Abstract
The escalating penetration of distributed photovoltaics (DPV) frequently triggers voltage violations and degrades the static voltage stability within distribution networks. To effectively mitigate these challenges, this paper proposes a dynamic hosting capacity (DHC) assessment method that incorporates the static voltage stability margin (SVSM). First, a set of key stress scenarios (KSS)—specifically capturing extreme conditions of “maximum reverse power flow” and “maximum forward heavy loading”—is generated utilizing non-parametric probabilistic forecasting. Next, an optimization model designed to maximize PV hosting capacity is formulated, innovatively enforcing the SVSM as a hard constraint to guarantee system robustness during heavy-load transitions. To overcome the intensive computational burden inherent to traditional physical verification, a hybrid solution framework is introduced. This framework integrates a deep neural network (DNN) surrogate model with an advanced particle swarm optimization (PSO) algorithm. By combining the offline learning of nonlinear SVSM boundaries with online rapid predictions and dual physical validation, the method achieves quasi-real-time capacity assessment. Case studies conducted on a modified IEEE 33-bus system demonstrate that the proposed framework not only uncovers the latent limitations imposed by evening peak voltage stability risks—which are typically neglected by conventional approaches—but also significantly accelerates computational efficiency while preserving rigorous evaluation accuracy.
1 Introduction
Driven by the deepening implementation of “dual carbon” and “whole-county advancement” strategies, establishing a novel power system dominated by renewable energy has emerged as an inevitable trend in the energy transition (). As a vital form of renewable energy development, distributed photovoltaics (DPV) have experienced a surging penetration rate in distribution networks owing to their short construction cycles and capability for local energy accommodation. However, as distribution networks evolve from traditional passive, radial topologies to active, multi-source configurations, the strong stochasticity, volatility, and intermittency of DPV output, coupled with their power electronic grid-connection characteristics, have profoundly altered the power flow distribution and voltage profiles (). Consequently, the contradiction between the randomness of PV generation and the weak supporting capability of terminal grids has become increasingly prominent (). Therefore, scientifically and accurately assessing the accommodation capability of distribution networks for DPV—known as the hosting capacity (HC)—has become a critical and urgent technical challenge for grid planning and operation departments (; ; ).
Existing methodologies for hosting capacity assessment are primarily categorized into deterministic and probabilistic approache (). Deterministic methods typically perform power flow calculations based on worst-case scenario snapshots, neglecting the temporal complementary characteristics of sources and loads, which often leads to overly conservative assessment results (; ). Conversely, probabilistic methods employ Monte Carlo simulations or chance-constrained programming to handle uncertainty, enabling a more comprehensive quantification of operational risks (). Nevertheless, when constructing constraint systems, these aforementioned studies predominantly focus on steady-state voltage deviations and equipment thermal stability limits, generally overlooking the degradation of static voltage stability caused by the high integration of power electronic devices (). Addressing voltage violation issues in distribution networks with high-penetration DPV, Reference () proposed a zonal coordinated optimal control strategy, verifying the vital role of active management in enhancing voltage stability. Reference () pointed out that under specific conditions of heavy load and low PV output, although the distribution network may not violate steady-state lower voltage limits, the operating point could approach the nose of the P-V curve, hovering on the verge of voltage collapse. Due to the lack of quantitative evaluation of the static voltage stability margin (SVSM), traditional methods are prone to overestimating the grid’s accommodation capability during heavy-load periods, thereby concealing severe safety risks (). Furthermore, incorporating complex nonlinear SVSM constraints involves checking the singularity of the Jacobian matrix. This causes the computational burden of iterative optimization based on full physical models to grow exponentially, rendering it inadequate for quasi-real-time assessment requirements (; ). Addressing the complex planning problem of PV integration in distribution networks, Reference () adopted an optimization strategy based on a multi-population genetic algorithm, demonstrating the superiority of evolutionary intelligent algorithms in handling nonlinear constraints and multi-objective optimization. While Reference () considered voltage stability constraints in transmission network planning, its application at the distribution network level remains scarce, and studies combining it with dynamic temporal assessment are exceptionally rare.
To address the aforementioned challenges, this paper proposed a dynamic hosting capacity (DHC) assessment method for distribution networks that explicitly accounts for static voltage stability. First, non-parametric probabilistic forecasting is utilized to construct a set of key stress scenarios (KSS), encompassing both maximum reverse power flow and maximum forward heavy loading, to achieve an effective envelope of the system’s safety boundaries. Subsequently, an optimization model aimed at maximizing the hosting capacity is established, innovatively integrating the SVSM as a hard constraint within the solution framework to guarantee the system’s disturbance resistance under heavy-load scenarios. Finally, to balance evaluation accuracy and computational efficiency, a hybrid solution framework based on a deep neural network (DNN) surrogate model and an improved particle swarm optimization (PSO) algorithm is proposed. Based on a modified IEEE 33-bus system, this framework achieves quasi-real-time DHC assessment through the offline learning of nonlinear SVSM boundaries, coupled with an online proxy pre-judgment and physical verification mechanism.
