In recent years, the rapid integration of renewable energy sources, such as wind and solar power, into power grids has posed significant challenges to frequency stability and power quality. The inherent variability and intermittency of these sources can lead to frequency deviations that threaten grid reliability. To address this, the use of battery energy storage systems for ancillary services, particularly secondary frequency regulation, has gained traction. In this study, I explore an optimal capacity allocation method for battery energy storage systems participating in secondary frequency modulation, leveraging the antlion optimization algorithm. The goal is to maximize net benefits over the system’s lifecycle while ensuring effective frequency support. The proposed approach involves constructing a comprehensive evaluation model that accounts for costs, revenues, and operational constraints, followed by solving it using a novel metaheuristic technique. This work aims to provide a practical framework for deploying battery energy storage systems in grid frequency regulation, enhancing economic viability and grid resilience.
The core of this research lies in developing a capacity optimization configuration model for battery energy storage systems. This model integrates multiple factors, including battery lifespan estimation, cost analysis, revenue streams, and frequency regulation performance. The battery energy storage system is tasked with responding to Area Control Error (ACE) signals, which indicate the imbalance between generation and load. By decomposing ACE signals using Empirical Mode Decomposition (EMD), high-frequency components are allocated to the battery energy storage system, while low-frequency parts are handled by conventional generators like thermal plants. This allocation strategy optimizes the use of the battery energy storage system’s rapid response capabilities, reducing wear and tear and improving efficiency. The antlion algorithm is then employed to optimize key variables, such as the rated power and signal selection threshold, to achieve maximum net benefits. Through simulation case studies, I demonstrate the effectiveness of this method in determining optimal capacity and operational strategies for battery energy storage systems in secondary frequency modulation applications.
To begin, I outline the fundamental models underpinning the optimization framework. The battery energy storage system’s lifecycle net benefit serves as the objective function, calculated as the difference between total revenues and total costs over its operational period. This requires accurate modeling of battery degradation, which is critical for cost projections. The lifespan of the battery energy storage system is estimated based on state-of-charge (SOC) patterns using rainflow counting to derive equivalent depth-of-discharge (DOD) cycles. The relationship between DOD and cycle life for lithium-ion batteries is expressed with a fitted power function. For instance, the cycle life $$ N(DOD) $$ can be approximated as:
$$ N(DOD) = 4122 \times DOD^{-1.562} – 458.5 $$
where DOD ranges from 0 to 1. The equivalent annual cycles $$ N_d $$ are computed by summing contributions from all discharge events, and the battery lifespan $$ T_{life} $$ in years is given by:
$$ T_{life} = \frac{4000}{N_d} $$
assuming a rated cycle life of 4000 at full discharge. This approach allows for realistic degradation modeling, essential for optimizing the battery energy storage system’s capacity.
The cost model for the battery energy storage system encompasses four main components: investment, operation and maintenance, recycling, and other penalties. Investment costs cover power conversion systems and battery packs, formulated as:
$$ C_{inv} = K_{PCS} P_N + \sum_{k=1}^{n} K_{BESS} E_N (1 + r)^{-[kT_L/(n+1)]} $$
where $$ P_N $$ is the rated power in MW, $$ E_N $$ is the rated capacity in MWh, $$ K_{PCS} $$ and $$ K_{BESS} $$ are unit costs, $$ r $$ is the discount rate, and $$ T_L $$ is the lifecycle in years. Operation and maintenance costs include periodic expenses for both components:
