-30.

Vigneshwar, AS; Shunmugalatha, A; Madhan, A. Power quality enhancement in hybrid sustainable energy systems grid-connected scheme by modified non-dominated sorting genetic algorithm. Journal of Electrical Engineering & Technology; 2024; 19, 4 pp. 2029-2046.

Wang, B; Zhang, P; He, Y; Wang, X; Zhang, X. Scenario-oriented hybrid particle swarm optimization algorithm for robust economic dispatch of power system with wind power. Journal of Systems Engineering and Electronics; 2022; 33, 5 pp. 1143-1150.

Wichitkrailat, K; Premrudeepreechacharn, S; Siritaratiwat, A; Khunkitti, S. Optimal sizing and locations of multiple BESSs in distribution systems using crayfish optimization algorithm. IEEE Access; 2024; 12, pp. 94733-94752.

Yasuda, Y; Kumagai, W; Tamura, K; Yasuda, K. A multi-objective evolutionary algorithm based on bilayered decomposition for constrained multi-objective optimization. IEEJ Transactions on Electrical and Electronic Engineering; 2025; 20, 2 pp. 244-262.

Zhang, D; Cui, X; Xu, C; Lv, S; Zhao, L. Multi-objective interval planning for 5G base station virtual power plants considering the consumption of photovoltaic and communication flexibility. IET Smart Grid; 2024; 7, 6 pp. 800-811.