HBA-VSG Joint Optimization of Distribution Network Voltage Control Under Cloud-Edge Collaboration Architecture

Abstract

High-penetration integration of distributed photovoltaics (PV) into distribution networks introduces significant challenges regarding voltage limit violations and fluctuations. To address these issues, this manuscript proposes a hierarchical coordinated voltage control strategy for medium- and low-voltage distribution networks utilizing a cloud-edge collaboration architecture. The research methodology involves constructing a multi-objective optimization model at the cloud layer to minimize network losses and voltage deviations, solved via an improved Honey Badger Algorithm (HBA). Simultaneously, at the edge layer, a multi-mode coordinated control strategy incorporating Virtual Synchronous Generator (VSG) technology is developed to provide fast reactive power support and inertial response. Through simulation analysis on an IEEE 33-node test system, the findings demonstrate that the proposed strategy significantly mitigates voltage fluctuations and enhances the hosting capacity of distributed energy resources. The study concludes that the cloud-edge framework effectively decouples control time-scales, ensuring both global economic operation and local transient stability. These results are significant for advancing the resilient operation of active distribution networks with high renewable penetration.

1. Introduction

With the large-scale integration of high-penetration distributed renewable energy sources into distribution networks, the power fluctuation and bidirectional power flow characteristics have become increasingly significant, leading to frequent occurrences of voltage limit violations [1,2,3]. In recent years, significant progress has been made in voltage control strategies based on cloud-edge collaboration architectures. Reference [4] proposed an adaptive resource allocation method for edge computing resources to enhance cloud-edge collaborative security. Reference [5] introduced a machine learning-based optimization algorithm to minimize data processing delay and energy consumption, thereby improving distribution network resilience. Furthermore, Reference [6] proposed a GCN-based evaluation network to reduce the solution space for voltage control, while Reference [7] developed a reactive power voltage cooperative control method based on Edge X Foundry to alleviate cloud-side computing pressure.

Building on these foundations, recent studies have further expanded hierarchical control strategies. Reference [8] proposed a dual-layer optimization model based on regional autonomy, effectively reducing voltage violations through active–reactive coordination. Similarly, Reference [9] introduced a hierarchical framework combining centralized control with distributed execution, which significantly enhances system scalability under communication constraints. However, most of these cloud-edge strategies predominantly focus on steady-state economic dispatch, often overlooking the millisecond-level transient fluctuations caused by rapid PV variability. To bridge this gap, VSG technology [10,11] has been widely adopted to provide essential inertial support for weak grids, though its effective coordination with cloud-level optimization remains limited. Regarding global optimization, HBA [12,13] has recently demonstrated superior robustness and convergence speed in complex optimal power flow problems compared to traditional metaheuristics like PSO and GA.

Therefore, although significant progress has been made, current methods still face challenges in dynamic resource scheduling and multi-objective collaborative optimization. To address these limitations, this paper proposes a hierarchical cloud-edge collaborative control scheme that integrates the fast inertial response of VSG with the robust global optimization capability of HBA.

First, to address the lack of inertial support in existing edge-side methods, VSG strategy is introduced to suppress millisecond-level voltage fluctuations. Second, to overcome the premature convergence of traditional algorithms in high-dimensional optimization, an Improved HBA with Gaussian mutation is proposed to enhance robustness. Third, the Artstein transformation is integrated to explicitly compensate for communication delays, resolving the latency issues often neglected in prior frameworks.

2. Hierarchical Control Strategy for Distribution Networks Under Cloud-Edge Collaboration Architecture

The proposed architecture, as illustrated in Figure 1, adopts a “Cloud-Edge-Terminal” hierarchical framework designed to decouple control tasks across distinct temporal scales. The cloud layer functions as the global decision-making center, orchestrating the Improved HBA on a minute-level time-scale to minimize network losses and voltage deviations globally. The edge layer, deployed at distribution transformers, addresses millisecond-level transient voltage fluctuations by leveraging local inverter resources. It acts as a critical interface, aggregating terminal data for upload to the cloud while dispatching real-time control commands to the terminal layer, comprising PV inverters, energy storage systems, and controllable loads, via secure communication links.

To ensure effective coordination, the architecture adopts a time-scale decomposition. The edge layer operates on a millisecond time-scale, prioritizing transient stability via VSG inertial response and local Q(U) control. Conversely, the cloud layer functions on a minute-level time-scale (10–15 min), utilizing HBA for global optimization of network losses and voltage deviations. This decoupling balances global economic efficiency with local system stability.