2 Construction of dynamic assessment scenarios considering uncertainty
2.1 Probabilistic forecasting and interval modeling of source and load
Distributed photovoltaic output is significantly influenced by meteorological factors such as irradiance and cloud cover, exhibiting strong intermittency. To comprehensively capture the fluctuation range of source and load outputs, a non-parametric quantile regression method is employed for probabilistic forecasting (). For any given time within the assessment period, let the PV output random variable be denoted as and the load random variable as . Defining the quantile function as the inverse of the cumulative distribution function, the predicted value at a confidence level can be expressed as Equation 1:where represents the predicted quantile value of the random variable (either PV or load) at time corresponding to the cumulative probability , and denotes the probability measure. To balance risk coverage and computational efficiency, the 10th percentile () and the 90th percentile () are selected as the lower and upper bounds of the confidence interval. Here, the 90th percentile represents a high-output scenario for PV or a heavy-load scenario, and vice versa.
An 80% prediction interval (P10/P90) is adopted because it captures the dominant operational risks while avoiding the excessive conservatism that a wider band (e.g., P5/P95) would introduce when its extreme quantiles are directly combined to form the key stress scenarios.
2.2 Definition of KSS
After obtaining the probabilistic forecasting intervals for source and load, employing Monte Carlo simulations for full-scenario verification would incur an excessive computational burden, failing to meet the timeliness requirements of online assessment. By analyzing the operational characteristics of distribution networks, it is evident that the system’s safety boundaries are typically determined by the extreme states of source-load combinations (). Consequently, a KSS set, denoted as , is directly constructed, comprising two complementary extreme operating conditions: Scenario A and Scenario B.
Scenario A is defined as the Maximum Reverse Power Flow and Overvoltage scenario (MRPO). This scenario corresponds to the operating condition where PV output is at a high level while the load is at a low level. Under these circumstances, a massive amount of PV power cannot be accommodated locally and is fed back to the upstream grid, which is highly likely to cause upper-limit voltage violations at the feeder ends and reverse overloading of the lines. The source-load state under this scenario is defined as Equation 2:where represents the 90th percentile of PV output at node and time ; represents the 10th percentile of the load at node and time ; and is the set of system nodes.
Scenario B is defined as the Maximum Forward Power Flow and Undervoltage scenario (MFPU). This scenario corresponds to the operating condition where PV output is at a low level while the load is at a high level, typically occurring during the evening peak hours after sunset or on rainy days. During this period, the active support capability of PV is weakened, and heavy loading leads to increased line voltage drops. This not only easily triggers low-voltage issues but also, more severely, pushes the operating point close to the system’s static voltage stability limit. The source-load state under this scenario is defined as Equation 3:where represents the 10th percentile of PV output at node and time ; represents the 90th percentile of the load at node and time . These two scenarios constitute an envelope around the safety boundaries of the distribution network, capable of covering the most severe operational risks with minimal computational cost.
It is worth noting that while high PV + high load scenarios also exist in distribution networks, the local accommodation effect of the load ensures that the line loading rates and nodal voltage deviations in such cases are generally superior to those in the aforementioned two extreme scenarios. Scenario A constitutes the boundary for upper voltage limits and reverse overloading, whereas Scenario B constitutes the boundary for lower voltage limits and voltage collapse. Therefore, the constructed KSS set forms a valid envelope of the distribution network’s safe operating region, inherently capable of covering other intermediate states.
3 Evaluation model considering SVSM constraints
3.1 Objective function
The assessment of DHC is formulated as a multi-period constrained optimization problem. Under the premise of satisfying all operational security constraints, the primary objective is to maximize the total newly accommodated photovoltaic capacity across the distribution network at each time interval . The objective function is defined as Equation 4:where represents the total dynamic hosting capacity of the entire network at time ; denotes the pre-selected set of candidate nodes for PV integration; and is the newly added capacity allocation at node and time .
To avoid blind searching within the high-dimensional solution space and to align with standard engineering practices, a weighted allocation strategy based on the existing installed capacity is introduced as an optimization benchmark. Assuming the baseline PV capacity at node is , the allocation of newly added capacity must satisfy the following proportional relationship as Equation 5:where is the capacity scaling factor of the entire network at time , serving as the primary decision variable of the model to represent the expansion magnitude relative to the baseline maximum integration node; is the normalized weight factor designed to ensure that the capacity allocation matches the resource endowment and construction conditions of each specific node. For nodes without any existing installation, the weight can be assigned a baseline default value or set to zero depending on planning preferences.
