Two-Layer Optimal Power Allocation of a Vanadium Flow Battery Energy Storage System Based on Adaptive Simulated Annealing Multi-Objective Harris Hawks Optimizer

Abstract

The power allocation in Vanadium Redox Flow Battery (VRB) energy storage systems faces a conflict between long-term lifespan and real-time power coupling. Using a single-layer optimization method to directly address multiple objectives simultaneously may lead to conflicts among these objectives. Therefore, this paper presents a multi-objective two-layer optimization allocation strategy. Its core is hierarchical scheduling for long/short-term goals to optimize multi-attribute objectives precisely. A two-layer model comprising an initial allocation layer and an operational optimization layer is constructed to ensure the prioritization of long-term lifespan objectives based on a predefined hierarchical structure. The initial allocation layer focuses on the long-term objective of energy storage capacity lifespan, by prioritizing minimal capacity degradation. A differential evolution algorithm is then applied to perform preliminary allocation of the total power demand. The operational optimization layer aims to achieve optimal State of Charge (SOC) balance across all units and minimize power losses. An Adaptive Multi-Objective Harris Hawks Optimizer (ASAMOHHO) based on adaptive simulated annealing is established to find the Pareto optimal solution set, and ultimately determining the real-time power allocation plan for each unit. Comparative simulations with conventional methods were conducted, and the results demonstrate that the proposed strategy provides an efficient and practical solution for efficient VRB scheduling.

1. Introduction

Against the backdrop of a global push for energy transition and vigorous development of renewable energy, China’s new energy power generation has experienced rapid growth. Wind and solar power, with their clean and sustainable characteristics, have seen their share in the energy mix steadily increase. Experts predict that by 2060, wind and solar power will account for over 50% of China’s electricity generation [1], becoming the primary sources of power supply. However, the intermittent and fluctuating nature of wind and solar power increases the instability of the electricity supply. Large-scale energy storage plays a crucial role in mitigating these fluctuations, effectively enhancing power supply quality and reliability. Among current energy storage technologies, the VRB stands out in the field of large-capacity storage due to its high safety, long cycle life, high charge and discharge efficiency, and scalability [2,3].

Given the limited capacity of individual energy storage units, the simplest strategy to meet the high-power, high-capacity demands of energy storage systems in power grids, electric vehicles, and other applications is to combine multiple units in series or parallel configurations to scale capacity and power [4]. However, this approach often overlooks variations in internal battery parameters during power distribution, potentially leading to uneven power allocation among units. Even for the same energy storage unit, differences in their SOC during connection can lead to uneven current distribution among them (some units will carry more current). Prolonged uneven charging can cause some units to local current overload, potentially resulting in a short circuit. It could cause the entire system to shut down, resulting in economic losses. Therefore, a rational power distribution strategy for energy storage systems is a core factor determining their performance. Such a strategy not only optimizes operational costs but also enhances the stability and flexibility of the entire energy system [5,6].

In-depth research on power allocation in energy storage systems and the proposal of efficient and rational allocation methods represent one of the current research hotspots with critical practical significance. Xu et al. [7] proposed a power allocation method for energy storage systems based on battery health state prediction. Aiming to maintain consistency in the SOC across the battery pack, it employs an entropy weighting approach to determine the charge and discharge sequence of battery. This result stems from the design of the initial allocation strategy: the strategy ensures that the operating frequency of each energy storage unit matches its current health state. Simulation examples demonstrate its effectiveness in reducing battery health degradation. Duan et al. [8] proposed enhancing SOC consistency among storage units by combining charge and discharge priority sequencing with an adaptive variable particle swarm optimization algorithm. Yan et al. [9] introduced a power control strategy for energy storage systems considering battery pack consistency, incorporating the variation in battery pack voltage spread to adjust power under SOC constraints, effectively narrowing the SOC variation range of the battery pack. Zhao et al. [10] proposed a power allocation strategy based on a variable interval window method which designs the charging and discharging power and SOC of the energy storage system as an exponential function relationship, reinforcing the difference in allocation coefficients between energy storage systems at different SOC values. This enables the SOC of battery units within the energy storage power station to rapidly converge toward consistency. The aforementioned references all focus solely on SOC balancing as the single objective, without comprehensively considering issues such as economic efficiency and operational effectiveness.