2.1. Edge Control Strategy

Traditional voltage regulation devices, such as mechanical capacitor banks, feature slow response and low adjustment accuracy, making them difficult to adapt to the randomness and rapid changes in distributed power sources. In contrast, power electronic inverters, with their millisecond-level reactive power regulation capability, have become a core technology in modern smart grids by stabilizing voltage through real-time control of their reactive power output. Against the backdrop of large-scale application of photovoltaic and energy storage devices, this paper proposes a voltage control method based on photovoltaic and energy storage inverters [14].

2.1.1. Voltage Control Method Based on Inverter Reactive Power

Q(U) control is a common reactive power control method, and its control curve is shown in Figure 2. When the grid-connected point voltage U is lower than the minimum value of the target voltage, the inverter delays the decrease in the node voltage by injecting reactive power; when the grid-connected point voltage is higher than the maximum value of the target voltage, the inverter delays the increase in the node voltage by absorbing reactive power; if the grid-connected point voltage reaches the preset upper limit U_max (or lower limit U_min) of the voltage, the inverter absorbs or injects reactive power according to the maximum reactive capacity [15].

2.1.2. Voltage Control Method Based on Inverter Active Power

Active power curtailment is an effective strategy to mitigate overvoltage and line overloads. Under normal conditions, PV inverters operate in MPPT mode to maximize energy efficiency, as illustrated by the output characteristics in Figure 2. However, when the nodal voltage exceeds the allowable limit U_max, the system overrides MPPT. Instead, it reduces active power injection according to a predefined voltage–active power curve, thereby effectively suppressing overvoltage and restoring stability [16].

2.1.3. Voltage Control Method Based on VSG

The VSG control strategy enhances power system stability by emulating the inertia and damping characteristics of synchronous generators. By incorporating virtual coefficients into dynamic equations to simulate electromechanical transients, this method enables the inverter to autonomously and precisely regulate voltage and frequency in response to grid disturbances [17]. The control principle of VSG is shown in Figure 3.

Drawing reference from the mechanical torque regulation principle of real synchronous generator sets, the rotor speed regulation equation of a VSG can be expressed as follows:

$$J{\displaystyle \frac{d\omega }{dt}}=T_m-T_e-D_p(\omega -{\omega }_0)$$

(1)

In the equation, J is the moment of inertia (kg·m²), determining the intensity of the inertial response; ω is the VSG output angular frequency (rad/s); ω₀ is the rated angular frequency of the power grid; T_m is the virtual mechanical torque (Nm), generated from the active power reference value; T_e is the virtual electromagnetic torque (Nm), related to the output active power; and D_p is the active damping coefficient (Nm·s/rad), suppressing frequency oscillations.

When operating in grid-connected mode, the VSG can automatically follow changes in grid frequency to achieve power tracking. In the event of grid frequency disturbances, the proportional feedback coefficient D_p enables it to emulate the primary frequency modulation process of real generator sets, automatically adjusting the magnitude of active power injected into the grid. The introduction of rotational inertia endows its regulatory dynamic process with inertial characteristics, while the adjustability of virtual parameters allows the inverter to achieve a wider regulation range compared to real synchronous generator sets [18,19].

With reference to the excitation regulation principle of actual synchronous generator sets, the stator voltage regulation equation of the VSG can be expressed as follows:

$$K{\displaystyle \frac{dV}{dt}}=Q_{\mathrm{ref}}-Q_{\mathrm{out}}-D_q(U-U_{\mathrm{ref}})$$

(2)

In the equation: K is the excitation integral coefficient (s), which determines the voltage regulation speed; U is the amplitude of the VSG output voltage (V); U_ref is the reference voltage; Q_ref is the reactive power reference value (var); Q_out is the output reactive power (var); and D_q is the reactive damping coefficient (s⁻¹), which suppresses voltage oscillations.

When operating in grid-connected mode, the VSG can track reactive power. In the event of disturbances to U, it can similarly emulate the primary voltage regulation process of real synchronous generators, proactively participating in the regulation of grid reactive power. The magnitude of its regulation is determined by K, and the regulation speed can be adjusted through K.

In the coordinated control strategy, the VSG handles millisecond-level voltage fluctuations, while the Q(U) control based on the voltage–reactive power relationship focuses on minute-level voltage regulation. The two eliminate control conflicts through time-scale decoupling. The structure of the VSG reactive power control loop is optimized: the traditional voltage PI controller is replaced with a Q(U) droop formula based on voltage sensitivity, achieving a synergistic improvement in dynamic response characteristics and steady-state regulation capability.

$$Q_{ref}=Q_0-K_v\left(U_{meas}-U_{ref}\right)$$

(3)

In the equation: U_meas is the real-time measured voltage at the grid-connected point; K_v is the adaptive adjustment coefficient based on grid impedance. When K_v is relatively high, rapid voltage regulation can be achieved, but the VSG damping needs to be increased to avoid oscillations. When K_v is low, the regulation is gently performed relying on the VSG inertia, which is suitable for high-fluctuation scenarios.