It is worth noting that this proportional relationship serves merely as an initial reference distribution or an ideal benchmark guidance. Within the actual optimization model, the capacity of each candidate node is permitted to independently adjust within reasonable bounds to fully satisfy the global network safety constraints, thereby exploring the maximum latent accommodation potential of the grid. Consequently, the ultimate decision vector of this model is the newly added PV capacity vector across all candidate nodes, denoted as .
3.2 Conventional steady-state safety constraints
Under any given key stress scenario, the system states must satisfy standard steady-state operational boundaries. First, the power flow constraints dictate that the active and reactive power balance must be maintained at every node within the network according to Kirchhoff’s laws as Equation 6:where and denote the active and reactive power injected by the substation/slack bus, respectively (which are identically zero for all non-slack buses); and represent the net PV active power injection and the baseline load demand under the specific scenario , respectively; is the voltage magnitude at node ; is the voltage angle difference between node and node ; and denote the real and imaginary parts of the -th element in the bus admittance matrix, respectively.
Second, the nodal voltage deviation constraints dictate that the voltage magnitudes at all buses must be strictly maintained within the regulatory statutory boundaries as Equation 7:where and are the lower and upper safety limits of the operational voltage, specified as 0.93 p. u. and 1.07 p. u., respectively.
Finally, the branch thermal stability constraints require that the apparent power flow along any transmission line must not exceed its long-term continuous thermal rating, incorporating a conservative operational security buffer as Equation 8:where represents the apparent power flowing through branch under scenario ; is the rated long-term carrying capacity of the branch; and is the operational margin coefficient set to 0.8 to account for measurement errors and short-term overloading protection. represents the comprehensive set of all lines in the network.
3.3 SVSM core constraints
Conventional hosting capacity evaluations routinely neglect the voltage stability performance of distribution networks under heavy-load conditions. However, with the massive integration of distributed renewable energy interfaces, the equivalent short-circuit ratio of the system degrades significantly, leading to a diminished disturbance rejection capability. To safeguard the network against such operational risks, the SVSM is explicitly embedded into the constraint matrix.
For any given evaluation interval , the system is strictly required to maintain a solvable power flow state when subjected to a small-magnitude load disturbance increment of times (where the security margin threshold is set to ). This condition guarantees that the current operational point does not cross over the critical “nose” point of the system P-V curve, thereby preventing catastrophic voltage collapse. This stability criterion can be mathematically formulated as follows: for any stress condition within the KSS set, when the global net load level is scaled up by times, the corresponding system power flow Jacobian matrix must remain non-singular, and the stressed voltage magnitudes at critical buses must stay above the lower critical collapse threshold as Equations 9–11:where represents the comprehensive set of nonlinear power flow equations governing the distribution network; denotes the baseline complex load power vector; represents the active power injection vector from the integrated PV units; is the resulting bus voltage magnitude vector under the stressed loading state; and is the minimum critical security voltage threshold defined to preclude impending voltage collapse.
4 Solution strategy based on DNN surrogate model and hybrid intelligent algorithm
Since the SVSM constraints inherently require singularity checks of the Jacobian matrix, directly embedding them into an iterative optimization process would lead to an exponential growth in computational time, rendering the approach impractical for operational use (). To overcome this bottleneck, an “offline training - online surrogate” hybrid solution strategy is proposed. This approach employs a DNN to approximate the high-dimensional, nonlinear safety boundaries, which is then integrated with an improved PSO algorithm to achieve quasi-real-time assessment of the DHC.
4.1 Construction and offline training of the physical constraint surrogate model
To achieve millisecond-level discrimination of the system’s safety boundaries, a DNN binary classifier is constructed to act as a surrogate model for the rigorous physical verification process. The input feature vector of the model is composed of the forecasted source-load quantiles at the current time step and the candidate PV integration capacities. Its mathematical expression is given by Equation 12:where and represent the load and PV forecast values under different confidence levels, respectively; is the newly added capacity at the -th candidate node; and denotes the total number of candidate nodes.
The output of the model is defined as a binary variable . The output if and only if the input capacity scheme simultaneously satisfies the upper voltage limit constraints under Scenario A and the SVSM constraints under Scenario B (as detailed in Section 3); otherwise, .