Power allocation in energy storage systems is predominantly a nonlinear multi-objective problem. With advances in computer technology, intelligent optimization algorithms have been extensively applied in power allocation strategies. Chen et al. [11] proposed considering power losses within vanadium flow batteries and balancing the SOC storage units as objectives. Employing a virtual particle-adaptive differential evolution algorithm, it achieves reduced losses and optimized power allocation in the energy storage system. Chen et al. [12] comprehensively considered the lifespan and operational losses of vanadium flow batteries. Using multiple parallel VRB battery groups as energy storage units, it employs a simulated annealing particle swarm optimization algorithm for simulation comparisons, verifying the practicality of the optimization algorithm. Liu et al. [13] established a model with the objective of maximizing the electricity sales revenue of the energy storage system. Utilizing an Ant Lion optimization algorithm, it achieves a reasonable optimization and allocation of power for wind power combined with energy storage systems.

Research on power allocation in vanadium flow battery energy storage systems has evolved from single-objective optimization toward multi-objective coordination. For instance, the aforementioned literature simultaneously considers multiple objectives such as economic viability, capacity degradation, battery health, and SOC balancing. However, these studies exhibit limitations in addressing multi-objective problems: On one hand, the prevalent linear weighting approach converts complex objectives into a single-objective problem. The selection of weighting coefficients is subjective, making it difficult to balance multiple goals effectively. This approach may lead to conflicting objectives and potentially overlook optimal solutions. On the other hand, conventional single-layer optimization frameworks fail to distinguish between long-term lifespan objectives (capacity degradation) and real-time operational objectives (SOC balancing, power loss), leading to forced coupling that limits optimization effectiveness. Therefore, a two-layer optimization control strategy is proposed that comprehensively considers VRB battery capacity degradation, power loss, and SOC balancing among storage units. The upper layer focuses on long-term energy storage lifespan, while the lower layer concentrates on real-time dynamic balancing. This hierarchical architecture decouples multiple optimization objectives. The upper layer considers charge and discharge priorities, dynamically adjusts unit output, and aims to minimize capacity loss. It leverages the global search capability of the differential evolution algorithm to derive an initial allocation scheme. The upper layer’s initial solution provides a feasible solution space for the lower layer, reducing computational complexity. The lower layer then pursues dual objectives: minimizing SOC imbalance and power loss. The multi-objective Harris hawks algorithm, incorporating adaptive and simulated annealing mechanisms, determines the final power allocation scheme, achieving dynamic adaptation between long-term and real-time objectives. The innovations of this paper are as follows: (1) A decoupling strategy combining “upper-layer global initial allocation and lower-layer fine-grained multi-employs objective optimization” is proposed. The upper layer performs initial power allocation using a differential evolution algorithm integrated with SOC priority, setting constraints such as power balance and SOC boundaries. The lower layer employs an improved intelligent algorithm for fine-grained tuning, pursuing dual objectives of SOC equilibrium and power loss to achieve synergistic optimization of economic efficiency and stability. (2) A multi-objective Harris hawks algorithm (MOHHO) integrated with adaptive simulated annealing is proposed. An adaptive exploration/exploitation switching mechanism is designed to dynamically adjust the energy factor parameter, addressing the original algorithm’s imbalance between exploration and exploitation. Concurrently, a simulated annealing local search module with adaptive temperature is embedded during the exploitation phase. A probabilistic acceptance mechanism enhances the algorithm’s ability to escape local optima. Using the ZDT4 multi-peak test function, the improved MOHHO demonstrates superior convergence results compared to the unmodified algorithm.

2. Vanadium Redox Flow Battery Energy Storage System Architecture

2.1. Vanadium Redox Flow Battery

The main components of a vanadium flow battery consist of a stack, two independent positive and negative electrolyte tanks, circulation pumps, and piping systems [14], as shown in Figure 1. The circulation pumps facilitate the flow and circulation of electrolyte between the stack and the storage tanks. Within the positive and negative electrolytes, vanadium ions in different oxidation states undergo electrochemical reactions, thereby enabling the conversion between electrical and chemical energy.

2.2. Vanadium Flow Battery Energy Storage System Architecture

In practical applications, energy storage systems require the series and parallel connection of multiple small-capacity battery units. Vanadium flow batteries can be configured in either series or parallel arrangements. The series structure involves connecting multiple stacks in series, with all units sharing a common electrolyte tank for circulation. The drawback of a series connection is the increased complexity of piping connections; failure in any single unit can disrupt the entire energy storage system’s operation.

The VRB parallel expansion structure is shown in Figure 2. When adopting a parallel structure, each energy storage unit can receive commands independently, facilitating flexible control of each unit. This approach is more conducive to reducing variations among units. Therefore, this paper considers vanadium flow batteries with a parallel structure for power distribution.

A microgrid is an independent power system comprising distributed power sources (such as photovoltaic and wind power generation), energy storage systems, and loads (power-consuming equipment) [15]. It enables the timely utilization of renewable energy generation, thereby promoting renewable energy adoption [16,17]. This study investigates a DC microgrid incorporating wind and solar power generation, with multiple parallel VRB batteries serving as the energy storage system. The system architecture is illustrated in Figure 3.