The selection of control parameters is critical for system stability. The virtual inertia J and damping coefficient D are determined based on the small-signal stability analysis of the inverter. J is tuned to provide sufficient inertial support without introducing excessive oscillatory modes, while D is selected to ensure the system damping ratio $\zeta \ge 0.707$ for rapid oscillation suppression. Furthermore, the adaptive coefficient K_v in Equation (3) is designed based on voltage sensitivity analysis. In stiff grid conditions, a larger K_v is permitted to enhance regulation speed. Conversely, in weak grid conditions, K_v is dynamically reduced to prevent control loop instability. This adaptive tuning ensures robust performance across varying grid strengths.

2.1.4. Multi-Mode Coordinated Control Strategy

A multi-mode coordinated control system is constructed, with the photovoltaic grid-connected point voltage U as the core control variable, and the integration of three control methods is realized through a multi-time-scale coordination mechanism.

Real-time monitoring is performed on the grid-connected point voltage U, photovoltaic active power P_pv, reactive power Q_pv, and state parameters of the energy storage inverter; voltage control intervals are set as follows:

(1)

Normal Operation Zone, $\left[U_{ref}-\Delta U,U_{ref}+\Delta U\right]$:

U_ref is the target voltage and ΔU is the allowable voltage fluctuation deviation.

The active power control of the photovoltaic inverter adopts the MPPT strategy, and the maximum P_mp is achieved by adjusting the duty cycle of the converter. The reactive power control is adaptively regulated based on the Q(U) curve to maintain $Q_{pv}=k\left(U_{ref}-U\right)$, where k is the reactive power regulation coefficient satisfying $\left|Q_{pv}\right|\le Q_{max}$. The energy storage inverter operates in the VSG pre-synchronization mode, tracking the system frequency and voltage phase in real time, and maintaining an active power reserve capacity of no less than 20% of the rated power.

(2)

Early Warning Regulation Zone, $U\in \left(U_{min},U_{ref}-\Delta U\right)\cup \left(U_{ref}+\Delta U,U_{max}\right)$:

U_min/U_max are the voltage safety limits.

(a) Low-Voltage Condition $\left(U<U_{ref}-\Delta U\right)$

Reactive Power Compensation:

The inverter switches to the VSG voltage control mode, providing inertial response via the Virtual Synchronous Generator equation, $T_{J\omega }=P_m-P_e-D\left(\omega -{\omega }_0\right)$, synchronously compensating for reactive power deficits and slowing down the voltage drop rate.

Active Power Auxiliary Regulation:

If the voltage fails to recover within (t₁ = 50 ms), initiate active power derating control by reducing the grid-connected power as $P_{red}=P_{mp}\left(1-\alpha \left(U_{ref}-U\right)\right)$ (where α is the active power regulation coefficient, 0 < α < 1).

(b) High-Voltage Condition, $\left(U>U_{ref}+\Delta U\right)$:

Reactive Power Control with Energy Storage:

The inverter enters the VSG voltage control mode, adjusting reactive power output while the energy storage system absorbs excess active power. The virtual damping coefficient D suppresses voltage oscillations.

Flexible Active Power Curtailment:

If the voltage does not decline within t₁ = 50 ms, implement stepwise power curtailment according to the voltage–active power curve:

First-stage curtailment: $\Delta P_1=10\%P_{mp}$

Subsequent stages: Each additional curtailment of $10\%P_{mp}$ with a time interval of t₂ = 100 ms.

This hierarchical control strategy ensures rapid response to voltage deviations while maintaining system stability through coordinated reactive and active power regulation.

(3)

Emergency Control Zone, $\left(U\le U_{min}\mathrm{or}U\ge U_{max}\right)$:

The inverter starts the VSG emergency support mode, invokes all available reactive power capacity, and forcibly raises the voltage to the safe range through the droop control equation $U=U_0-k_p\left(Q-Q_0\right)$.

Active power auxiliary reduction: The active power is reduced to $P_{safe}=50\%P_{mp}$ (anti-islanding protection threshold).

Coordination mechanism: When the voltage exceeds the limit for more than the set time, a voltage limit violation alarm is triggered, real-time data is uploaded to the cloud server, and a request is sent to the superior dispatching to coordinate neighboring distributed power sources to participate in voltage regulation.

Analysis of Strategy Advantages:

• Multi-time-scale Coordination:

The VSG handles millisecond-level voltage fluctuations, while the Q(U) control is responsible for minute-level regulation. Time-scale separation avoids control conflicts, ensuring that dynamic and steady-state adjustments operate synergistically without interference.

2.