$$ C_{OM} = K_{POM} P_N \frac{(1+r)^{T_L} – 1}{r(1+r)^{T_L}} + \sum_{t=1}^{T_L} K_{EOM} W (1+r)^{-t} $$
with $$ K_{POM} $$ and $$ K_{EOM} $$ as unit costs and $$ W $$ as annual energy throughput. Recycling costs account for end-of-life disposal:
$$ C_{rc} = K_{Prc} P_N (1+r)^{-T_L} + \sum_{j=1}^{n+1} K_{Erc} E_N (1+r)^{-jT_L/(n+1)} $$
Other costs arise from SOC deviations, such as shortage or excess energy penalties:
$$ C_{ot} = \sum_{t=1}^{T_L} (K_{lack} E_{lack} + K_{loss} E_{loss}) (1+r)^{-t} $$
The total cost $$ C $$ is the sum of these components. On the revenue side, the battery energy storage system earns from frequency regulation energy, reserve capacity, and environmental benefits. The energy revenue is:
$$ B_{ele} = \sum_{t=1}^{T_L} K_{DAY} K_{ele} W_s (1+r)^{-t} $$
where $$ W_s $$ is daily energy output, $$ K_{ele} $$ is frequency regulation price, and $$ K_{DAY} $$ is annual operating days (e.g., 300). Reserve revenue comes from up- and down-regulation capacity:
$$ B_{cap} = \sum_{t=1}^{T_L} \frac{1}{2} \left[ \sum_{i=1}^{N} (K_u P_m^{(up)}(i) + K_d P_m^{(dn)}(i)) \right] K_{DAY} (1+r)^{-t} $$
with $$ K_u $$ and $$ K_d $$ as reserve prices, and $$ P_m^{(up)} $$ and $$ P_m^{(dn)} $$ as reserve powers. Environmental revenue quantifies reduced emissions:
$$ B_{env} = \sum_{t=1}^{T_L} \left[ \sum_{i=1}^{n} (K_{ai} M_{ai}) W^+ \right] K_{DAY} (1+r)^{-t} $$
where $$ W^+ $$ is annual discharge energy. Total revenue $$ B $$ aggregates these streams. Thus, the net benefit $$ P $$ is:
$$ P = B – C $$
and the optimization objective is to maximize $$ P $$ by adjusting variables like $$ P_N $$ and the ACE signal allocation threshold.
The output model simulates the battery energy storage system’s response to ACE signals. ACE represents the power imbalance in the grid, and for secondary frequency regulation, it is decomposed using EMD into intrinsic mode functions (IMFs) of varying frequencies. The high-frequency IMFs are assigned to the battery energy storage system for fast response, while low-frequency ones go to traditional generators. Mathematically, for a given ACE sequence, the EMD yields:
$$ \sum_{u=1}^{9} \text{imf}_u = \text{EMD}(ACE) $$
The portion allocated to the battery energy storage system, denoted $$ ACE_{BESS} $$, is determined by a selection variable $$ k_h $$ (from 1 to 9):
$$ ACE_{BESS} = \sum_{u=1}^{k_h} \text{imf}_u $$
This allocation leverages the battery energy storage system’s agility for high-frequency adjustments, reducing strain on conventional units. The output power of the battery energy storage system is then calculated per sampling point, considering SOC limits (e.g., 0.1 to 0.9) and rated power constraints. A capacity margin is incorporated to handle uncertainties, derived from SOC standard deviations. This model feeds into the evaluation framework, providing operational data for cost and revenue computations.
The antlion optimization algorithm is employed to solve the capacity optimization problem. Inspired by antlions’ hunting behavior in nature, this metaheuristic efficiently explores high-dimensional search spaces. It involves two main agents: ants (solutions) and antlions (elite solutions). Initially, positions are randomly initialized, and fitness (net benefit) is evaluated. The algorithm iteratively updates positions via random walks around antlions, with selection based on fitness. For our problem, each ant represents a candidate solution with variables $$ P_N $$ (rated power) and $$ k_h $$ (signal selection). The optimization process minimizes the negative net benefit:
$$ f(P) = \min(-P) $$
subject to constraints: $$ 0 < P_N \leq 90 \text{ MW} $$ and $$ 1 \leq k_h \leq 9 $$. The antlion algorithm’s robustness in handling non-linear, multi-modal functions makes it suitable for optimizing the battery energy storage system configuration, ensuring global or near-global optima.