3. Two-Layer Optimization Model

To effectively ensure the reliability of energy storage system operation and reduce capacity degradation and power losses during operation of vanadium flow batteries, a dual-layer power optimization allocation strategy is proposed. This strategy targets capacity degradation mitigation, SOC balancing among storage units, and power loss reduction, comprising an initial allocation layer and an operational optimization layer. The overall framework is illustrated in Figure 4.

In the initial allocation layer of the dual-layer optimization structure, after the power command is issued, units are first selected for priority operation based on their actual SOC values. Aiming to minimize capacity degradation of the vanadium flow battery, the allocation results are obtained using a differential evolution algorithm. The optimization operation layer establishes a multi-objective optimization model to achieve optimal SOC balance across vanadium flow battery energy storage units and minimize power loss. The adaptive simulated annealing multi-objective Harris hawks algorithm is employed to derive the optimal solution.

3.1. Initial Allocation Layer

To better achieve SOC balancing across energy storage units, a preprocessing sorting step is incorporated during initial allocation. First, charging and discharging priorities are assigned based on each unit’s SOC level. When issuing charging power commands, units with lower SOC are selected first according to the sorting order. For discharging, units with higher SOC are prioritized. During initial allocation, capacity degradation of vanadium flow batteries is given priority consideration. Subsequently, the power allocation scheme is further optimized using a differential evolution algorithm. These operations reduce battery aging rates during charge/discharge cycles, thereby lowering replacement costs and frequency.

Initial Allocation Layer Optimization Objectives

Vanadium flow batteries experience capacity loss during charge–discharge cycles, primarily due to concentration changes caused by vanadium ion diffusion in the electrolyte (excluding hydrogen evolution side reactions). Capacity degradation correlates with the number of cycles $N_s$. $r_{100}$ is the capacity decay rate of a storage unit during a single full charge–discharge cycle. Since not every cycle achieves 100% depth of discharge (DOD), the cumulative loss $d$ should be calculated based on the actual DOD values across all cycles. The total capacity loss $L_{\mathrm{z}}$ is determined by summing the loss values L from each cycle. When the available capacity of a vanadium flow battery degrades to 80% of its initial rated capacity, the battery is deemed to have reached end-of-life. Therefore, the parameter δ is set to 0.8, and $k\mathrm{p}$ is a constant derived from experimental data fitting, with a value of 0.85 [18].

$${(1-r_{100})}^{N_s}=\delta$$

(1)

$$L_{\mathrm{z}}=\sum _{t}L=\sum _t(0.5d^{k\mathrm{p}}{r}_{100})$$

(2)

The initial allocation layer employs a differential evolution algorithm for solution determination. Drawing inspiration from biological evolution principles, this algorithm treats the solution set as a population and achieves iterative optimization through genetic operations such as mutation and crossover. Throughout the optimization process, strict adherence to boundary conditions ensures that the generated allocation scheme satisfies both power requirements and constraint conditions, thereby yielding the optimal power allocation solution at the initial allocation layer.

3.2. Operational Optimization Layer

After the initial allocation layer completes power allocation, the resulting optimal power allocation scheme serves as an initial individual in the Harris hawks population within the operational optimization layer, providing a favorable starting solution. The operational optimization layer further refines this power allocation scheme based on this initial solution. It employs an adaptive simulated annealing multi-objective Harris hawks optimization algorithm within the neighborhood to comprehensively optimize the scheme by balancing both SOC equilibrium and power loss objectives.

Operational Optimization Layer Optimization Objectives

(1) Lowest Power Loss: Power losses in vanadium flow battery energy storage systems primarily consist of battery operational losses and DC-DC converter losses. Battery operational losses comprise internal resistance losses and parasitic losses. DC-DC converter losses are categorized into active losses during operation and standby losses during idle states.

The total loss is recorded as $P_{\mathrm{loss}}$. According to the equivalent circuit model of vanadium flow batteries established in [19], the losses in vanadium flow batteries include internal resistance losses, parasitic losses, and DC-DC converter losses.

$$P_{\mathrm{loss}}={\displaystyle \sum _{j=1}^N\left(P_{{\mathrm{loss}}_{-}\mathrm{R},\mathrm{j}}+P_{\mathrm{loss}\_\mathrm{B},\mathrm{j}}+P_{{\mathrm{DC}}_{-}\mathrm{loss},\mathrm{j}}\right)}$$

(3)