Hierarchical Capacity Utilization:

The strategy prioritizes the use of the inverter’s reactive power capacity, followed by invoking the active power regulation margin of energy storage, and finally curtails photovoltaic active power. This hierarchical approach maximizes equipment utilization efficiency by fully exploiting available resources in a sequential manner.

3.

Fault-tolerant Design:

Through threshold grading and time-delay criteria, frequent switching of control strategies caused by transient voltage fluctuations is avoided. This enhances system robustness, ensuring stable operation even under intermittent or short-term voltage disturbances.

2.2. Centralized Optimization and Scheduling in the Cloud

To effectively reduce the computational and communication burden of coordinated control in the cloud-side distribution network, a centralized–distributed scheduling method based on the Honey Badger Algorithm is proposed. In the day-ahead stage, a centralized optimization calculation is performed by the cloud. In the real-time stage, the coordinated regulation of each transformer supply area is transformed into distributed control within the distribution transformer supply area group. The centralized–distributed scheduling method can solve the high computational pressure problem of centralized control methods and also overcome the global limitations of distributed control methods.

(1)

Basic optimization algorithm [20]:

The high-dimensional and nonlinear nature of cloud centralized scheduling challenges traditional algorithms like PSO and GA, which often suffer from premature convergence and complex parameter tuning. To address this, HBA is introduced. Mimicking the ‘smelling’ and ‘digging’ foraging behaviors, HBA dynamically balances global exploration and local exploitation. Its adaptive parameter mechanism ensures rapid convergence and robustness against local optima, making it well-suited for complex distribution network optimization [21]. The algorithm flowchart is shown in Figure 4

(2)

Improved Honey Badger Algorithm

In the original algorithm, both modes of the Honey Badger Algorithm are determined by the decision coefficient c. To enable a smoother and more reasonable transition between the two search modes for the honey badger population, a time control mechanism is introduced, where the function c(t) is expressed as follows:

$$c\left(t\right)=\left|\left({\displaystyle \frac{t_{\max }-t}{t_{\max }}}\right)\left(2r_1-1\right)\right|$$

(4)

In the formula: t and t_max represent the current number of iterations and the maximum number of iterations, respectively; r₁ is a random number within the range [0, 1].

It is assumed that when c(t) > 0.5, the honey badger population is in the honey-gathering stage, which enables it to effectively search for the global optimum at the beginning of iterations. When c(t) < 0.5, the population gradually approaches the optimal value, and the honey badger population enters the digging stage, thereby enhancing the local search capability of the algorithm.

To enhance the population diversity of the Honey Badger Algorithm and prevent the algorithm from falling into local optima, while improving the decision coefficient c, the population position update is also modified by introducing a Gaussian distribution function. Its probability density function is expressed as follows:

$$f\left(y|{\mu }_2,{\sigma }_2\right)={\displaystyle \frac{1}{\sqrt{2\pi }\sigma }}{e}^{{\displaystyle \frac{-\left(y-\mu \right)2}{2{\sigma }^2}}}$$

(5)

Assume that the expected value of the Gaussian distribution function is 0, which can generate a disturbance factor y following the Gaussian distribution. Let r₂ and r₃ be random numbers within the range [0, 1]; then

$$y=\sqrt{2{\sigma }^2\log \left({\displaystyle \frac{1}{1-l}}\right)}\cos \left(2\pi r_2\right)$$

(6)

In the formula: l is related to the number of algorithm iterations, and its expression is

$$l=\left[1-{\left({\displaystyle \frac{t}{t_{\max }}}\right)}^2\right]r_3$$

(7)

To enhance population diversity and prevent premature convergence—a common drawback in traditional metaheuristics—a Gaussian mutation mechanism is introduced. This mechanism balances the algorithm’s Exploration and Exploitation phases:

Exploration: When the perturbation y is large, it forces individuals to jump out of local optima, exploring new areas of the solution space.

Exploitation: As the iteration count l increases, the variance of the Gaussian distribution decays, allowing the algorithm to fine-tune the solution around the discovered global optimum.

Equation (7) utilizes this perturbation to maintain population diversity throughout the iterative process. After introducing the Gaussian distribution disturbance factor, the overall position update formula of the Honey Badger Algorithm is expressed as follows:

$$x_{\mathrm{new}}=\left\{\begin{array}{c}\{x_{\mathrm{prey}}+F\times \beta \times I\times x_{\mathrm{prey}}+F\times r_3\times \alpha \times d_i\times \\ |\cos (2\pi r_4)\times \left[\begin{array}{c}1-\cos (2\pi r_5)\end{array}\right]|\}\times y,c<0.5,\\ (x_{\mathrm{prey}}+F\times r_6\times \alpha \times d_i)\times y,c⩾0.5,\end{array}\right.$$

(8)

In the formula: x_new represents the new position of an individual honey badger; x_prey is the position of the beehive; r_i is a random number within [0, 1]; β is a constant; I denotes the attraction intensity factor; and F is used to change the search direction and expand the search range.