During the offline training phase, to ensure the discriminatory accuracy of the surrogate model near the feasibility boundaries, the Latin Hypercube Sampling (LHS) technique is utilized to generate a massive training dataset within the solution space. The Pandapower physical engine is then invoked to perform rigorous power flow calculations and SVSM verification to obtain the ground-truth physical labels. The network architecture is designed as a fully connected feed-forward network (Multilayer Perceptron). The hidden layers employ the ReLU activation function to enhance nonlinear fitting capabilities, while the output layer utilizes a Sigmoid function to output the feasibility probability. The model is trained using the Adam optimizer by minimizing the binary cross-entropy loss function, iterating until the model accurately approximates the decision boundary of the physical equations. Furthermore, to eliminate dimensional discrepancies among input features and accelerate convergence, Z-score standardization is applied to the input data.
To prevent overfitting on the 1500 offline samples, the dataset was partitioned into training and validation subsets, and early stopping was applied by monitoring the validation loss; training was terminated once the validation loss ceased to improve. The 500-epoch value in Table 1 therefore serves as the maximum training budget rather than a fixed training length.
TABLE 1
| Module | Parameter name | Value |
|---|---|---|
| DNN surrogate model | Offline training samples | 1500 |
| Hidden layer structure | [100, 50] | |
| Maximum training epochs | 500 | |
| Improved PSO algorithm | Population size | 40 |
| Maximum iterations | 30 | |
| Learning factors () | 1.5 | |
| Inertia weight range | [0.4, 0.9] |
Algorithm parameter configuration.
4.2 Online dynamic optimization based on improved PSO
In the online assessment phase, the well-trained DNN model replaces the computationally expensive power flow calculations to serve as the fitness evaluator for the PSO algorithm. Addressing the unique characteristic of the DHC optimization problem—where the feasibility region is typically anchored near zero capacity—a hybrid initialization strategy is proposed. Specifically, 50% of the population’s particles are initialized within the core feasibility region at lower capacities to guarantee the presence of feasible “seed” solutions; the remaining 50% are randomly distributed across the entire decision space to maintain population diversity and global exploration capability. Let the position vector of the -th particle be ; its fitness function is formulated as Equation 13:where represents the probability output from the DNN; is the decision threshold (typically set to 0.5); and is a sufficiently large penalty factor. If the DNN evaluates the current scheme as feasible, the total capacity is assigned as the fitness value; otherwise, a massive penalty is imposed, forcing the particle swarm to migrate back toward the feasible region. The updates of particle velocities and positions strictly follow standard PSO evolutionary rules, and the inertia weight employs a linearly decreasing strategy to balance global exploration and local exploitation ().
4.3 Dual physical verification and fallback correction mechanism
Although the DNN surrogate model exhibits extremely high prediction accuracy, minor misclassifications may still occur when processing rare samples situated near highly complex nonlinear boundaries. Relying solely on the surrogate model could inadvertently output pseudo-optimal solutions that technically violate physical constraints, posing latent safety hazards to grid operations. To guarantee absolute operational safety for engineering applications, a dual verification mechanism is incorporated.
Once the PSO algorithm converges and outputs the global optimal solution , this solution is not immediately accepted. Instead, it is fed back into the rigorous physical model for ultimate validation. If the global optimum successfully passes this physical verification, it is recorded as the final DHC evaluation result for time . If the physical verification fails, it indicates a false-positive misclassification by the surrogate model near that operating point. In this event, the algorithm triggers an adaptive fallback correction mechanism: a scaling factor is introduced (initially set to 0.95, for instance) to construct a corrected solution , which is then re-submitted for physical verification. If the constraint is still violated, α is iteratively decayed and the solution re-verified, or a more granular local search is initiated, until the maximum capacity solution that strictly satisfies all physical operational boundaries is secured.
5 Case studies
5.1 Simulation setup
To validate the effectiveness of the proposed methodology, simulation analysis is performed on a modified IEEE 33-bus distribution system, as illustrated in Figure 1. To simulate the impact of distributed PV integration on the hosting capacity, distributed PV units and static var generators (SVGs) are integrated into selected nodes across the network (excluding the slack bus). The specific installation nodes and their corresponding capacities are detailed in Table 2. The thermal stability limit of the main feeder, corresponding to the long-term allowable continuous current-carrying capacity, is specified as 12 MW (approximately 550 A) to represent the grid capacity boundary under a high penetration of renewable energy.
FIGURE 1
TABLE 2
| Device type | Node index (bus no.) | Installed capacity |
|---|---|---|
| DPV 1 | 13 | 0.3 MW |
| DPV 2 | 17 | 0.2 MW |
| DPV 3 | 24 | 0.4 MW |
| DPV 4 | 30 | 0.2 MW |
| DPV 5 | 32 | 0.1 MW |
| SVG 1 | 17 | ±0.8 Mvar |
| SVG 2 | 32 | ±0.4 Mvar |
| Load model | Total system peak load | 3.715 MW + j2.3 Mvar |
Configuration of PV units and SVGs.