Internal resistance losses primarily include those generated by R₁ and R₂($P_{{\mathrm{loss}}_{-}\mathrm{R}}$), while parasitic losses are caused by the pump and parasitic resistance R₃($P_{\mathrm{loss}\_B}$). $P_{{\mathrm{DC}}_{-}\mathrm{loss}}$ is the DC-DC converter losses. The respective expressions are as follows:

$$\begin{array}{c}{P}_{{\mathrm{loss}}_{-}\mathrm{R},j}={\left(I_{\mathrm{d},j} - {\displaystyle \frac{U_{\mathrm{d},j}}{R_{3,j}}}-I_{\mathrm{p},j} \right)}^2 R_{2,j} +\\ {\displaystyle \frac{{\left(U_{\mathrm{d},j}-\left(I_{\mathrm{d},j}-\frac{U_{\mathrm{d},j}}{R_{3,j}}-I_{\mathrm{p},j}\right)R_{2,j}-V_{\mathrm{s},j}\right)}^2}{R_{1,j}}}\end{array}$$

(4)

$$P_{{\mathrm{loss}}_{-}\mathrm{B},j}={\displaystyle \frac{U_{\mathrm{d},j}^2}{R_{3,j}}}+U_{\mathrm{d},j}{I}_{\mathrm{p},j}$$

(5)

In the equation: $R_{1,j}$ and $R_{2,j}$ denote the internal impedance of the j-th vanadium electrolyte flow cell, comprising mass transfer impedance, diaphragm impedance, solution impedance, electrode impedance, and bipolar plate impedance; $R_{3,j}$ represents the parasitic resistance of the j-th VRB; $U_{\mathrm{d},j}$ is the terminal voltage across the j-th vanadium electrolyte flow cell; $I_{\mathrm{d},j}$ denotes the charge and discharge current; $V_{\mathrm{s},j}$ denotes the open-circuit voltage; $I_{\mathrm{p},j}$ denotes the pump loss current [20].

The losses in a DC-DC converter are categorized into operating losses $P_{{\mathrm{DC}}_{-}\mathrm{loss},j}$ and standby losses $P_{{\mathrm{DC}}_{-}\mathrm{S},j}$, with their respective calculation formulas shown in Equations (6)–(8).

$$P_{{\mathrm{DC}}_{-}\mathrm{loss},j}={\lambda }_j{P}_{{\mathrm{DC}}_{-}\mathrm{W},j}+\left(1-{\lambda }_j\right)P_{{\mathrm{DC}}_{-}\mathrm{S},j}$$

(6)

$$P_{{\mathrm{DC}}_{-}W,j}=\left(1-{\eta }_{\mathrm{dc}}\right)\left|P_j\right|$$

(7)

$$P_{{\mathrm{DC}}_{-}S,j}=0.5\%\times P_{\mathrm{DCN},j}$$

(8)

The decision variable ${\lambda }_j$ characterizes the operational state of the j-th energy storage unit. When ${\lambda }_j$ = 1, the DC-DC converter is in active mode; when ${\lambda }_j$ = 0, it is in standby mode. $P_j$ is the actual allocated power of the j-th VRB energy storage unit. $P_{\mathrm{DCN},j}$ and ${\eta }_{\mathrm{dc}}$ denote the rated power and operating efficiency of the DC-DC converter, respectively. According to [21], the DC-DC converter’s operating efficiency is 95%.

(2) SOC Balance: The SOC balance of an energy storage system is represented by the variance in SOC values among its storage units. A smaller variance indicates superior SOC balance within the system.

$${\mathrm{S}}_{balance}(t)={\displaystyle \frac{1}{N}}\sum _{j=1}^N{\left({\mathrm{SOC}}_j(t)-{\displaystyle \frac{1}{N}}\sum _{j=1}^N{\mathrm{SOC}}_j(t)\right)}^2$$

(9)

${\mathrm{SOC}}_j(t)$ for the j-th energy storage unit in the cycle, where N is the total number of energy storage units.

3.3. Power Distribution Constraints

(1) The charging and discharging power of each energy storage unit must not exceed its rated power. At each time step, its charging and discharging power must satisfy:

$$P_{\min }\le P_j\le P_{\max }$$

(10)

$P_{\max }$ is maximum discharge power, $P_{\min }$ denotes minimum charge power, When $P_j$ > 0, it represents charge power; When $P_j$ < 0, it represents discharge power.

(2) The total power balance constraint for all VRB energy storage units. $P_j(t)$ is the charge–discharge power of the j-th unit during the cycle, $P_{\mathrm{load}}(t)$ represents the power demand within the microgrid during the cycle.

$${\displaystyle \sum _{j=1}^{\mathrm{n}}{P}_j(t)}=P_{\mathrm{load}}(t)$$

(11)

(3) VRB Energy Storage Unit Output Constraint Equation:

$$SO{C}_{\min }\le SO{C}_j\le SO{C}_{\max }$$

(12)

$SO{C}_{\max }$ and $SO{C}_{\min }$ denote the upper and lower bounds of the SOC of each storage unit, respectively. Imposing these SOC limits can effectively prevent overcharging and over-discharging.