2.3. Artstein Transformation

To address the communication delay issue when cloud-layer commands are delivered to edge nodes in the cloud-edge collaboration architecture, this paper introduces a predictive compensation mechanism based on the Artstein transformation in the edge control layer. This mechanism constructs a real-time state predictor:

Let τ be the average communication delay of the downlink from the cloud to the edge. The globally optimized command received by the edge node is $u_{cloud}\left(t-\tau \right)$. The virtual state quantity ${\widehat{x}}_p(t)$ is constructed through the inverse Artstein transformation.

$${\widehat{x}}_p(t)=x(t)+{\int }_{t-\tau }^t{e}^{A(t-s)}B{u}_{cloud}(s)ds$$

(9)

where $x(t)$ is the locally measured node voltage state, and A and B are the system state-space matrices. Based on this predicted state, an edge controller $u_{edge}\left(t\right)$ is designed as follows:

$$u_{edge}\left(t\right)=K\cdot \left[{\widehat{x}}_p\left(t\right)-x_{ref}\right]$$

(10)

In practical distribution networks utilizing 4G/5G wireless communication, the downlink latency τ typically ranges from fifty milliseconds to several hundred milliseconds. While the edge controller handles fast dynamics locally, the global dispatch commands from the cloud are subject to this delay. Without compensation, using outdated state information, $x(t-\tau )$ can degrade control performance.

The Artstein transformation constructs a predictor $z(t)$ that estimates the future state $x(t)$ based on the system model and the known delay τ. This effectively ‘removes’ the delay from the feedback loop, allowing the controller to act on the predicted current state rather than the delayed historical state [22,23].

3. Computational Model of Hierarchical Control Strategy for Distribution Network with Cloud-Edge Collaboration

3.1. Analysis and Modeling of Low-Voltage Distribution Areas

(1)

Distributed Photovoltaic Power Generation:

The power generation of photovoltaic power is related to light intensity, temperature, and efficiency:

$$P_{pv}(t)={\eta }_{pv}\cdot A_{pv}\cdot G(t)\cdot (1-0.005(T(t)-25))$$

(11)

In the formula: G(t) is the solar irradiance at time t (W/m²); T(t) is the ambient temperature at time t (°C); A_pv is the area of the photovoltaic panel (m²); and η_pv is the comprehensive efficiency.

(2)

Energy Storage:

The state of charge (SOC) of energy storage varies with time:

$$SOC(t+1)=SOC(t)+{\displaystyle \frac{{\eta }_{ch}\cdot P_{ch}\left(t\right)-P_{dis}\left(t\right)/{\eta }_{dis}}{E_{ess}}}\cdot \mathsf{\Delta }t$$

(12)

In the formula: P_ch(t) and P_dis(t) are the charging and discharging powers at time t (kW), respectively; η_ch and η_dis are the charging and discharging efficiencies, respectively; and E_ess is the rated capacity of the energy storage (kWh).

(3)

Electric Vehicle Charging Piles:

Assume that the total power of the charging pile cluster is expressed as follows:

$$P_{ev}\left(t\right)=\sum _{i=1}^{N_{ev}}{x}_i\left(t\right)\cdot P_{ev}^{\mathrm{rate}}$$

(13)

In the formula: x_i(t) represents the charging state of the i-th vehicle at time t (0 indicates inactive, 1 indicates active); P_ev^rate is the charging power of a single pile (kW).

3.2. Optimization Scheduling Model for Cloud-Edge Collaborative Medium- and Low-Voltage Distribution Networks

3.2.1. Objective Functions

The overall control objectives of the medium- and low-voltage distribution network are to minimize the network loss P_loss, minimize the voltage deviation rate U_D of each node, and maximize the output of distributed generation P_DG, which are expressed as follows:

$$\left\{\begin{array}{c}\min P_{\mathrm{loss}}=\min {\displaystyle \sum _{i=1}^N}(U_i^2+U_j^2-2U_i{U}_j\cos {\theta }_{ij})g_{ij}\\ \min U_{\mathrm{D}}=\min {\displaystyle \sum _{i\in N_{\mathrm{MV}}}}{\displaystyle \frac{{(U_i-U_{\mathrm{N}})}^2}{{(U_{\max }-U_{\min })}^2}}\\ \max P_{\mathrm{DG}}=\max {\displaystyle \sum _{i=1}^n}{P}_{\mathrm{DG},i}\end{array}\right.$$

(14)

In the formula: i and j denote the starting and ending nodes of the line, respectively; N represents the number of system nodes; U_i and U_j are the voltages at the starting and ending nodes of the line, respectively; cosθ_ij is the cosine of the phase angle difference between node voltages; g_ij stands for the line conductance; U_N is the rated value of node voltage; U_max and U_min represent the maximum and minimum allowable values of node voltage, respectively; N_MV denotes all nodes in the medium- and low-voltage distribution network; n is the number of distributed generators; and P_DG,i is the grid-connected active power of the distributed photovoltaic system connected to node i.

