DynaG Algorithm-Based Optimal Power Flow Design for Hybrid Wind–Solar–Storage Power Systems Considering Demand Response

Abstract

With a high proportion of renewable energy sources connected to the distribution network, traditional optimal power flow (OPF) methods face significant challenges including multi-objective co-optimization and dynamic scenario adaptation. This paper proposes a dynamic optimization framework based on the Dynamic Gravitational Search Algorithm (DynaG) for a multi-energy complementary distribution network incorporating wind power, photovoltaic, and energy storage systems. A multi-scenario OPF model is developed considering the time-varying characteristics of wind and solar penetration (low/medium/high), seasonal load variations, and demand response participation. The model aims to minimize both network loss and operational costs, while simultaneously optimizing power supply capability indicators such as power transfer rates and capacity-to-load ratios. Key enhancements to DynaG algorithm include the following: (1) an adaptive gravitational constant adjustment strategy to balance global exploration and local exploitation; (2) an inertial mass updating mechanism constrained to improve convergence for high-dimensional decision variables; and (3) integration of chaotic initialization and dynamic neighborhood search to enhance solution diversity under complex constraints. Validation using the IEEE 33-bus system demonstrates that under 30% penetration scenarios, the proposed DynaG algorithm reduces capacity ratio volatility by 3.37% and network losses by 1.91% compared to non-dominated sorting genetic algorithm III (NSGA-III), multi-objective particle swarm optimization (MOPSO), multi-objective atomic orbital search algorithm (MOAOS), and multi-objective gravitational search algorithm (MOGSA). These results show the algorithm’s robustness against renewable fluctuations and its potential for enhancing the resilience and operational efficiency of high-penetration renewable energy distribution networks.

1. Introduction

Optimal power flow (OPF) has long been recognized as a fundamental tool for the economic operation of power systems [1,2]. Its primary objective is to achieve the lowest-cost operating strategy through centralized optimization, while satisfying the key system requirements such as power balance, voltage stability, and load constraints. Traditional OPF models were developed predicated on a solitary fossil energy system, and its steady-state optimization framework has been extensively validated as an efficacious instrument for the economic dispatch of power systems in historical development [3]. However, with the rapid advancement of the global energy transition and the increasing penetration of renewable energy sources, the operational characteristics of the distribution networks have undergone significant changes [4]. The co-optimization of current and supply capacity in a high-penetration multi-energy complementary distribution network is a challenging endeavor that faces a multitude of difficulties [5]. These include conflicts among multiple objectives, high-dimensional nonlinear constraints, and the need for real-time coordination [6,7]. Furthermore, the intermittent nature of renewable energy sources necessitates coordination across temporal domains [8]. The cross-energy coupling of wind and solar photovoltaic (PV) systems further exacerbates the modeling complexity, impeding the balance between dynamic equilibrium and computational efficiency in conventional methods.

Traditional power system planning methodologies face dual challenges in high-penetration renewable scenarios. At the modeling level, conventional approaches predominantly adopt fragmented strategies for renewable penetration rate optimization [9]. For instance, the literature demonstrates discrete optimization of power sources and grid structures, yet neglects the dynamic coupling relationships between these elements [10]. This leads to frequent grid expansion adjustments under renewable fluctuations, as evidenced by the literature’s findings on flexibility reserve deficiencies during extreme weather events. Specifically, fluctuations in wind/solar output increased voltage limit violations by 37%, directly causing OPF model convergence failures [11]. At the algorithmic level, existing multi-objective optimization methods exhibit critical limitations. For example, the non-dominated sorting genetic algorithm (NSGA-II) is reliant upon static crowding distance for the purpose of diversity maintenance, a factor that has been demonstrated to result in Pareto front clustering in high-dimensional spaces, while the particle velocity update mechanism of multi-objective particle swarm optimization (MOPSO) fails to filter noise interference, resulting in ±5% penetration rate confidence interval fluctuations under volatile wind–solar conditions [12]. Moreover, the multi-objective grey wolf optimizer (MOGWO) suffers from a rigid social hierarchy, which has been experimentally shown to reduce swarm diversity by 40%, thereby accelerating premature convergence in complex, target-coupled scenarios. Relevant works (e.g., ref. [13] on NSGA-III’s improved variants, ref. [14] on MOGWO’s adaptive hierarchy, ref. [15] on GSA-based hybrid optimizers) further validate these limitations and explore mitigation strategies for high-renewable OPF scenarios. Collectively, these limitations undermine the robustness, scalability, and engineering applicability of conventional power system planning methodologies in high-renewable contexts [16,17].

To address these challenges, this paper proposes a Dynamic Gravitational Search Algorithm (DynaG)-based framework for optimizing distribution networks under high renewable energy penetration. The framework is designed to tackle three core challenges: (1) dynamic multi-scenario adaptation, (2) high-dimensional objective coordination, and (3) algorithm robustness enhancement [18,19]. A multi-scenario OPF model is developed, incorporating the characteristics of wind/PV penetration rates, seasonal load patterns, and demand response participation [20]. With the dual objectives of minimizing network losses and optimizing operational costs, the framework simultaneously optimizes power supply capability indicators including transfer rates (TR) and capacity–load ratio (CLR) while taking battery energy storage system (BESS) into account for optimization [21,22]. The DynaG algorithm is enhanced through several key mechanisms: (1) a dynamic gravitational constant adaptive adjustment mechanism to balance global exploration and local exploitation capabilities; (2) an inertia mass updating strategy constrained to improve convergence precision of high-dimensional decision variables; and (3) integration of chaos initialization and dynamic neighborhood search algorithms to enhance the diversity and distribution characteristics of Pareto-optimal solutions under complex constraints. Compared to MOGWO’s fixed social hierarchy, DynaG’s dynamic neighborhood search mechanism adaptively balances the exploration capability through a radius decay factor λ₂ to avoid entering a local optimum.

The main contributions and innovations of this paper are as follows:

• By simulating celestial gravitational interactions, particle masses are dynamically correlated with multi-objective fitness metrics. Guided by gravitational acceleration, this mechanism accelerates Pareto frontier exploration, resolving convergence inefficiencies caused by multi-objective conflicts in traditional algorithms.
• The integration of the anti-noise mechanism with exponentially decaying gravitational constants, in conjunction with the sequential increase in renewable penetration from 10% to 30%, has been demonstrated to be effective in mitigating the sensitivity to fluctuating wind and solar output. This is achieved through the utilization of a sample-simulated renewable energy fluctuation model.
• This study addresses the planning redundancy problem caused by the segmentation of renewable energy penetration modeling in traditional power system planning. A dynamic coupled optimization mechanism is introduced to reflect real-world conditions, incorporating an optimal tidal current solution while accounting for the interactions between tidal quality indicators, wind and solar generation, and energy storage.