The operation optimization layer must adhere to the unit power constraint, total power constraint, and upper and lower limits of unit SOC during the optimization process, and must not exceed the specified power allocation framework.

3.4. Evaluation Indicators

To evaluate the effectiveness of the proposed dual-layer optimized power allocation, two indicators commonly used in vanadium redox flow energy storage systems are adopted as the evaluation metrics, which are listed as follows:

3.4.1. Number of Charge–Discharge Cycles for Energy Storage Units

An energy storage unit operates in three modes: charging, discharging, and standby. A transition from charging to discharging, or from discharging back to charging, is considered a single charge–discharge mode switch. More frequent switching leads to additional energy losses and adversely affects the unit’s service life. Let $N_j$ denote the total number of switches, $N_j(t-1)$ represents the number of switches in the previous cycle, and let $P_j\left(t\right)$ and $P_j(t-1)$ represent the charging and discharging power of unit $j$ in the current period $t$ and the previous period, respectively. If the product of these two power values becomes negative over two consecutive periods, it indicates a charge–discharge mode switch, and the total number of switches is increased by one.

$$N_j=\left\{\begin{array}{c}{N}_j(t-1)+1\hspace{1em}\hspace{1em}{P}_j(t-1)P_j(t)<0\\ N_j(t-1)\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}{P}_j(t-1)P_j(t)⩾0\hfill \end{array}\right.$$

(13)

3.4.2. Charge–Discharge Balance of Energy Storage Units

The charge–discharge balance index $B_i$ of a storage unit range within the closed interval $[-1,1]$. When $B_i$ approaches 1, it indicates that the discharging performance of the unit is dominant, while its charging capability is restricted; the smaller the value, the stronger the charging capability of the unit. When $B_i$ is close to 0, the storage unit achieves the best overall charge–discharge performance. $SO{C}_{\mathrm{ref}}$ denotes the reference SOC of the storage unit and is taken as the average of the upper and lower SOC limits.

$$B_i\left(t\right)={\displaystyle \frac{SO{C}_j\left(t\right)-SO{C}_{\mathrm{ref}}}{\left(SO{C}_{\max }-SO{C}_{\min }\right)/2}}$$

(14)

4. Power Allocation Solution Method

4.1. Multi-Objective Optimization

Within the operational optimization layer, two objectives must be optimized simultaneously: balancing the SOC among energy storage units and minimizing power losses. This requires a multi-objective optimization approach for solution.

$$\left\{\begin{array}{l}\min F=\min \{P_{\mathrm{loss}},{\mathrm{S}}_{balance}(t)\}\\ \mathrm{s}.\mathrm{t}.g_i(x)⩽0,i=1,2,\dots ,k\\ h_j(x)=0,j=1,2,\dots ,p\end{array}\right.$$

(15)

The optimization variables are the charging and discharging power of each cell within each cycle. The optimization objectives represent the total power loss and the variance in the SOC values across all cells. Both objective functions must simultaneously achieve their minimum values, with $h_j(x)$ and $g_i$ representing the power and SOC constraints mentioned earlier. In multi-objective optimization problems, due to the simultaneous presence of multiple objective functions, it is impossible to directly obtain a single optimal solution as in single-objective problems. Instead, a set of Pareto solutions exists. If solution X₁ is not inferior to another solution X₂ in any objective and is superior to X₂ in at least one objective, then X₁ dominates X₂ [22]. In Figure 5, each dot is dominated by at least one square, while the squares themselves are non-dominated relative to each other. The curve formed by the set of squares constitutes the Pareto frontier.

4.1.1. Multi-Object Harris Hawks Algorithm

Harris Hawks Optimization (HHO) is a swarm-based intelligent optimization algorithm proposed by Heidari et al. [23] in 2019. It simulates the behavior of Harris hawks hunting rabbits and their ambush-based hunting strategy. The algorithm’s optimization process comprises three stages: global exploration, exploration-capture transition, and capture. During optimization, each Harris hawks represents a candidate solution, while the prey rabbit serves as the optimal candidate solution for each iteration. To adapt HHO for multi-objective problems, non-dominated sorting, crowding degree calculation, and an elite strategy are introduced to enhance search capabilities: Non-dominated sorting organizes the population hierarchy based on dominance relationships, filtering out the Pareto optimal solution set; Crowding degree reflects the density of individuals surrounding a candidate, with higher values indicating sparser surroundings, thereby ensuring solution set diversity; during iteration, individuals with lower non-dominated levels and higher crowding degrees are prioritized. The elite retention strategy prevents the loss of superior solutions, maintaining convergence by accumulating historically optimal solutions.