The case study utilizes measured PV output and load demand data from a specific distribution network in Northwest China for the year 2025 as the evaluation baseline, with a time resolution of 15 min. A typical day characterizing both high PV generation and a double-peak load profile is extracted via K-means clustering, as depicted in Figures 2, 3. The PV output curve exhibits a sharp drop between 11:30 and 13:00 to simulate convective cloud shading events, which aims to test the robustness of the algorithm against severe power volatility at the source side. The load curve reaches its evening peak at 19:00, a period when the PV output drops to zero and the system lacks sufficient reactive voltage support. This constitutes a critical boundary condition to validate the effectiveness of the proposed SVSM constraints under heavy-load scenarios.
FIGURE 2
FIGURE 3
The core parameters of the proposed solution framework are configured as listed in Table 1. The DNN surrogate model adopts a fully connected feed-forward architecture, where the input layer receives the forecasted source-load quantiles and the candidate integration capacity features, and the output layer consists of a single neuron to yield the feasibility probability. The hidden layers employ the ReLU activation function, and the output layer utilizes the Sigmoid activation function. Model training is executed via the Adam optimizer with a binary cross-entropy loss function. For the improved PSO algorithm, a linearly decreasing strategy is applied to the inertia weight, which decays from 0.9 to 0.4 as the iterations progress. To accommodate the solution space characteristics of the DHC assessment problem, a hybrid initialization strategy is implemented: 50% of the population’s particles are initialized within the core feasible region at lower capacities, while the remaining 50% are randomly distributed across the entire decision space. This setup significantly enhances the search efficiency for boundary solutions while maintaining population diversity. All data processing tasks are implemented in a Python 3.10 environment using scientific libraries such as Pandas and NumPy, running on a computer equipped with an Intel Core i7 processor and 16 GB RAM.
5.2 Impact of source-load uncertainty and SVSM constraints on assessment results
To investigate the corrective effects of uncertainty modeling and voltage stability constraints on the DHC assessment, two benchmark scenarios are established for comparison with the proposed method (which utilizes probabilistic interval inputs): Scheme 1 (Deterministic Input), which adopts only the predicted expected values () of PV and load without considering prediction error intervals; and Scheme 2 (Industrial Approach), which executes the static assessment method in strict accordance with the DL/T 2041—2019 standard.
Figures 4, 5 illustrate the temporal evolution trajectories of the DHC under Scheme 1, Scheme 2, and the proposed method on a typical day. Scheme 1 fails to recognize the overvoltage risk triggered by cloud shading because it neglects prediction residuals during high-volatility PV periods, leading to an overestimation of the theoretical DHC by approximately 3.6 MW. In contrast, the proposed method successfully eliminates this risky capacity by constructing key stress scenarios, thereby achieving robust avoidance of dual source-load uncertainties.
FIGURE 4
FIGURE 5
During the evening peak heavy-load period (17:30–20:00), Scheme 2 maintains a high DHC of 10.8 MW due to the lack of quantitative evaluation of the system’s disturbance rejection capability. However, the proposed method successfully captures the potential risk of the operating point approaching the nose of the P-V curve, forcing the safe operational boundary to converge tightly to 4.4 MW via the SVSM constraints. The traditional method exhibits a deviation of up to 145% under heavy-load conditions, demonstrating that the SVSM constraint has become the decisive factor limiting PV accommodation during this critical period.
5.3 Performance comparison analysis of different solution algorithms
To verify the effectiveness of the proposed dynamic hosting capacity assessment method, simulations are conducted based on the relevant parameters from
Section 5.1, and the following three evaluation solution algorithms are set up for comparison:
Scheme 3 (Physical Benchmark-PSO): Adopts the same improved PSO algorithm as this paper but without integrating the DNN surrogate model; each iteration directly calls the continuation power flow program for SVSM and steady-state constraint verification.
Scheme 4 (Classical Genetic Algorithm-GA): Employs a standard genetic algorithm for optimization, with fitness evaluation also based on the physical model; the population size and maximum iterations are kept consistent with the proposed method.
Scheme 5 (Step-by-Step Approximation Method): A heuristic rule method commonly used in the industry, which uses the existing installed capacity as weights and gradually increases the PV capacity of each node with a fixed step size (set to 0.1 MW) until any constraint limit is violated.
Independent solutions are executed for all 96 time cross-sections throughout the day to calculate the computational time consumption, optimization accuracy (using the physical optimal solution of Scheme 3 as the benchmark), and convergence characteristics of each algorithm. Table 3 summarizes all five comparison schemes and the proposed method used in this study, clarifying their key differences in input type, solution algorithm, SVSM constraint handling, and surrogate model integration.