The comprehensive objective function (Equation (15)) is constructed as a weighted sum:

$$\min f={\omega }_1{P}_{\mathrm{loss}}^{\ast }+{\omega }_2{U}_{\mathrm{D}}^{\ast }-{\omega }_3{P}_{\mathrm{DG}}^{\ast }$$

(15)

In the formula: f is the comprehensive objective function; P^*_loss represents the per-unit value of the network loss objective function; U^*_D denotes the per-unit value of the node voltage deviation objective function; and P^*_DG. is the per-unit value of the distributed generation accommodation objective function in the medium-voltage distribution network; Here, the base values for per-unitization are the respective index values in the initial state. ω₁–ω₃ are the weights of each sub-objective, which can be determined according to the actual control requirements of the medium- and low-voltage distribution network in the optimization target area, with the constraint ω₁ + ω₂ + ω₃ = 1.

The weighting coefficients are determined using the AHP based on operational priorities. In this study, voltage stability is identified as the critical bottleneck, thus assigned the highest priority. Renewable accommodation is secondary, followed by loss reduction.

3.2.2. Constraints

The control constraints of medium- and low-voltage distribution networks mainly include power flow constraints, branch current constraints, node voltage constraints, node photovoltaic output constraints, and adjustable low-voltage distribution area output constraints.

(1)

The power flow balance constraint is

$$\left\{\begin{array}{c}{P}_i-U_i{\displaystyle \sum _{j\in i}}{U}_j(G_{ij}\cos {\theta }_{ij}+B_{ij}\sin {\theta }_{ij})+P_{\mathrm{DG}.i}=0\\ Q_i-U_i{\displaystyle \sum _{j\in i}}{U}_j(G_{ij}\sin {\theta }_{ij}-B_{ij}\cos {\theta }_{ij})=0\end{array}\right.$$

(16)

In the formula: i and j represent the node numbers; G_ij and B_ij represent the conductance and susceptance values of the line between nodes i and j, respectively; P_i and Q_i represent the active power and reactive power of the node, respectively; U_i and U_j represent the node voltages; and θ_ij is the voltage phase difference between nodes i and j.

(2)

The branch current constraint is

$$I_{ij.\min }\le I_{ij}\le I_{ij.\max }$$

(17)

In the formula: I_ij is the branch current between nodes i and j; and I_ij.min and I_ij.max are the minimum and maximum allowable currents of branch ij, respectively.

(3)

The node voltage constraint is

$$U_{\min }\le U_i\le U_{\max }$$

(18)

(4)

The photovoltaic power output constraint is

$$P_{\mathrm{DG}.i\min }\le P_{\mathrm{DG}.i}\le P_{\mathrm{DG}.i\max }$$

(19)

In the formula: P_DG.imin and P_DG.imax are the minimum and maximum power outputs of the distributed photovoltaic connected to node i, respectively.

(5)

The power output constraint of the adjustable low-voltage distribution area is

$$P_{i.\min }\le P_i\le P_{i.\max }$$

(20)

In the formula: P_i.min and P_i.max are the minimum and maximum power outputs of the adjustable low-voltage distribution area connected to node i, respectively.

(6)

The energy storage power output constraint is

$$\left\{\begin{array}{c}0\le P_{ch}\left(t\right)\le P_{ess}^{\max }\\ 0\le P_{dis}\left(t\right)\le P_{ess}^{\max }\\ SO{C}^{\min }\le SOC(t)\le SO{C}^{\max }\\ P_{ch}\left(t\right)\cdot P_{dis}\left(t\right)=0\end{array}\right.$$

(21)

(7)

The electric vehicle charging pile constraint is

$$\left\{\begin{array}{c}{\displaystyle \sum _{t=S_i}^{D_i}}{x}_i\left(t\right)\cdot \mathsf{\Delta }t\ge E_i^{\mathrm{req}}\\ x_i\left(t\right)=0,t\notin \left[S_i,D_i\right]\end{array}\right.$$

(22)

3.3. Control Flow

Based on the cloud-edge collaborative architecture and hierarchical control model, a hierarchical coordinated voltage control flow for medium- and low-voltage distribution networks is proposed, as shown in Figure 5.