4.1.2. Adaptive Simulated Annealing Multi-Objective Harris Hawks Algorithm

In the original Harris hawks strategy, the transition between exploration and exploitation phases is determined by the prey’s escape energy. The core parameter E of escape energy is a random value within the fixed range of −1 to 1, without considering the actual state of the solution set during optimization. This approach is prone to trapping the algorithm in local optima during the development phase.

To address the aforementioned issues, this paper proposes a multi-objective Harris hawks algorithm incorporating adaptive and simulated annealing mechanisms. It retains the global search capability of the simulated Harris hawks hunting algorithm while embedding SA as a local optimization operator. By leveraging SA’s Metropolis criterion—which accepts suboptimal solutions with a certain probability—the algorithm escapes local optima. An adaptive mechanism dynamically adjusts the energy factor and search strategy, enabling autonomous balancing between “global exploration” and “local exploitation” The adaptive mechanism first calculates the variance in the crowding distance of the current elite solution set to obtain a dynamic escaping energy $E$, which replaces the randomly generated $E$ in the original code. When the diversity of the solution set is low, $E$ is increased to enhance global exploration; when the diversity is high, $E$ is reduced to focus on local exploitation. The escaping energy $E$ is calculated as shown in Equation (16). $m$ denotes the current iteration number of the algorithm, $M$ is the maximum number of iterations, $m/M$ represents the iteration progress, $E_{\mathrm{m}\mathrm{a}\mathrm{x}}$ and $E_{\mathrm{m}\mathrm{i}\mathrm{n}}$ are the maximum and minimum values of the energy factor, respectively, and $d_{\mathrm{norm}}$ is the normalized diversity index (the variance in the crowding distance of the non-dominated solutions).

$$E=E_{\max }-\left(E_{\max }-E_{\min }\right)d_{\mathrm{norm}}(0.6+0.4(1-{\displaystyle \frac{m}{M}}))$$

(16)

During the local development phase, when the iteration progress exceeds 50% or the variance in solution set crowding falls below the threshold, SA is introduced to enhance local convergence accuracy.

SA generates new solutions by perturbing non-dominated solutions, guiding the algorithm to explore regions beyond the current local optimum. New solutions are compared with non-dominated solutions, with a key feature being the probabilistic acceptance of non-dominated solutions. This helps the algorithm escape local optima traps, while the perturbation magnitude dynamically adjusts with temperature. The initial temperature is dynamically calculated based on the variance in solution set crowding, with the temperature decay process also accounting for iteration progress. If a new solution generated by SA dominates the current solution, it is directly accepted. If the new solution is non-dominated relative to the current solution, reasonable screening of non-dominated solutions maintains solution set diversity. If the new solution is dominated by the current solution, acceptance occurs only with low probability when the temperature exceeds a set threshold, preserving the possibility of escaping local optima.

The settings for adaptive temperature and temperature decay coefficient are shown in Equation (17) and Equation (18), respectively.

$$T_0=T_{\max }{e}^{\left(-\partial d_{\mathrm{norm}}\right)}(1-{\displaystyle \frac{m}{M}})$$

(17)

$$T_{j+1}=\beta T_j$$

(18)

$T_{\max }$ is the maximum initial temperature, and $T_0$ is the actual initial temperature. $\partial$ is an adjustment coefficient selected based on experience. A lower $d_{\mathrm{norm}}$ results in a larger $T_0$, allowing more inferior solutions to escape from local optima. $\beta$ is the temperature decay coefficient, typically set to 0.95.

The incorporation of an adaptive simulated annealing strategy enables dynamic state adjustment, mitigating the premature convergence issue caused by fixed parameters in the original algorithm. This approach enhances solution diversity and distribution uniformity, compensating for MOHHO’s tendency to become stuck in local optima during the development phase. The primary workflow of the multi-objective Harris Hawks algorithm with adaptive simulated annealing is shown in Figure 6.

4.1.3. Algorithm Performance Validation

The ZDT4 test function is used to evaluate the performance of ASAMOHHO. As a classic test function for multi-objective optimization, ZDT4 exhibits multi-modal characteristics: it contains numerous local optima, challenging the algorithm’s global search capability and ability to escape local optima. Performance is evaluated using Inverted Generational Distance (IGD) and Hypervolume (HV). The IGD metric focuses on algorithm convergence by calculating the average distance between the solution set and the true Pareto frontier, thereby measuring how closely the solution set approaches the optimal solution. A smaller value indicates superior convergence performance. HV measures the extent of the Pareto frontier covered by the solution set. A larger value indicates a broader solution set and superior diversity. As shown in Figure 7. The core optimization settings of the algorithm are as follows: The population size was set to 50, the maximum number of iterations was 200. The adaptive range of was 0.3~1.7; for the Simulated Annealing local search, the initial temperature was 20, the temperature attenuation coefficient was 0.85, the low-temperature threshold was 5, the maximum number of steps for a single SA local search was 2, and the SA trigger threshold was 0.1. The interval for diversity enhancement strategy was 8 generations, and the probability of polynomial mutation was 0.15.