The remainder of this paper is structured as follows:

Section 1 introduces the operational challenges of high-penetration renewable energy in distribution networks and proposes the DynaG algorithm; Section 2 develops a multi-scenario OPF model integrating time-varying demand response and BESS dynamics with dual objectives of minimizing network loss and operational costs; Section 3 details the DynaG algorithm’s core innovations—chaotic initialization, adaptive gravitational adjustment, and dynamic neighborhood search—to enhance solution diversity under complex constraints; Section 4 validates the approach through seasonal case studies on the IEEE 33-bus system, demonstrating DynaG’s superior performance in voltage stability, load fluctuation control, and economic efficiency compared to NSGA-III, MOPSO, MOAOS, and MOGSA under 30% renewable penetration; and Section 5 concludes with the algorithm’s robustness in improving grid resilience against renewable uncertainties.

2. Multi-Objective Demand Side Response and Energy Storage Modeling

With the transformation of the power system to an increasing penetration of renewable energy and the growing participation of flexible user-side resources, the coordinated optimization of demand response (DR) and energy storage system has become a critical strategy to improve system flexibility [23,24], reduce operating costs, and ensure the reliability of power supply. This study develops an integrated modeling framework encompassing the entire chain of “demand response—energy storage regulation—system optimization”. The proposed model addresses the limitations of traditional approaches, such as linear assumption and constant efficiency, and proposes a refined, multi-objective modeling system that considers the economic efficiency, technical constraints, and reliability requirements. These traditional limitations are twofold: (1) conventional OPF models rely on linear assumptions for the DR load–price interaction, which cannot capture the nonlinear correlation between time-of-use (TOU)/real-time pricing (RTP) signals and user load adjustment; and (2) they treat energy storage charge–discharge efficiency as constant, neglecting dynamic efficiency variations in BESS in practical operation.

2.1. Demand Response

OPF is a fundamental optimization tool for the economic and safe operation of power systems [25,26]. Its objective is to minimize the system operating cost or network loss by adjusting control variables such as generator output, transformer ratio, and reactive power compensation [27] under the premise of meeting the physical constraints of the power grid. Conventional OPF models are based on nodal power balance equations and incorporate constraints such as generator output limit, line transmission capacities, and voltage magnitude constraints, resulting in a nonlinear programming problem [28]. Detailed formulations of the objective function and constraints can be found in the relevant literature [29].

Demand response is a key mechanism for guiding users to adjust their electricity consumption behavior and achieve spatial and temporal load shifting [30]. The foundation of DR modeling lies in the precise quantification of the nonlinear relationship between the electricity price signals and the user load response, while considering demand-side dynamic boundary on the demand side. This modeling framework supports the coordinated optimization of energy storage and demand-side flexibility.

The time-of-use (TOU) and real-time pricing (RTP) structure is employed in this study to model DR, using dynamic tariffs defined as follows:

$$\left\{\begin{array}{c}{e}_{\mathrm{p}}(t)=e_0(1+{\alpha }_{\mathrm{p}}),t\in T_{\mathrm{p}}\\ e_{\mathrm{f}}(t)=e_0(1+{\beta }_{\mathrm{f}}),t\in T_{\mathrm{f}}\\ e_{\mathrm{v}}(t)=e_0(1+{\gamma }_{\mathrm{v}}),t\in T_{\mathrm{v}}\end{array}\right.$$

(1)

where $e_{\mathrm{p}}(t)$, $e_{\mathrm{f}}(t)$, and $e_{\mathrm{v}}(t)$ represent the dynamic tariffs at peak, flat, and valley periods, respectively; $e_0$ is the base tariff; ${\alpha }_{\mathrm{p}}$, ${\beta }_{\mathrm{f}}$, ${\gamma }_{\mathrm{v}}$ are the amplitude parameters, which are used to regulate the peak-valley price difference; and $T_{\mathrm{p}}$, $T_{\mathrm{f}}$, and $T_{\mathrm{v}}$ are the time-domain parameters, which are used to determine the period by fitting the seasonal load curves.

In high-renewable penetration scenarios, DR plays a critical role by reshaping the load curve in response to price signals or incentive mechanisms. In this paper, the user load response $\Delta P_D$ is modeled in Equation (2). It is explicitly incorporated into the OPF optimization variables, enabling a tighter coupling between load flexibility and renewable energy variability.

$$\Delta P_D(t)={\alpha }_{\mathrm{T}\mathrm{O}\mathrm{U}}(t)\cdot P_{\mathrm{base}}(t)+{\beta }_{\mathrm{T}\mathrm{O}\mathrm{U}}(t)\cdot \Delta \rho (t)$$

(2)

where ${\alpha }_{\mathrm{T}\mathrm{O}\mathrm{U}}(t)$ denotes TOU price elasticity coefficient matrix, which captures the long-term impacts of peak-valley price differences on load demands, reflecting how sustained variations in time-differentiated tariffs shape users’ habitual electricity consumption patterns across different periods; ${\beta }_{\mathrm{T}\mathrm{O}\mathrm{U}}(t)$ represents the real-time price elasticity coefficient, quantifying the short-term regulatory capability of instantaneous price fluctuations on loads, enabling dynamic adjustments of flexible resources in response to minute-level price changes; $P_{\mathrm{base}}(t)$ stands for the baseline load profile, serving as the reference trajectory of natural load demands under a no-price-intervention scenario, typically derived from historical data or typical load patterns to isolate the effects of exogenous factors; and $\Delta \rho (t)$ is the price increment signal, defined as the deviation between the real-time tariff $\rho (t)$ and the baseline price, acting as the direct input to drive load variations through interactions with the elasticity coefficients, thereby bridging the gap between price policies and load responses.

The charging and discharging behavior of energy storage is characterized by energy balance equations with power constraints:

$$\left\{\begin{array}{c}{E}_{SOC}(t)=E_{SOC}(t-1)+{\eta }_{\mathrm{c}\mathrm{h}}\cdot P_{\mathrm{BESS}}^{\mathrm{c}\mathrm{h}}(t)\cdot \Delta t-{\displaystyle \frac{P_{\mathrm{BESS}}^{\mathrm{dis}}(t)\cdot \Delta t}{{\eta }_{\mathrm{dis}}}}\\ E_{\mathrm{S}\mathrm{O}\mathrm{C}}^{\min }\le E_{SOC}(t)\le E_{\mathrm{S}\mathrm{O}\mathrm{C}}^{\max }\end{array}\right.$$

(3)

$$P_{\mathrm{BESS}}^{\mathrm{c}\mathrm{h}}(t)\le P_{\mathrm{B}\mathrm{E}\mathrm{S}\mathrm{S}}^{\mathrm{m}\mathrm{a}\mathrm{x}},P_{\mathrm{BESS}}^{\mathrm{dis}}(t)\le P_{\mathrm{B}\mathrm{E}\mathrm{S}\mathrm{S}}^{\mathrm{m}\mathrm{a}\mathrm{x}}$$

(4)

where $E_{SOC}(t)$ denotes state of charge (SOC) at time $t$, representing the ratio of the remaining energy stored in ESS to its rated capacity; $E_{SOC}(t-1)$ is SOC at the previous time step $(t-1)$, enabling recursive state updates; ${\eta }_{\mathrm{c}\mathrm{h}}$ and ${\eta }_{\mathrm{dis}}$ are the charging and discharging efficiencies, respectively, quantifying energy losses during power conversion processes; $P_{\mathrm{BESS}}^{\mathrm{c}\mathrm{h}}(t)$ and $P_{\mathrm{BESS}}^{\mathrm{dis}}(t)$ represent the charging and discharging powers of ESS at time $t$, with positive/negative signs indicating charging/discharging modes; $\Delta t$ is the time interval between consecutive time steps, facilitating energy calculation from power values; $E_{\mathrm{S}\mathrm{O}\mathrm{C}}^{\min }$ and $E_{\mathrm{S}\mathrm{O}\mathrm{C}}^{\max }$ define the lower and upper bounds of SOC, respectively, to prevent over-discharging and over-charging; and $P_{\mathrm{B}\mathrm{E}\mathrm{S}\mathrm{S}}^{\mathrm{m}\mathrm{a}\mathrm{x}}$ specifies the maximum allowable magnitude of both charging and discharging powers, safeguarding against equipment overload.