• Determine the control time-scale for the cloud-based centralized control of the medium-voltage distribution network. Considering the communication and computing capabilities between the cloud and edge terminals, this control time-scale is set to 10 min.
• Acquire load parameters of each node in the medium-voltage distribution network, as well as parameters of each controlled object—such as distributed photovoltaics, energy storage reactive compensation capacity, and adjustable low-voltage distribution areas in the network. The parameters of adjustable low-voltage distribution areas include the output range data of low-voltage distribution areas fed back in step 6.
• Solve the model using the HBA based on the cloud control model.
• Issue the solved control power commands to the adjustable edge-terminal low-voltage distribution areas and activate edge-terminal control.
• Obtain data of each node in the low-voltage distribution network, including parameters such as distribution transformer taps, distributed photovoltaics, energy storage, and loads.
• Perform real-time control of the edge terminal using adjustable variables such as inverters and capacitor bank switching based on the edge-terminal hierarchical control model.
• Determine whether the edge-terminal control range is exceeded. If yes, edge devices upload real-time data to the cloud, and return to step 2; if not, edge devices issue control commands to terminal devices to complete the control flow.

4. Case Analysis

4.1. Case Analysis of Multi-Mode Cooperative Joint Control

To verify the performance of the proposed control strategy, a typical station area was selected as the simulation object, which includes photovoltaic devices and energy storage devices. The key parameter settings are shown in

Table 1. Simulation parameter table.

Parameter Category	Parameter Name	Parameter Value
System parameters	Target voltage Uref	1.0 p.u.
	Normal operating range ΔU	0.04 p.u.
Photovoltaic parameters	Maximum power Pmp	0.5 MW
	Maximum reactive power capacity Qmax	0.1 MVar
Energy storage parameters	Capacity Ecap	1.0 MWh
	Maximum active power Pes_max	0.2 MW
	Maximum reactive power capacity Qes_max	0.1 MVar
VSG control parameters	Virtual inertia J	1.0
	Damping coefficient D	15
Control parameters	Voltage recovery gain	2.2
	Emergency trigger threshold	0.03 p.u.
	Anticipatory control gain	2.5

.

To comprehensively evaluate the control strategy, four test scenarios, as illustrated in Figure 6d, simulating distinct working conditions were designed:

• Normal operating state (0–10 s): The load is maintained stable at a reference value of 0.5 MW, with a rate of change of zero.
• Low-voltage state (10–25 s): The load gradually increases from 0.5 MW to 1.1 MW over a period of 15 s, simulating a peak load ramp event.
• High-voltage state (25–40 s): The load is gradually shed from 1.1 MW down to 0.1 MW, simulating a light-load condition with a negative rate of change.
• Fluctuation Scenario (40–60 s): Periodic load fluctuations are applied, oscillating between −0.5 MW and 0.7 MW with a frequency of approximately 0.25 Hz, simulating dynamic intermittency in the actual grid.

Analysis of Simulation Results:

• Normal operating state (0–10 s): The system voltage is stably maintained at 1.00 p.u., and the control mode remains in a normal state continuously
• Low-voltage state (10–25 s): The voltage drops from 1.00 p.u. to a minimum of 0.954 p.u. When the predicted voltage approaches Uref-ΔU, the control mode enters low-voltage warning, with the maximum reactive power output. When it approaches Umin, it enters low-voltage emergency mode, maintaining the active power of the photovoltaic while appropriately discharging the energy storage. The maximum voltage deviation observed is −0.046 p.u.
• High-voltage state (25–40 s): The voltage rises to a maximum of 1.031 p.u. and the predicted voltage approaches U_ref + ΔU, the control mode enters the high-voltage warning mode, with the maximum reactive power output and the active power decreasing in a stepwise manner.
• Fluctuation scenario (40–60 s): The voltage fluctuation range is between 0.981 p.u. and 1.045 p.u., and the mode switches rapidly between early warning and emergency.

Under the same test conditions, the voltage variation curve under the traditional droop control is shown in the following Figure 7. The comparison of indicators is shown in

Table 2. Indicator comparison table.

Indicator	Optimal Control Strategy	MPC Control	Traditional Droop Control
Maximum voltage deviation	0.045 p.u.	0.055 p.u.	0.077 p.u.
Voltage qualification	87%	82%	71%

below.

As illustrated in Table 2, the proposed optimal control strategy exhibits superior performance compared to existing methods. Specifically, it limits the maximum voltage deviation to a mere 0.045 p.u., which is significantly lower than the 0.055 p.u. achieved by MPC and the 0.077 p.u. by traditional droop control. Furthermore, the proposed strategy elevates the voltage qualification rate from 71% (under traditional control) to 87%, highlighting its distinct advantages in enhancing both voltage stability and the overall operational quality of the system.