As shown in Figure 7a, ASAMOHHO converges more rapidly to IGD (approaching 0 within 20 iterations), indicating that the solution set increasingly approximates the true frontier. In contrast, standard MOHHO converges slowly due to its lack of localized fine-grained search capabilities. As shown in Figure 7b, ASAMOHHO exhibits significant fluctuations in its early HV values because the algorithm is in the global exploration phase and converges rapidly to high HV values. In contrast, the standard MOHHO’s exploration strategy initially clusters the solution set in local regions, resulting in limited coverage and a slow upward trajectory. The IGD and HV metrics indicate that the ASAMOHHO algorithm effectively enhances the diversity and coverage of the solution set.

4.2. Power Allocation Strategy

4.2.1. Single-Layer Optimization Strategy

The single-layer power optimization strategy does not distinguish between primary and secondary objectives. Instead, it places capacity degradation, SOC balancing among energy storage units, and power loss as three objectives at the same optimization level, simultaneously balancing each objective. The optimal solution is derived using the standard MOHHO algorithm.

4.2.2. Dual-Layer Optimization Strategy

The dual-layer optimization strategy proposed in this paper establishes a hierarchical structure: the initial allocation layer prioritizes long-term capacity goals, while the operational optimization layer employs the ASAMOHHO algorithm to solve short-term real-time objectives.

4.3. Conventional Power Allocation Strategy

Conventional power distribution strategies, which allocate power based on the total proportion of SOC across all units, are simple in principle and easy to implement.

$$T_{SOC}={\displaystyle \sum _{i=1}^{n}SOC(i)}$$

(19)

$$W(i)={\displaystyle \frac{SOC(i)}{T_{SOC}}}$$

(20)

$$P(i)=W(i)P(t)$$

(21)

Here, $T_{\mathrm{SOC}}$ is the sum of the SOC values of all units, and $P\left(t\right)$ is the power demand at time $t$. For the $i$th storage unit, $W\left(i\right)$ denotes its SOC proportion, and its output power is $P\left(i\right)$.

5. Case Study Simulation and Analysis

5.1. Case Configuration

This study utilizes a storage system composed of five parallel vanadium flow battery units, each with a rated power of 100 kW. The power demand data of a microgrid over one day is selected as input, comprising 60 scheduling cycles with 15 min intervals. The specific details are shown in Figure 8. The initial SOCs for Units 1–5 are set to 0.2, 0.25, 0.3, 0.4, and 0.5, respectively. The upper and lower limits for the remaining SOC of the vanadium flow battery units are set at 0.8 and 0.2, respectively. Both the proposed dual-layer optimization allocation strategy and the conventional power allocation strategy are applied to solve the problem, with the final allocation results compared.

5.2. Comparison Results Between Strategy

Figure 9 shows the SOC variation of each unit over 60 cycles under the conventional power distribution strategy. The balance of SOC among units varies significantly. Furthermore, since all units require charging and discharging in each cycle, charging occurs even when the remaining SOC of energy storage units is high, while discharging is still necessary when the remaining SOC is low. This leads to the risk of excessive charging and discharging, potentially causing the system to cease operation.

Figure 10 illustrates the SOC changes in each unit under the single-layer multi-objective Harris hawks optimization strategy. Even with differing initial SOC values across energy storage units, the optimization process converges their SOC changes toward consistency, enabling synchronized charging and discharging. However, significant fluctuations occur between scheduling cycle 25 and 40. This arises because the strategy must simultaneously balance two additional objectives, preventing isolated consideration of SOC equilibrium during power allocation. Figure 11 illustrates the SOC changes for each unit under the ASAMOHHO optimization strategy proposed in this paper. Compared to the single-layer optimization strategy, the SOC change curves for each unit are more concentrated, exhibiting better balance and smaller stable fluctuations during charging and discharging.

The power distribution results of the conventional power allocation strategy are shown in Figure 12. A positive value indicates charging power, while a negative value indicates discharging power. It can be observed that all cells output power proportionally over 60 cycles, with cell 5 (high SOC) and cell 1 (low SOC) contributing power in every cycle. The results of single-layer and dual-layer optimized power allocation are shown in Figure 13 and Figure 14, respectively. Since both strategies incorporate charge/discharge priority settings, power output across cells is uneven within each cycle. For instance, during the first ten scheduling cycles, power absorption is primarily from cells 1 and 2 because they had the lowest initial SOC and were prioritized for charging. Conversely, during the 20th scheduling cycle when total power is negative, cell 5 is discharging. This occurs because the initial allocation layer sets charge/discharge priorities for each cell, and cell 5 with the highest SOC is prioritized for discharge.