Energy storage reduces the system operation cost directly by transferring the surplus electricity from high-price periods to low-price periods through the “low storage and high discharge” strategy [31]. Meanwhile, the rapid response capability of energy storage can smooth out the fluctuations in the output of wind and solar power, relieve line congestion, and indirectly reduce network losses.

2.2. Objective Function and Decision Variables

The decision variables include integer variables for BESS installation nodes $x_n\in N_{\mathrm{nodes}}$, BESS rated capacity $E_{\mathrm{rated},\mathrm{BESS}}$, BESS charge/discharge power $P_{\mathrm{ch},\mathrm{BESS}}(t)$, $P_{\mathrm{di},\mathrm{BESS}}\left(t\right),$ and BESS of SOC across four BESS units.

The lower-layer function minimizes the levelized lifecycle cost (LCC):

$$\mathrm{m}\mathrm{i}\mathrm{n}{F}_1={\displaystyle \frac{V_{\mathrm{I}\mathrm{C}\mathrm{C},\mathrm{B}\mathrm{E}\mathrm{S}\mathrm{S}}+V_{\mathrm{M}\mathrm{C},\mathrm{B}\mathrm{E}\mathrm{S}\mathrm{S}}+V_{\mathrm{O}\mathrm{C},\mathrm{B}\mathrm{E}\mathrm{S}\mathrm{S}}+V_{\mathrm{R}\mathrm{C},\mathrm{B}\mathrm{E}\mathrm{S}\mathrm{S}}}{52}}$$

(5)

$$\left\{\begin{array}{c}\\ V_{\mathrm{M}\mathrm{C},\mathrm{B}\mathrm{E}\mathrm{S}\mathrm{S}}={\delta }_{\mathrm{C}\mathrm{R}\mathrm{F},\mathrm{B}\mathrm{E}\mathrm{S}\mathrm{S}}\left({ϵ}_{\mathrm{M}\mathrm{C},\mathrm{b}\mathrm{a}}{\sigma }_{\mathrm{b}\mathrm{a}}{E}_{\mathrm{B}\mathrm{E}\mathrm{S}\mathrm{S},i}+{ϵ}_{\mathrm{M}\mathrm{C},\mathrm{i}\mathrm{n}}{\sigma }_{\mathrm{i}\mathrm{n}}{P}_{\mathrm{B}\mathrm{E}\mathrm{S}\mathrm{S},i}\right)\hfill \\ V_{\mathrm{O}\mathrm{C},\mathrm{B}\mathrm{E}\mathrm{S}\mathrm{S}}=52{\sum }_{t=1}^T\left[e_{\mathrm{s}\mathrm{e}\mathrm{l}\mathrm{l}}\left(t\right)P_{\mathrm{BESS}}^{\mathrm{c}\mathrm{h}}\left(t\right)-e_{\mathrm{p}\mathrm{u}\mathrm{r}}(t)P_{\mathrm{BESS}}^{\mathrm{dis}}(t)\right]\hfill \\ V_{\mathrm{R}\mathrm{C},\mathrm{B}\mathrm{E}\mathrm{S}\mathrm{S}}={\delta }_{\mathrm{C}\mathrm{R}\mathrm{F},\mathrm{B}\mathrm{E}\mathrm{S}\mathrm{S}}{\sum }_{g_{\mathrm{b}\mathrm{a}}=1}^{R_{\mathrm{b}\mathrm{a}}}{\displaystyle \frac{(1-{\alpha }_{\mathrm{B}\mathrm{E}\mathrm{S}\mathrm{S}}{)}^{{\sigma }_{\mathrm{b}\mathrm{a}}{T}_{\mathrm{b}\mathrm{a}}}}{(1+d{)}^{{\sigma }_{\mathrm{b}\mathrm{a}}{T}_{\mathrm{b}\mathrm{a}}}}}\hfill \end{array}\right.$$

(6)

$${\delta }_{\mathrm{C}\mathrm{R}\mathrm{F},\mathrm{B}\mathrm{E}\mathrm{S}\mathrm{S}}={\displaystyle \frac{d}{{(1+d)}^{T_{\mathrm{B}\mathrm{E}\mathrm{S}\mathrm{S}}}-1}}+d$$

(7)

where $V_{\mathrm{I}\mathrm{C}\mathrm{C},\mathrm{B}\mathrm{E}\mathrm{S}\mathrm{S}}$, $V_{\mathrm{M}\mathrm{C},\mathrm{B}\mathrm{E}\mathrm{S}\mathrm{S}}$, $V_{\mathrm{O}\mathrm{C},\mathrm{B}\mathrm{E}\mathrm{S}\mathrm{S}}$, and $V_{\mathrm{R}\mathrm{C},\mathrm{B}\mathrm{E}\mathrm{S}\mathrm{S}}$ correspond to the core expenditure in the construction period, operation and maintenance period, operation period, and decommissioning period, respectively; ${\sigma }_{\mathrm{b}\mathrm{a}}$ represents the battery cell cost; ${\sigma }_{\mathrm{i}\mathrm{n}}$ denotes the converter cost; $E_{\mathrm{B}\mathrm{E}\mathrm{S}\mathrm{S},i}$ and $P_{\mathrm{B}\mathrm{E}\mathrm{S}\mathrm{S},i}$ are the rated capacity and rated charge/discharge power of the $i$-th BESS unit, respectively; $N_{\mathrm{B}\mathrm{E}\mathrm{S}\mathrm{S}}$ indicates the total number of installed BESS units; ${ϵ}_{\mathrm{M}\mathrm{C},\mathrm{b}\mathrm{a}}$ and ${ϵ}_{\mathrm{M}\mathrm{C},\mathrm{i}\mathrm{n}}$ represent the proportions of battery maintenance and converter maintenance relative to the initial investment, respectively; $e_{\mathrm{s}\mathrm{e}\mathrm{l}\mathrm{l}}$ and $e_{\mathrm{p}\mathrm{u}\mathrm{r}}$ are the electricity selling price and electricity purchasing price, respectively; $R_{\mathrm{b}\mathrm{a}}$ denotes the battery replacement frequency; $T_{\mathrm{b}\mathrm{a}}$ and ${\alpha }_{\mathrm{B}\mathrm{E}\mathrm{S}\mathrm{S}}$ represent the single battery replacement time and battery cost decline rate, respectively; ${\sigma }_{\mathrm{b}\mathrm{a}}$ is the number of battery replacements; $T_{BESS}$ is the design life of the BESS; and $d$ shows the discount rate.