TABLE 3
| Scheme | Name | Input type | Solution algorithm | SVSM constraint | Surrogate model |
|---|---|---|---|---|---|
| Scheme 1 | Deterministic input | Deterministic (P50) | Improved PSO | Yes | No |
| Scheme 2 | Industrial approach | Deterministic (P50) | Static enumeration (DL/T 2041-2019) | No | No |
| Scheme 3 | Physical benchmark-PSO | Probabilistic (P10/P90) | Improved PSO | Yes | No |
| Scheme 4 | Classical genetic algorithm-GA | Probabilistic (P10/P90) | Standard GA | Yes | No |
| Scheme 5 | Step-by-step approximation | Probabilistic (P10/P90) | Heuristic rule | Yes | No |
| Proposed method | Hybrid DNN-PSO | Probabilistic (P10/P90) | Improved PSO | Yes | Yes (DNN) |
Overview of comparison schemes.
5.3.1 Comparison of computational efficiency and real-time performance
Table 4 presents the average time consumption statistics of the four algorithms for a single cross-section and the full-day task. For Schemes 3 and 4, since the SVSM constraints involve the singularity check of the Jacobian matrix or continuation power flow calculations, a single fitness evaluation requires a massive amount of computation. PSO and GA need to invoke the physical model thousands of times during the optimization process, resulting in a full-day assessment taking over 7 min, which is difficult to meet the quasi-real-time refresh requirements of grid dispatching.
TABLE 4
| Algorithm name | Surrogate model | Physical verification mechanism | Average time per step (s) | Total daily time (s) | Acceleration ratio |
|---|---|---|---|---|---|
| Physical Benchmark-PSO | None | Every iteration | 4.80 | 460.8 | 1.0x |
| Classical GA | None | Every iteration | 5.12 | 491.5 | 0.9x |
| Step-by-step approximation | None | Every iteration | 1.25 | 120.0 | 3.8x |
| Proposed method | Yes | Only final validation | 0.05 | 4.8 | 96.0x |
Comparison of computational efficiency among different algorithms.
Benefiting from the millisecond-level inference capability of the DNN surrogate model, the proposed method transforms complex physical field verifications into simple matrix operations, calling the physical model only once for dual verification when outputting the final solution. Compared with the physical benchmark (Scheme 3), the proposed method achieves an acceleration ratio of nearly 96 times, completing the full-day calculation in approximately 5 s, thereby demonstrating high value in engineering practicability.
5.3.2 Comparison of optimization accuracy and global search capability
Figure 6 compares the DHC assessment result deviations of the four algorithms at 12:00 (a typical high PV output moment) and 19:00 (a typical heavy-load moment). Because Scheme 3 utilizes a full physical model combined with the PSO algorithm for high-intensity iterative optimization, its calculation results are theoretically regarded as the physical limit capacity of the system at that moment.
FIGURE 6
During the high PV output period at 12:00, Scheme 5, limited by its fixed step-size strategy and the lack of a global search mechanism, yields an assessment result of only 4.50 MW, exhibiting significant conservatism. In contrast, the optimization result of the proposed method is 5.57 MW, approaching the physical limit. Particularly at the 19:00 heavy-load moment, the feasible region of the system is severely compressed by the SVSM constraint; however, the proposed method can still accurately converge to 4.45 MW, which is consistent with Scheme 3, whereas Scheme 5 fails to search for a valid solution under this strict operating condition. This indicates that the proposed method possesses excellent global optimization capabilities in conventional scenarios and can accurately lock onto the physical safety boundary in tightly constrained scenarios.
5.3.3 Comparison of convergence characteristics
To deeply analyze the optimization mechanism and convergence behavior of intelligent algorithms when solving the dynamic hosting capacity problem, Figure 7 plots the fitness evolution curves of the proposed method, Scheme 3, and Scheme 4 at the typical moment of 12:00. Given that Scheme 5 is a deterministic heuristic rule without the iterative convergence characteristics typical of swarm intelligence algorithms, this section only conducts a comparative analysis of the convergence trajectories of the first three intelligent optimization algorithms.
FIGURE 7
The convergence trajectories show that Scheme 4 is easily affected by random mutation operations, leading to premature convergence and poor final solution quality. Although Scheme 3 can converge to the optimal solution, its lack of prior guidance results in lower search efficiency. Benefiting from the hybrid initialization strategy, the proposed method exhibits a steep upward trend in fitness in the early stage of iteration, locking into a high-quality solution space after approximately 10 iterations. It is particularly noteworthy that the adaptive fallback feature presented in the later stage of the curve verifies the effectiveness of the dual verification mechanism: when the surrogate model makes minor misjudgments, the physical verification step can forcibly revert the solution back to the absolutely safe physical boundary. This eliminates the potential prediction risk of the surrogate model while simultaneously ensuring highly efficient optimization.