Figure 15 shows the variation in charge–discharge balancing capabilities of each energy storage unit under the conventional power allocation strategy. The results reveal an uneven distribution of values across the five units under the conventional power allocation method. Since optimal charge and discharge capability occur near zero, negative values for Units 1 and 2 indicate charging tendencies, while Units 4 and 5 exhibit pronounced discharging tendencies—a scenario detrimental to system stability. Changes in energy storage unit charge and discharge balancing capability under single-layer and dual-layer optimization strategies are shown in Figure 16 and Figure 17, respectively. After optimization by both strategies, the values of all five units gradually become uniform after the 20th cycle. Following single-layer optimization, the range of values remains between [0, 0.35], while after dual-layer optimization, the range for each unit falls within [0, 0.25]. Compared to the single-layer strategy, the five units exhibit improved charge–discharge balance, and the result curves indicate that the variation in charge–discharge balancing capability is more concentrated.

The charge–discharge switching cycles for each energy storage unit are shown in

Table 1. Charging and discharging switching times of energy storage units.

Energy Storage Unit	Number of Charge–Discharge Cycles
	Single-Layer Optimization Strategy	Dual-Layer Optimization Strategy
1	15	11
2	11	10
3	9	9
4	10	9
5	10	9

. Under the dual-layer optimization strategy, during 60 scheduling cycles, only Unit 3 exhibited the same switching frequency as the single-layer strategy. Units 1, 2, 4, and 5 all demonstrated lower charge–discharge switching cycles compared to the single-layer allocation strategy.

The capacity degradation of each energy storage unit, calculated using the degradation formula, is shown in

Table 2. Capacity Degradation of Energy Storage Units.

Energy Storage Unit	Capacity Loss %
	Single-Layer Optimization Strategy	Two-Layer Optimization Strategy
1	0.041	0.027
2	0.026	0.025
3	0.022	0.022
4	0.014	0.017
5	0.009	0.011

. Under the dual-layer strategy, the total degradation rate of the five units was reduced by 8.77% compared to that under the single-layer strategy.

6. Conclusions

This paper addresses potential conflicts arising from the mutual coupling of multiple objectives in power allocation for vanadium flow battery energy storage systems. It proposes a two-layer optimization strategy for power allocation: the initial allocation layer prioritizes minimizing capacity degradation as the primary objective, generating an initial power allocation scheme through priority ranking combined with a differential evolution algorithm; The operation optimization layer generates an initial solution centered on the initial allocation scheme from the initial allocation layer. It employs an adaptive simulated annealing multi-objective Harris hawks algorithm to solve a multi-objective optimization model that considers optimal SOC balance and minimizes total power loss, thereby obtaining the optimal solution.

• Conventional allocation strategies simply distribute power proportionally. Compared to the dual-layer optimization strategy employing intelligent algorithms, conventional allocation strategies yield poorer results in simulations, both in terms of SOC balance and charge and discharge equilibrium capability. The disparity in SOC levels between individual units grows increasingly pronounced over time, which will ultimately compromise the overall system performance. The dual layer optimization strategy proposed in this article has the core advantages of hierarchical collaboration and balancing global and local aspects, resulting in better performance in various objective results.
• Compared with the single-layer multi-objective Harris hawks optimization strategy, the proposed two-layer combined ASAMOHHO allocation strategy achieves superior SOC consistency among energy storage units after optimization, without SOC fluctuations arising from inter-objective conflicts. After stabilizing the charge–discharge balance across all five units, the range remained within [0, 0.25], indicating superior charge–discharge capability compared to the single-layer strategy. Furthermore, the total number of charge–discharge switching cycles across energy storage units decreased by eight cycles compared to the single-layer strategy, representing a 14.3% reduction in total switching frequency and an 8.77% relative decrease in degradation rate. This contributes to extending the lifespan of energy storage units. These results demonstrate that the proposed two-layer optimization strategy achieves outcomes more aligned with target requirements. Additionally, since single-layer optimization lacks a prioritized hierarchy among objectives, the computation of three objectives becomes more complex, leading to longer simulation runtime.
• Future research may consider incorporating additional objective functions and comparative analyses of alternative approaches to further validate the effectiveness of the dual-layer allocation strategy presented herein. Improvements to the multi-objective Harris hawks algorithm could also be explored to enhance its performance, thereby enabling its application to more complex real-world systems.