The minimum value of voltage fluctuation is calculated by minimizing the root mean square error (RMSE), and its mathematical expression is as follows:

$${\min F}_2=\sqrt{{\displaystyle \frac{1}{N_{\mathrm{nodes}}T}}{\sum }_{i=1}^{N_{\mathrm{nodes}}}{\sum }_{t=1}^{\mathrm{T}}{\left(V_i(t)-{\overline{V}}_i\right)}^2}$$

(8)

where $N_{\mathrm{nodes}}$ is the number of nodes; $V_i$ is the predicted value of the $i$-th node at time $t$; ${\overline{V}}_i$ is the actual value of the $i$-th node; and $\mathrm{T}$ is 168 h.

Equation (8) uses the RMSE of node voltage deviation to quantify and minimize voltage fluctuations because RMSE preserves the dimensional consistency of voltage (in p.u.) and amplifies the weight of larger deviations, which is consistent with the goal of ensuring voltage stability in the OPF model. Unlike mean square error, RMSE more intuitively reflects the actual magnitude of voltage deviation at nodes and over time, making it a more practical indicator for evaluating and optimizing the voltage quality of wind–solar complementary systems.

The multi-objective weighting in this study adopts an adaptive rule that dynamically adjusts weights based on seasonal scenario characteristics—for example, increasing the weight of minimizing voltage fluctuation in winter high-load periods and prioritizing operational cost optimization in high-renewable-penetration seasons—to align with the hybrid wind–solar–storage OPF model’s practical operational demands.

2.3. Optimization Model

The proposed optimization model integrates three-layered objective framework: economic dispatch, load regulation, and power supply capability enhancement, while explicitly considering multi-energy complementarity and grid resilience [32]. The core OPF constraints are retained, with extension to incorporate renewable energy intermittency and demand-side elasticity.

(1)

Power Balance Constraint

The fundamental power balance at each time step t is expressed as follows:

$$P_{\mathrm{grid}}(t)+P_{\mathrm{BESS}}(t)+P_{\mathrm{P}\mathrm{V}}(t)+P_{\mathrm{WT}}(t)=P_{\mathrm{load}}(t)$$

(9)

where $P_{\mathrm{grid}}(t)$ denotes the power exchanged with the external power grid at time $t$, indicating whether power is imported from or exported to the grid; $P_{\mathrm{BESS}}(t)$ stands for the power output or input of BESS, reflecting its charging or discharging status; $P_{\mathrm{P}\mathrm{V}}(t)$ represents the power generated by PV system, which relies on solar irradiance; $P_{\mathrm{WT}}(t)$ is the power output from the wind energy system, dependent on wind speed and turbine efficiency; and $P_{\mathrm{load}}(t)$ refers to the power consumed by the load demand at time $t$.

This equation ensures energy balance by matching supply from all sources with demand at each time step.

(2)

Voltage and line flow constraints

To ensure voltage stability and avoid line overloads, especially under high PV penetration, voltage limits and line flow constraints are enforced as follows [33]:

$$V^{\mathrm{m}\mathrm{i}\mathrm{n}}\le V_i(t)\le V^{\mathrm{m}\mathrm{a}\mathrm{x}}$$

(10)

$$|P_{L,k}(t)|\le (1+\sigma )\cdot P_{L,k}^{\mathrm{m}\mathrm{a}\mathrm{x}}$$

(11)

where $V^{\mathrm{m}\mathrm{i}\mathrm{n}}$ denotes the minimum allowable voltage magnitude at a grid node, $V_m(t)$ represents the time-varying voltage at node $i$, and $V^{\mathrm{m}\mathrm{a}\mathrm{x}}$ specifies the maximum permissible voltage magnitude to maintain operational safety; augmented with dynamic voltage sensitivity analysis, this constraint addresses fluctuations induced by high PV penetration by quantifying how voltage deviations respond to changes in PV generation. $P_{L,k}(t)$ is the instantaneous active power flowing through line $L$ (indexed by $k$), $\sigma$ denotes the predefined margin factor accounting for uncertainty or forecast errors, and $P_{L,k}^{\mathrm{m}\mathrm{a}\mathrm{x}}$ stands for the thermal or operational capacity limit of line $L$, $k$ to prevent overload and ensure long-term equipment integrity.

(3)

Reliability and supply capacity constraints

To ensure supply adequacy and resilience under (N − 1) contingency conditions, the model incorporates a capacitance ratio and transfer capacity constraint. Reliability is quantified via Monte Carlo simulation across seasonal scenarios. The TR constraint in the model is as follows:

$$\left\{\begin{array}{c}{P}_s={\displaystyle \frac{1}{N_{\mathrm{MC}}}}{\sum }_{k=1}^{N_{\mathrm{MC}}}{\displaystyle \frac{{\sum }_{i\in N_{\mathrm{load}}}\mid P_{\mathrm{foult},i,t}^{(k)}\mid }{{\sum }_{i\in N_{\mathrm{load}}}\mid P_{\mathrm{original},i,t}\mid }}\times 100\%\\ P_{\mathrm{combined}}={\displaystyle \frac{{\sum }_s^4{D}_s\cdot P_s}{{\sum }_s^4{D}_s}}\ge P_{\min }\end{array}\right.$$

(12)

where $N_{\mathrm{MC}}$ denotes the number of Monte Carlo simulation runs; $N_{\mathrm{load}}$ represents the set of load nodes; $P_{\mathrm{foult},i,t}^{(k)}$ is the active power deficit at load node $i$ during time $t$ under the $k$-th simulation scenario of $(N-1)$ contingencies; $P_{\mathrm{original},i,t}$ is the original active power demand at load node $i$; $P_s$ is the capacitance ratio (in percentage) for each scene, reflecting load restoration performance under fault conditions of the scene; $D_s$ is the weighting factor for each of the four seasons; and $P_{\min }$ is the minimum required combined capacitance ratio to satisfy system-wide reliability criteria across all scenes which are for seasons.

(4)

Capacity load ratio constraint

To assess the adequacy of generation and storage relative to peak loads across seasons, the power transfer constraint is formulated as follows:

$$R_{\mathrm{capacit}}={\displaystyle \frac{{\sum }_{k=1}^{\mathrm{K}}{P}_k^{\mathrm{gen}}}{{\max }_{s\in S}{L}_s^{\mathrm{peak}}}}$$

(13)

where $R_{\mathrm{capacit}}$ is the system CLR; $\mathrm{K}$ denotes the number of generation unit types, including wind power, PV, and BESS; $P_k^{\mathrm{gen}}$ represents the installed capacity of each generation unit, encompassing the installed capacities of wind power and photovoltaic systems as well as the rated power of BESS; $s$ is the set of seasons; and $L_s^{\mathrm{peak}}$ indicates the maximum load in scene $s$.

3. Design of DynaG

3.1. Gravitational Search Algorithm

The gravitational search algorithm (GSA) is a nature-inspired metaheuristic optimization technique that models the search process based on Newtonian laws of gravity and motion [34]. In this framework, candidate solutions are analogized as celestial bodies within a multi-dimensional search space. Each particle exerts an attractive gravitational force on others, where the magnitude of attraction is directly proportional to the product of their masses and inversely proportional to the square of the distance between them. Importantly, a particle’s mass is not static, but is dynamically determined by its fitness value relative to the population—solutions exhibiting superior fitness are assigned greater mass [35,36].