5.4 Spatiotemporal distribution characteristics of DPV hosting capacity
To reveal the spatiotemporal heterogeneity of the distribution network’s carrying capacity, a visual analysis is performed on the DHC assessment results and their corresponding voltage states across all system nodes on a typical day. As illustrated in Figure 8, the DHC distribution exhibits significant spatial divergence. Nodes at the head of the feeder (e.g., N2, N6) possess higher accommodation capabilities because their short electrical distances and low line impedances result in a weaker voltage rise effect. Conversely, the hosting capacity of terminal nodes (e.g., N28, N33) is strictly locked at a low level due to cumulative line voltage drops and voltage deviation constraints.
FIGURE 8
In the temporal dimension, during the midday period of high PV generation, the overall voltage baseline of the system is elevated, causing the accommodation margins of certain nodes to contract noticeably compared to the light-load periods of dawn and dusk. Furthermore, Figure 9 plots the 3D surface map of nodal voltage magnitudes across the entire network under the condition of accommodating the maximum calculated capacity. During the period from 10:00 to 15:00, although the voltage magnitudes at the terminal ends of the distribution network approach the upper safety limit of 1.07 p. u., no violations occur. This demonstrates that the proposed method can precisely delineate the grid security boundary, fully exploiting the voltage safety margin of the network without breaching physical constraints, thereby achieving an optimal balance between asset utilization and operational security.
FIGURE 9
5.5 Sensitivity analysis of the static voltage stability margin threshold
5.5.1 Nonlinear constraint mechanism of safety margins on capacity
The static voltage stability margin threshold is a critical parameter for balancing grid security with PV accommodation capability. The default value was previously set to . To explore the specific impact of this parameter on the DHC results, a sensitivity analysis is conducted at the evening peak heavy-load moment (18:00) by setting to 5%, 10%, 15%, and 20%, respectively.
Figure 10 illustrates the variations in the hosting capacity of the distribution network during the heavy-load period under different settings. As gradually increases from 5% to 20%, the DHC of the distribution network during the evening peak heavy-load period exhibits a significant nonlinear decay trend. When a lower standard of is applied, the system is permitted to operate in a region approaching the nose of the P-V curve, yielding a DHC of up to 8.5 MW. However, when the standard is raised to 20%, the capacity space is severely compressed to approximately 3.5 MW, representing a sharp decrease of up to 58%.
FIGURE 10
The physical essence of this phenomenon lies in the fact that under heavy-load conditions, the system operating point is situated in the highly nonlinear and steep region of the P-V curve, where a minor incremental demand for the stability margin leads to a rapid depletion of voltage stability reserves. Figure 11 visually illustrates this “crowding-out effect”: as safety standards rise, the green region representing physical accommodation capability is forced to shrink significantly to accommodate the mandatory reserved stability margin (yellow region), revealing the inherent trade-off between essential security and capacity benefits.
FIGURE 11
5.5.2 Recommendations for adaptive operational strategies
Based on the aforementioned sensitivity patterns, it is recommended that the dispatch department establish a risk-aware dynamic adaptive adjustment mechanism. During steady-state periods with favorable meteorological conditions and high load forecasting accuracy, the threshold can be appropriately lowered (e.g., to 5%–8%) to maximize the release of the grid’s physical potential and enhance renewable energy consumption. Conversely, during high-risk periods characterized by extreme weather, severe load fluctuations, or system maintenance configurations, should be proactively raised to 15%–20%. Through this time-of-use and zonal, dynamically quantified management strategy, certain PV outputs can be restricted in exchange for the system’s intrinsic safety, thereby maximizing the utilization value of grid assets under controllable risk conditions.
6 Conclusion
This paper proposes a DHC assessment method for distribution networks with high penetrations of distributed photovoltaics (PV), explicitly accounting for static voltage stability constraints and source-load temporal correlations. By integrating the SVSM into the constraint framework, the proposed method effectively mitigates the operational risk of neglecting voltage stability during evening peak heavy-load periods, which is a common oversight in traditional deterministic or static approaches. Through the construction of KSS and the development of a hybrid solution framework combining a DNN surrogate model with an improved PSO algorithm, an efficient and precise evaluation of dynamic hosting capacity is achieved, successfully resolving the computational bottleneck associated with complex nonlinear constraints. Furthermore, case studies quantify the nonlinear restrictive patterns of safety margins on the grid’s accommodation capability, providing a rigorous decision-making basis for grid dispatchers to balance operational safety and economic efficiency.