A typical topology example of medium- and low-voltage distribution networks based on IEEE33 node networks is shown in Figure 8. The IEEE33 node system incorporates high-penetration distributed photovoltaic and energy storage devices.

Using the system-rated voltage as the base value, the voltage per-unit value at node 0 is set to 1. The PV curve and daily load fluctuation curve are shown in the Figure 9 and Figure 10 below.

To verify the performance of the Improved HBA in solving the cloud optimization model, its convergence characteristics are first compared with the Particle Swarm Optimization algorithm and the Genetic Algorithm. The three algorithms solve the multi-objective optimization model independently under the same conditions, and the convergence curve comparison is shown in Figure 11 and

Table 3. Comparison of different algorithms.

Algorithm	Avg. Convergence Iteration	Execution Time
HBA	15	3.42
AOA	28	4.15
GA	60	5.68
PSO	67	3.85

.

Figure 11 presents the convergence performance comparison of HBA against AOA, GA, and PSO over 100 independent runs. In terms of convergence speed, HBA exhibits the steepest descent slope during the initial phase, which is visibly faster than that of AOA, GA, and PSO. Regarding optimization accuracy, as iterations proceed, HBA eventually converges to the lowest objective function value, outperforming the other three comparative algorithms. Furthermore, the standard deviation range represented by the shaded areas indicates that HBA has the narrowest bandwidth, particularly in the later stages of iteration where its fluctuation amplitude is significantly smaller than that of PSO and GA, demonstrating higher consistency and stability across multiple runs.

The following scenarios are selected for simulation:

Scene 1: Reactive power regulation is carried out on all medium- and low-voltage photovoltaic inverters using the traditional drooping control curve without cloud-edge connection.

Scene 2: The cloud-edge collaborative medium- and low-voltage double-layer optimization strategy proposed in this paper is adopted for optimization and solution.

The simulation results are shown in the following Figure 12.

The voltage curves under different states are shown in Figure 11. In this case study, voltage limit violations in the target area are relatively frequent, power quality is poor, and the demand for voltage optimization is urgent. Therefore, ω₁, ω₂, and ω₃ are assigned values of 0.2, 0.5, and 0.3, respectively, based on actual needs. The indicator values of the objective function are shown in

Table 4. The index values of the objective function.

Operating Scene	Power Loss/kW	Voltage Deviation/p.u.	Distributed Power Consumption/kW
Scene 1	22,575.04	18.49	144,000
Scene 2	16,465.59	10.46	147,450.39

. The simulation results show that compared with the traditional method, the proposed optimization control algorithm reduces network loss by 27%, reduces the voltage deviation rate by 43%, increases distributed energy consumption by 23%, and results in a better overall system operating state.

4.2. Sensitivity and Robustness Analysis

The robustness of the proposed strategy is evaluated under varying PV penetration levels, communication delays, and measurement noise intensities, as illustrated in Figure 13, Figure 14 and Figure 15.

Sensitivity analysis confirms the strategy’s robustness. Even at 90% PV penetration, the peak voltage is clamped at 1.048 p.u., strictly satisfying the 1.05 p.u. safety limit. Despite communication latency up to 600 ms causing minor ripples, the system avoids divergence, maintaining voltage within the [0.95, 1.05] p.u. safe zone. Moreover, precise voltage regulation is sustained under measurement noise intensities ranging from 0 to 0.06 p.u.

4.3. Discussion and Limitations

While the simulation results demonstrate the superiority of the proposed HBA-VSG strategy, certain limitations should be acknowledged. First, the validation is performed on the IEEE 33-node standard test system. While representative, actual distribution networks may possess more complex topologies and unbalanced phase conditions that require further verification. Second, the study relies on numerical simulations. Although realistic models are used, hardware-in-the-loop experiments would provide stronger evidence of the strategy’s real-time performance, particularly regarding communication delay compensation. Future work will focus on experimental validation and extending the model to three-phase unbalanced networks.

5. Conclusions

This paper proposes a hierarchical voltage control strategy that integrates the Improved HBA with VSG control to address voltage regulation challenges in high-penetration active distribution networks. The primary contributions and findings are summarized as follows:

• The established cloud-edge collaborative architecture successfully decouples control tasks across different time-scales, effectively reconciling the inherent trade-off between long-term economic operation and millisecond-level transient stability.
• Quantitative simulation results demonstrate the superior performance of the proposed strategy, which reduces active power losses by 27% and voltage deviation by 43% compared to traditional droop control. The Improved HBA also manifests enhanced convergence speed and global search capability over traditional algorithms.
• The strategy exhibits strong robustness against stochastic disturbances, maintaining system stability even under extreme conditions involving 80% PV penetration and communication delays of up to 500 ms.

These findings provide a viable technical pathway for accommodating high shares of renewable energy without compromising grid security.