The fundamental dynamics governing particle movement in GSA are derived from Newton’s laws:

$$F_{o,p}^d(t)=G(t){\displaystyle \frac{M_o(t)\times M_p(t)}{R_{o,p}(t)+ϵ}}(x_p^d(t)-x_o^d(t))$$

(14)

where $G(t)$ is the gravitational constant at time $t$, which typically decreases over time to control the search intensity; $M_o(t)$ and $M_p(t)$ are the masses of particles $o$ and $p$ at time $t$, respectively; $R_{o,p}(t)$ is the Euclidean distance between particles $o$ and $p$; $ϵ$ is a small constant to prevent division by zero; and $x_p^d(t)$ and $x_o^d(t)$ are the positions of particles $o$ and $p$ in dimension $d$ at time $t$.

The total force acting on particle $o$ is the weighted sum of forces from other particles. The acceleration $a_o^d(t)$ of particle $o$ in dimension $d$ is then given by Newton’s second law:

$$a_o^d(t)={\displaystyle \frac{F_o^d(t)}{M_o(t)}}$$

(15)

where $F_o^d(t)$ is the total gravitational force acting on particle $o$ in dimension $d$ at time $t$.

Finally, the velocity and position of each particle are updated based on its acceleration, as follows:

$$\left\{\begin{array}{c}{v}_o^d(t+1)={rand}_o\times v_o^d(t)+a_o^d(t)\\ x_o^d(t+1)=x_o^d(t)+v_o^d(t+1)\end{array}\right.$$

(16)

where $v_o^d(t)$ is the velocity of particle $o$ in dimension $d$ at time $t$; ${rand}_o$ is a random number uniformly distributed in [0, 1], introducing a stochastic element; $a_o^d(t)$ is the acceleration of particle $o$ in dimension $d$ at time $t$; and $x_o^d(t)$ is the position of particle $o$ in dimension $d$ at time $t$.

3.2. Coordinated Optimization Strategies of DynaG Algorithm and System Model

The proposed DynaG algorithm achieves coordinated optimization through a series of interlinked strategies. First, gravitational interactions among particles are simulated to emulate Newtonian attraction, guiding the search process. To enhance initial population diversity, a chaotic initialization mechanism is introduced, enabling broad exploration of the solution space. Thereafter, the gravitational constant $G$ is dynamically adjusted to balance global exploration and local exploitation. In addition, particle masses are updated based on iteration stages to emphasize superior solutions. Finally, a dynamic neighborhood search is integrated to enhance diversity under complex constraints. Collectively, these mechanisms empower DynaG to navigate the solution space efficiently, evaluate fitness based on conflicting objectives, and converge to optimal or near-optimal solutions that satisfy OPF constraints. This enhances the operational performance and stability of power distribution networks.

This proposed DynaG extends the traditional GSA by introducing a set of dynamic adaptive mechanisms. It optimizes high-dimensional decision variables by simulating gravitational interactions between celestial bodies. The GSA’s schematic diagram is shown in Figure 1, while the overall flowchart of the enhanced DynaG modeling algorithm is shown in Figure 2.

To enhance the diversity of the initial solution space and avoid local optima, the initial population is generated using chaotic mapping. The logistic map is employed here for its simplicity and good ergodicity, described as follows:

$$x_{n+1}=4x_n(1-x_n)$$

(17)

where $x_n$ is the $n$-th chaotic variable in [0, 1]; and $x_{n+1}$ is the next chaotic variable.

These chaotic variables are then mapped to the feasible region of the OPF problem using the following:

$$P_i=P_{\mathrm{m}\mathrm{i}\mathrm{n},i}+x_n\cdot (P_{\mathrm{m}\mathrm{a}\mathrm{x},i}-P_{\mathrm{m}\mathrm{i}\mathrm{n},i})$$

(18)

where $P_i$ is the initial active power of generator $i$; $P_{\mathrm{m}\mathrm{i}\mathrm{n},i}$ the minimum active power limit of generator $i$; and $P_{\mathrm{m}\mathrm{a}\mathrm{x},i}$ is the maximum active power limit of generator $i$.

The fitness of each particle is evaluated using three conflicting objectives: minimize the LLC, minimize net load fluctuation, and minimize voltage fluctuation, which are the above three objective functions [37]. At the same time, the introduction of a dynamically adjusted gravitational constant $G$, results in the establishment of an adaptive balance mechanism between global exploration and local exploitation capabilities [38]. The mathematical expression for this mechanism can be defined as follows:

$$G(It)=G_0\cdot \mathrm{e}\mathrm{x}\mathrm{p}\left(-{\lambda }_1\cdot {\displaystyle \frac{t}{{It}_{\mathrm{m}\mathrm{a}\mathrm{x}}}}\right)$$

(19)

where $G(It)$ is the gravitational constant at iteration $It$; $G_0$ is the initial gravitational constant; ${\lambda }_1$ is the decay factor, with the permissible fluctuation range set to [15,25]; t is the current iteration; and ${It}_{\mathrm{m}\mathrm{a}\mathrm{x}}$ is the maximum number of iterations.

The exponential decay form is selected for the gravitational constant $G(It)$ (instead of linear or step-wise decay) because it enables the DynaG algorithm to flexibly transition from global exploration to local exploitation—avoiding the rapid loss of exploration capability with linear decay and search disruptions from step-wise decay, which better adapts to the hybrid wind–solar–storage OPF model’s requirement for stable convergence to optimal solutions.

Particle mass is updated according to the iteration stage to emphasize the influence of superior particles, calculated as follows:

$$m_i(It)={\displaystyle \frac{f_{\mathrm{w}\mathrm{o}\mathrm{r}\mathrm{s}\mathrm{t}}(It)-f_i(It)}{f_{\mathrm{w}\mathrm{o}\mathrm{r}\mathrm{s}\mathrm{t}}(It)-f_{\mathrm{b}\mathrm{e}\mathrm{s}\mathrm{t}}(It)+\psi }}$$

(20)

where $m_i(It)$ is the mass of particle $i$ at iteration $It$; $f_{\mathrm{w}\mathrm{o}\mathrm{r}\mathrm{s}\mathrm{t}}(It)$ is the worst fitness value in the population at iteration t; $f_i(It)$ is the fitness value of particle $i$ at iteration $It$; $f_{\mathrm{b}\mathrm{e}\mathrm{s}\mathrm{t}}(It)$ is the best fitness value in the population at iteration $It$; and $\psi$ is a small positive number to avoid division by zero.

The mathematical expression for this mechanism can be defined as follows:

$$r_i(It)=r_0\cdot \mathrm{e}\mathrm{x}\mathrm{p}\left(-{\lambda }_2\cdot {\displaystyle \frac{It}{{It}_{\mathrm{m}\mathrm{a}\mathrm{x}}}}\right)$$

(21)

where $r_i(It)$ is the neighborhood radius of particle $i$ at iteration $It$; $r_0$ is the initial neighborhood radius; and ${\lambda }_2$ is the radius decay factor, with the permissible fluctuation range specified as [18,22].