Nevertheless, certain limitations remain in the current study. First, the methodology relies on a single-phase power flow model assumption, thereby neglecting the three-phase imbalance issues prevalent in low-voltage distribution networks, which may affect evaluation accuracy. Second, the assessment primarily focuses on the passive accommodation capability of the grid, without fully considering the active regulation capabilities of flexible resources such as energy storage systems and soft open points. Third, the training of the surrogate model depends on a fixed network topology, indicating a need for enhanced generalization capability to seamlessly adapt to network reconfigurations or topology alteration scenarios. Fourth, the loads are represented by a constant-power (PQ) model, which constitutes the most demanding case for voltage stability, as such loads maintain constant power consumption even as the voltage declines, thereby accelerating the approach to the P–V nose point. Adopting voltage-dependent (e.g., ZIP or constant-impedance) load models would partially relieve the effective demand at depressed voltages and enlarge the SVSM, which may yield a somewhat higher assessed hosting capacity during heavy-load periods; the results reported herein are therefore conservative in this respect.
Future research will pivot toward the following directions:
Incorporating a three-phase four-wire model to construct a refined assessment framework adapted to unbalanced operating conditions.
Shifting the assessment paradigm toward active management by exploring capacity enhancement strategies under source-network-load-storage coordinated control.
Exploring topology-aware algorithms, such as graph neural networks (GNNs), to bolster the model’s adaptability to network structure variations and further improve its engineering applicability.
Adopting voltage-dependent (ZIP) load models to quantify the sensitivity of the assessed dynamic hosting capacity to load characteristics and to relax the conservatism of the constant-power assumption.
Statements
Data availability statement
The original contributions presented in the study are included in the article/supplementary material, further inquiries can be directed to the corresponding author.
Author contributions
ZY: Conceptualization, Writing – review and editing, Writing – original draft, Methodology, Supervision. JL: Validation, Data curation, Methodology, Writing – review and editing, Writing – original draft. DZ: Writing – original draft, Validation, Formal Analysis, Visualization. DL: Investigation, Resources, Software, Writing – review and editing. YF: Formal Analysis, Methodology, Writing – review and editing, Visualization, Data curation. JS: Software, Funding acquisition, Resources, Writing – review and editing, Writing – original draft, Project administration.
Funding
The author(s) declared that financial support was received for this work and/or its publication. The APC was funded by Inner Mongolia Power Company 2025 Science and Technology Project “Research and Application of Key Technologies for Distributed PV Hosting Capacity Assessment and Voltage Active Support in Distribution Networks for New-Type Power Systems”.
Conflict of interest
Author(s) ZY, JL, DZ, DL, YF and JS were employed by Xuejiawan Power Supply Company of Inner Mongolia Electric Power (Group) Co., Ltd.
The author(s) declared that this work received funding from Inner Mongolia Power Company. The funder had the following involvement in the study: providing financial support for the research of the project’s technical route and software development.
Generative AI statement
The author(s) declared that generative AI was not used in the creation of this manuscript.
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All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.
Abbreviations
DHC, Dynamic Hosting Capacity; DNN, Deep Neural Network; DPV, Distributed Photovoltaics; GA, Genetic Algorithm; GNN, Graph Neural Network; KSS, Key Stress Scenarios; LHS, Latin Hypercube Sampling; MFPU, Maximum Forward Power Flow and Undervoltage; MRPO, Maximum Reverse Power Flow and Overvoltage; PSO, Particle Swarm Optimization; PV, Photovoltaic; SVG, Static Var Generator; SVSM, Static Voltage Stability Margin.
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Summary
Keywords
distributed photovoltaics, dynamic hosting capacity, key stress scenarios, probabilistic forecasting, static voltage stability margin
Citation
Yan Z, Liu J, Zhang D, Liu D, Feng Y and Shu J (2026) Dynamic hosting capacity assessment of photovoltaics in distribution networks considering static voltage stability margin. Front. Energy Res. 14:1916798. doi: 10.3389/fenrg.2026.1916798
Received
23 June 2026
Revised
30 June 2026
Accepted
20 July 2026
Published
03 September 2026
Volume
14 - 2026
Edited by
Xizhen Xue, Nanyang Technological University, Singapore
Reviewed by
Zhe-Kang Dong, Hangzhou Dianzi University, China
Kai Tao, Nanjing University of Posts and Telecommunications, China
Updates
Copyright
© 2026 Yan, Liu, Zhang, Liu, Feng and Shu.
This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.
*Correspondence: Zhihai Yan, xjw20262026@outlook.com
Disclaimer
All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article or claim that may be made by its manufacturer is not guaranteed or endorsed by the publisher.