Neighborhood particles are selected if their distance to the current particle is within the radius, as follows:

$$N_i(It)=\{j\mid \parallel x_i(It)-x_j(It)\parallel \le r_i(It)\}$$

(22)

where $N_i(It)$ is the set of neighborhood particles of particle $i$ at iteration $It$; $x_i(It)$, $x_j(It)$ are the positions of particles $i$ and $j$ at iteration $It$; and $\parallel x_i(It)-x_j(It)\parallel$ refers to computing the Euclidean norm.

The velocity and position of particles are updated to search for better solutions. The velocity update formula is as follows:

$$\begin{array}{c}{v}_i^d\left(It+1\right)=\omega v_i^d\left(It\right)+c_1{r}_1\left(p_i^d-x_i^d\left(It\right)\right)+c_2{r}_2\left(p_g^d-x_i^d\left(It\right)\right)\\ +G(It){\sum }_{j\ne i}{\displaystyle \frac{m_j(It)}{M(It)}}(x_j^d(It)-x_i^d(It))\end{array}$$

(23)

where $v_i^d(It+1)$ is the velocity of particle $i$ in dimension $d$ at iteration $(It+1)$; $\omega$ is the inertia weight; $c_1$ and $c_2$ are acceleration coefficients; $r_1$ and $r_2$ are random numbers in the range of [0, 1]; $p_i^d$ is the personal best position of particle $i$ in dimension $d$; $x_i^d(It)$ is the position of particle $i$ in dimension $d$ at iteration $It$; $p_g^d$ is the global best position; $M(t)={\sum }_{j=1}^N{m}_j(t)$ is the total mass of the population at iteration $It$; and $N$ is the population size.

$It$ denotes the current iteration number, which is consistent with the iteration definition in Equations (19) and (21); the particle position notation $x_i^d(It)$ in this equation aligns with the particle position at time (iteration) $t$ in Section 3.1, where $It$ corresponds to $t$ in Section 3.1 to avoid confusion.

$$x_i^d(It+1)=x_i^d(It)+v_i^d(It+1)$$

(24)

where $x_i^d(It+1)$ is the position of particle $i$ in dimension $d$ at iteration $(It+1)$.

It describes the position update mechanism of particles in the DynaG algorithm for the $d$-dimensional decision space, where the position of particle $i$ at the next iteration is obtained by superimposing its velocity at the next iteration onto its position at the current iteration. By dynamically updating particle positions, it enables the algorithm to adaptively explore feasible solutions that match time-varying renewable energy outputs, seasonal load changes, and BESS scheduling constraints, thereby providing effective support for minimizing network loss, optimizing operational costs, and improving power supply capacity indicators in OPF model. As the core execution step for the DynaG algorithm to search the solution space, the position update logic of Equation (24) is directly associated with the high-dimensional decision requirements of OPF in wind–solar–storage distribution networks. By superimposing the comprehensive velocity (integrating inertia, individual cognition, swarm collaboration, and gravitational interaction) calculated by Equation (23) onto the current particle position, candidate solutions can dynamically adapt to renewable energy fluctuations and seasonal load changes.

It is essential to ensure that the proposed solution complies with the system’s constraint conditions, including limits on voltage magnitude, active and reactive power, and branch apparent power, among others. The constraint-handling process is structured into three sequential phases. In the initial phase, a threshold solution density evaluation is conducted, followed by the application of weak perturbation to facilitate global search. In the intermediate phase, the threshold evaluation is repeated, with an adaptive perturbation mechanism applied to balance global search and local exploitation capabilities. Finally, constraint conflict detection is conducted, and strict perturbation filtering is implemented to ensure feasibility and solution robustness under complex operational limits.

4. Case Study

In this study, measured data from typical days across four seasons in a region of Kunming were employed to simulate the IEEE 33-bus distribution system. The corresponding network topology is illustrated in Figure 3, where nodes 14 and 21 are connected to wind turbine (WT) generators with an output of 0.5 MW, and nodes 17 and 31 are connected to PV generators producing 1.6 MW. The performance of the proposed optimization model is evaluated under four-season scenarios, with different wind penetration levels and load demands. During the simulation, both load characteristic data and WT output power data are derived from actual seasonal curves. The system parameters and algorithm settings are listed in

Table 1. System parameters.

Technical Specifications Table
Parameter	Value	Parameter	Value
WT installed capacity	0.5 MW/unit	SOC range	0.1–0.9
PV installed capacity	1.6 MW/unit	Charge/discharge efficiency	0.96, 1/0.96
Wind power penetration coefficient	0.897	Monte Carlo simulation runs	100
PV penetration coefficient	0.296	Single line failure probability	0.01
BESS capacity range	0–4 MWh	BESS power range	−4~4 MW
Economic Parameters Table
Valley electricity price	(1:00–6:00, 23:00–24:00): 290.5 CNY/MWh	Peak electricity price	(11:00–13:00, 18:00–22:00): 1443.5 CNY/MWh
Flat electricity price [39] (7:00–10:00, 14:00–17:00): 1023.0 CNY/MWh

and

Table 2. Algorithm Parameters.

Algorithm	Parameter	Value	Parameter Function
Common parameters	Maximum iterations	200	Ensure algorithm convergence under 30% renewable penetration; adapt to 8-dimensional OPF decision variables in IEEE 33-bus system.
	Population size	20
DynaG and MOGSA [15]	Initial gravitational constant	100	Cover output ranges of WT/PV (0–1.6 MW) and BESS; guarantee global exploration in early iterations.
	Decay coefficient	20	Control exponential decay of $G(t)$.
	Initial neighborhood radius	0.2	Cover 20% of decision space; maintain 3–5 neighborhood particles per node; lower CLR volatility.
	Radius decay coefficient	20	Decay neighborhood radius synchronously with ${\lambda }_1$; focus search on optimal regions.
	Constrained mass factor	10−6	Prevent numerical collapse in winter high-load scenarios; ensure stable particle mass calculation.
NSGA-III [13]	Crossover probability	0.8	Enhance chromosome diversity for OPF multi-objective conflicts; coordinate with 0.02 mutation rate to avoid Pareto clustering.
	Mutation rate	0.02	Stabilize economic objective; mitigate solution disturbance.
MOPSO [40]	Mutation probability	0.3	Reduce ±5% penetration confidence interval fluctuation.
	Inertia weight	0.7	Enhance global search; adapt to WT output fluctuations.
	Individual learning factor	1.5	Strengthen particle memory of historical optimal solutions; improve voltage stability.
	Swarm learning factor	1.5	Guide particles to global optimal; lower LCC.
MOAOS [14]	Encircling coefficient	2	Cover PV output fluctuations; improve renewable energy accommodation
	Chasing coefficient	1	Reduce solution oscillation; stabilize voltage deviation.
	Attacking coefficient	0.5	Prevent solutions from exceeding feasible regions; maintain constraint compliance.
	Decay coefficient	0.99	Maintain exploration range for seasonal load adaptation; ensure stable search.

, respectively. The IEEE 33-bus node topology is illustrated in Figure 3.

4.1. Simulation Setup and System Parameters

To evaluate the effectiveness of the proposed DynaG algorithm, a comparison analysis was conducted against four mainstream multi-objective optimization algorithms: NSGA-III, MOPSO, MOAOS, and MOGSA. The comparative results are summarized in Table 2, which shows the technical performance metrics for the solutions corresponding to the minimum average voltage level, observed in 30% of the test cases.

The simulations in this section are conducted using MATLAB R2023b, with its solver for modeling the IEEE 33-bus distribution network, wind/PV generators, and BESS, and the Optimization Toolbox for implementing the DynaG algorithm and comparative algorithms. This toolchain is selected for two core reasons: (1) it supports high-fidelity modeling of renewable energy intermittency and BESS dynamic behavior, which is consistent with the multi-scenario OPF model’s requirement to reflect real-world system dynamics; and (2) it enables efficient code integration and parallel computing for multi-season, multi-penetration (10–30%) simulation cases, ensuring accurate and timely verification of the algorithm’s performance.

4.2. Analysis of Results

4.2.1. Power Balance Analysis

In scenarios with a 30% renewable penetration rate, as illustrated in Figure 4, the power balance among WT, PV, power purchases, and BESS charging and discharging is analyzed over a typical day in four seasons. It has been demonstrated that, following optimization via the optimal tidal current model, the BESS is capable of cooperating with the power network to discharge effectively during peak load hours and charge actively during off-peak hours. This role is clear in the shaving of peaks and filling of valleys, thereby rendering the power curve of power purchasing smoother and reducing the system’s dependence on the external power grid.

The interplay between the four seasons gives rise to discrepancies in wind power and load characteristics, which in turn necessitate adaptation in the regulatory capabilities of the BESS. It is ensured that the grid load fluctuation is kept to a minimum through the use of BESS. During the off-peak periods, the BESS mitigates grid load fluctuations by absorbing excess energy. The co-optimization strategy, enabled by BESS participation, enhances the local utilization of renewable energy, while simultaneously optimizing the economy and stability of system operation.

4.2.2. Voltage Deviation Analysis

As illustrated in Figure 5, a comparison of voltage deviation at each node of the optimal trending before and after optimization is provided. The results indicate that, across all seasons, the execution of optimal power flow optimization leads to substantial reductions in voltage deviation across the network. Prior to optimization, nodes 17 and 33 exhibit particularly high voltage deviations. However, after applying the optimization model, the voltage deviation of each node in each season is effectively smoothed, with a marked reduction in both the overall voltage deviation level and the disparity between nodes. The comparison curves further show that DynaG algorithm consistently outperforms MOAOS and MOGSA, particularly under high-penetration renewable scenarios. DynaG achieves closer alignment with the ideal voltage profile, indicating improved voltage stability and better adaptability to seasonal and spatial variations in network conditions.

Comparing the voltage deviation optimization effects of various algorithms, DynaG achieves the best voltage deviation control, particularly at key nodes such as nodes 17 and 33. Specifically, DynaG achieves voltage deviations that are 2.7% lower than NSGA-III and 2.7% lower than MOPSO at these key nodes. This performance gap becomes even more pronounced under winter high-load conditions, where voltage stability is more challenging to maintain. These results validate the effectiveness of the dynamic gravitational mechanism in DynaG, which enhances the algorithm’s ability to suppress voltage fluctuations and maintain system stability under varying load and renewable penetration conditions.

4.2.3. Load Fluctuation Analysis

Figure 6 illustrates the fluctuations in load across the four seasonal scenarios, both before and after the implementation of optimal power flow. Prior to optimization, the load profiles exhibit significant amplitude fluctuations, particularly during peak and valley periods. Following the OPF optimization, there is a substantial decrease in load fluctuation amplitude, resulting in another and more stable load curves across all seasons.

A comparative analysis of algorithm performance demonstrates that the DynaG algorithm achieves superior smoothing effects. Specifically, peak load fluctuation is reduced by approximately 6.2% compared to NSGA-III and by 8.5% relative to MOPSO. These results highlight the effectiveness of BESS cooperative optimization in optimal flow, reducing the impact of sudden load changes on the power grid and enhancing the stability of power grid operation.

4.2.4. Network Loss Analysis

As illustrated in Figure 7, the network loss distribution of each node at 8:00 in spring is depicted. When the penetration of renewable energy sources remains below 30%, the average network loss before optimization is 9.7278 MWh/day. After applying the proposed trend optimization, this loss is reduced to 2.976 MWh/day, representing a 69.4% reduction. From the comparative curves, the network loss profile of DynaG is lower than those of NSGA-III and MOAOS at all nodes. This advantage is particularly pronounced in the loss-prone areas such as nodes 2–5. Consequently, the annual network loss is diminished from 329.55 MWh/year to 244.71 MWh/year. Furthermore, the figure shows that the network loss is at its maximum at node 2 while the minimum is observed at node 33.

4.2.5. Analysis of Power Supply Capacity

As demonstrated in Figure 7 and Figure 8, the four-season CLR and TR for the four scenarios are presented before and after optimal tidal optimization. It is evident from the figures that the power supply capacity of the grid is significantly augmented following the optimization. Specifically, the TR increases from 61.33% in the pre-optimization state to 76.14% post-optimization. The comparison curves show that DynaG exhibits steeper improvement slopes in both TR and CLR compared to other algorithms. Moreover, DynaG achieves greater CLR stability with a fluctuation of only 3.37%, outperforming NSGA-III (5.12%) and MOPSO (4.89%) under the 30% penetration scenario.

These improvements indicate that the optimization of flow enhances the grid’s load transfer capacity during faults or maintenance, thereby increasing reliability and operational flexibility. In terms of CLR, the value improves from 0.57 to 1.07 in the baseline scenario, indicating an enhancement in the capacity reserve of the grid equipment. This, in turn, facilitates better adaptation to load growth and fluctuations, reduces the risk of grid overload, and enhances the stability and adaptability of the grid. The collective enhancement of flow optimization has yielded substantial advancements in grid reliability and stability, supporting the safe, stable, and efficient operation of the grid.

4.2.6. Performance Evaluation of DynaG

As shown in Figure 9, the box-and-whisker plot of integrated cost demonstrates that DynaG consistently outperforms the comparison algorithms. The lower quartile, median, and upper quartile of DynaG are all significantly lower than those of MOAOS, MOPSO, MOGSA, and NSGA-III algorithms, which indicates that the DynaG has a clear advantage in integrated cost performance. Furthermore, the range of the fluctuation of the cost is relatively small, and the stability of the algorithm is better.

4.2.7. Multi-Objective Optimization Performance Comparison

The Pareto frontier of DynaG under 30% renewable penetration (Figure 10) shows better uniformity and dispersion in the objective function space than those of NSGA-III (Figure 11), MOAOS (Figure 12), MOPSO (Figure 13), and MOGSA (Figure 14)—without the aggregation seen in NSGA-III (due to static crowding distance) and MOPSO (due to velocity update noise—providing richer compromise solutions for multi-objective conflicts like minimizing network loss and operational cost. The solution set covers a wide range in the objective function space without obvious aggregation phenomenon, indicating that the algorithm can provide rich compromise solutions in multi-objective conflicts, such as minimizing network loss and reducing operational cost. The uniformity and dispersion of the solutions demonstrate the algorithm’s ability to provide a wide range of high-quality compromise solutions. This reflects both the robustness of DynaG in handling complex constraints and its effectiveness in solving multi-objective optimization problems under high-penetration scenarios.

4.2.8. Impact of Renewable Penetration Levels on Grid Performance

The quantitative results in

Table 3. Comparison results of multi-objective optimization algorithms.

Algorithm	Annual Costs (CNY Million/Year)	Voltage Deviation (p.u.)	Network Losses (MWh/Year)
DynaG	51.986	0.0179	248.25
NSGA-III	58.000	0.0184	250.77
MOPSO	60.356	0.0184	249.29
MOAOS	63.466	0.0190	263.34
MOGSA	67.756	0.0183	248.93

indicate that among all the comparison algorithms, DynaG has the lowest annual operating cost, which is 51.986 million yuan/year. This cost is 10.37% lower than NSGA-III (5.8 million yuan/year), 13.87% lower than MOPSO (60.356 million yuan/year), 18.09% lower than MOAOS (63.466 million yuan/year), and 23.27% lower than MOGSA (6.7756 million yuan/year). The excellent economic efficiency of DynaG is mainly attributed to its effective coordination of BESS scheduling and DR mechanisms, which collectively reduce the system’s dependence on purchasing high-cost electricity from external power grids during peak load periods.

The baseline scenario indicators on the IEEE 33 node system are: voltage deviation, 0.022320 p.u; annual network loss, 329.55 MWh/year; daily load fluctuation: 9.7278 MW/day. A comparative analysis was conducted on the performance data of DynaG in Table 3 and

Table 4. OPF for different penetration rates of DynaG.

Penetration Rate	Annual Costs (CNY Million/Year)	TR	CLR	Voltage Deviation (p.u.)	Load Fluctuation (MW/Day)	Network Losses (MWh/Year)
30%	51.986	72.16%	1.00	0.0179	6.9821	248.25
25%	74.338	74.93%	1.03	0.0196	6.6829	232.30
20%	75.026	75.02%	1.11	0.0152	6.3853	232.02
15%	93.484	75.76%	1.07	0.0140	6.0850	230.59
10%	108.027	78.04%	1.10	0.0127	5.7888	236.88

, and the results showed that the daily load fluctuation exhibited the greatest relative improvement. DynaG reduced this indicator to 6.9821 MW/day, a decrease of 28.23% compared to the baseline. This performance is also about 6.2% and 8.5% higher than NSGA-III and MOPSO, respectively. For other key technical indicators, compared to the baseline, DynaG’s annual grid losses decreased by 24.67% (from 329.55 MWh/year to 248.25 MWh/year), and voltage deviation decreased by 19.79% (from 0.022320 p.u. to 0.0179 p.u.).

Data analysis reveals significant correlations between wind penetration levels and key grid operation indicators. As penetration decreases from 30% to 10%, annual operational costs exhibit a stepwise increase, highlighting the economic challenges associated with reduced renewable integration. Technical parameters follow three patterns:

(1)

For reliability, TR progressively increases, confirming enhanced grid redundancy at lower penetration.

(2)

Regarding power quality, voltage deviation linearly improves with higher conventional power share.

(3)

In terms of operational efficiency, load fluctuation decreases, but network losses rebound to 236.88 MWh/year at 10% penetration, resonating with Section 2.2’s conclusion on “scheduling economy deterioration under low renewables”. Overall, the 25–30% penetration range demonstrates optimal balance between cost control and technical performance.

As illustrated in Table 4, the variation in network loss for each algorithm under 10–30% penetration is also examined. DynaG exhibits the smallest increase in network loss (only 1.2%), while NSGA-III and MOPSO increase by 3.5% and 4.1%, respectively. This finding suggests that the anti-volatile design of DynaG is particularly advantageous in high-penetration scenarios, and its stability advantage is further emphasized when the penetration rate exceeds 25%.

To systematically clarify the goal and results of the proposed wind–solar–storage OPF scenario, this study takes the IEEE 33-bus system as the test platform, focuses on the core goal of “minimizing network loss and operational cost while enhancing power supply capacity”, and verifies the effectiveness of the DynaG-based optimization framework through four-season typical day simulations (with 10–30% renewable penetration). Key results show that under the optimal 25–30% penetration range, the DynaG algorithm reduces annual operational cost to 51.986 million CNY/year (10.37–23.27% lower than NSGA-III, MOPSO, etc.), cuts annual network loss by 24.67% compared with the baseline, and stabilizes capacity–load ratio volatility at 3.37%; meanwhile, the multi-scenario OPF model, by integrating demand response (DR) and BESS dynamic scheduling, smooths load fluctuations by 28.23% and improves voltage stability (voltage deviation down to 0.0179 p.u.), fully achieving the coordinated optimization of economic efficiency and technical performance.

5. Conclusions

This paper first constructs a multi-scenario OPF model considering the time-varying characteristics of wind and solar penetration, four-season load patterns, and demand response participation. Based on this, a dynamic optimization frame DynaG is proposed for distribution networks with wind–solar–storage hybrid systems. The key findings are summarized as follows:

(1)

Through the coordinated optimization of the multi-scenario OPF model and DynaG, the distribution network can effectively reduce network loss, compress the peak-to-valley difference, and reduce the volatility of the capacity–load ratio. This improves the voltage stability and operational economy of the system and enhances the resilience of the distribution network against renewable energy fluctuations.

(2)

The superior performance of DynaG in handling complex optimization problems lies in its adaptive gravitational constant adjustment strategy, constrained inertia mass updating mechanism, and integrated chaotic initialization with dynamic neighborhood search. Therefore, these enhancements are crucial to balancing global exploration and local exploitation, and to improving the diversity and convergence of solutions in high-penetration renewable energy distribution networks.

(3)

The paper compares the performance of five optimization algorithms. Based on a comprehensive evaluation of economic and technical indicators, the MOGSA is identified as the optimal trending scheme. Economically, MOGSA ranked just below NSGA-III, but significantly outperforms MOAOS and MOGSA. In terms of power quality, its voltage deviation is comparable to that of MOGWO and better than that of MOPSO, while network loss is comparable to those of NSGA-III. Overall, the MOMA algorithm balances the economic strength of NSGA-III, with superior power quality optimization, making it the most suitable choice for the optimal allocation of distributed energy storage in distribution networks.

The future work will focus on three goals to further expand its applicability and sustainability. First, in response to the demand for aligning with broader sustainability goals, the multi-objective OPF model will integrate CO₂ emission reduction (e.g., quantifying carbon costs of conventional power generation) and renewable energy curtailment minimization as explicit optimization objectives, addressing the current focus on cost and technical performance alone. Second, the DynaG algorithm will be enhanced with a multi-timescale adaptive mechanism to adapt to ultra-short-term wind/PV fluctuations and long-term load evolution, improving its dynamic response to complex time-varying scenarios. Finally, the framework will be extended to multi-microgrid interconnected systems, incorporating distributed energy resources such as electric vehicles and demand-side flexible loads, to explore the coordinated optimization of regional multi-energy systems and further enhance grid resilience under high-renewable penetration.