A case study illustrates the application of this method. The ACE data from a regulation plant with a system capacity of 250 MW is sampled over one day at 1-minute intervals, as shown in a representative plot. Key parameters for the models are summarized in the following table:
| Parameter Type | Parameter Name | Value | Unit |
|---|---|---|---|
| Output Model | SOC Range (MIN~MAX) | 0.1~0.9 | – |
| Initial SOC | 0.5 | – | |
| Reference SOC Standard Deviation | 0.1, 0.3 | – | |
| Evaluation Model | Discount Rate (r) | 0.06 | – |
| Lifecycle (T_L) | 20 | years | |
| Annual Operating Days (K_DAY) | 300 | days | |
| PCS Unit Cost (K_PCS) | 900,000 | ¥/MW | |
| Battery Unit Cost (K_BESS) | 1,600,000 | ¥/MWh | |
| PCS O&M Cost (K_POM) | 68,400 | ¥/MW | |
| Battery O&M Cost (K_EOM) | 68.3 | ¥/MWh | |
| PCS Recycling Cost (K_Prc) | 6,800 | ¥/MW | |
| Battery Recycling Cost (K_Erc) | 6,800 | ¥/MWh | |
| Shortage Cost (K_lack) | 11,000 | ¥/MWh | |
| Excess Cost (K_loss) | 8,000 | ¥/MWh | |
| Frequency Regulation Price (K_ele) | Hourly variable | ¥/MWh |
Frequency regulation and reserve prices vary hourly, as depicted in graphs, with peaks during high-demand periods. The ACE signal is decomposed into nine IMFs via EMD, revealing high-frequency components (IMF1-IMF5) and low-frequency ones (IMF6-IMF9). Using the antlion algorithm with 100 antlions and 1000 iterations, the optimal variables are found: $$ P_N = 62.579 \text{ MW} $$ and $$ k_h = 3 $$. This corresponds to an optimal capacity $$ E_N = 10.999 \text{ MWh} $$ and a net benefit $$ P = 2.8038 \times 10^8 \text{ ¥} $$. The SOC trajectory under this configuration remains within bounds, with a standard deviation of 0.095, indicating effective utilization. The output power of the battery energy storage system primarily handles high-frequency ACE components, while traditional generators manage lower frequencies, as shown in output plots.
To analyze the impact of allocation strategies, comparative experiments are conducted by varying $$ k_h $$ while keeping $$ P_N $$ fixed at the optimal value. Two scenarios are considered: with capacity margin (A) and without (B). The results are summarized below:
| k_h | Scenario | Net Benefit P (10^8 ¥) | Capacity E_N (MWh) | Lifespan T_life (years) | SOC Standard Deviation |
|---|---|---|---|---|---|
| 1 | A (with margin) | 2.73 | 11.36 | 7.70 | 0.098 |
| B (without margin) | 2.70 | 6.36 | 3.09 | 0.175 | |
| 2 | A | 2.77 | 7.64 | 2.76 | 0.090 |
| B | 2.69 | 4.85 | 1.25 | 0.148 | |
| 3 | A | 2.80 | 10.99 | 3.84 | 0.095 |
| B | 2.79 | 7.99 | 2.32 | 0.131 | |
| 4 | A | 2.75 | 8.20 | 2.23 | 0.090 |
| B | 2.61 | 5.21 | 1.12 | 0.142 | |
| 5 | A | 2.69 | 16.46 | 5.88 | 0.097 |
| B | 2.61 | 10.46 | 2.85 | 0.152 | |
| 6 | A | 1.80 | 69.61 | 20 | 0.199 |
| B | 2.05 | 59.61 | 20 | 0.233 | |
| 7 | A | 1.93 | 63.77 | 20 | 0.198 |
| B | 1.98 | 61.77 | 20 | 0.204 | |
| 8 | A | 1.92 | 63.80 | 20 | 0.159 |
| B | 2.23 | 59.80 | 20 | 0.169 | |
| 9 | A | 1.92 | 64.12 | 20 | 0.159 |
| B | 2.02 | 60.12 | 20 | 0.170 |
The analysis reveals that allocating only high-frequency ACE signals ($$ k_h = 1-5 $$) to the battery energy storage system yields higher net benefits compared to including low-frequency signals ($$ k_h = 6-9 $$). This is because low-frequency components require larger capacity investments without proportional revenue gains, leading to underutilization and increased costs. For instance, at $$ k_h = 6 $$, the battery energy storage system’s output becomes imbalanced, causing SOC to drift and necessitating oversized capacity. Capacity margins prove beneficial for $$ k_h = 1-5 $$, extending lifespan and enhancing benefits by providing buffer for SOC fluctuations. However, for $$ k_h \geq 6 $$, margins add unnecessary costs, reducing net benefits. Thus, the optimal strategy involves assigning high-frequency regulation to the battery energy storage system with moderate capacity margins, maximizing economic returns while ensuring grid stability.
The antlion algorithm’s effectiveness in this context stems from its ability to handle non-linear constraints and multi-objective trade-offs. By optimizing both $$ P_N $$ and $$ k_h $$ simultaneously, it identifies configurations that balance rapid response with cost efficiency. The algorithm’s iterative refinement process avoids local optima, ensuring robust solutions for the battery energy storage system deployment. This metaheuristic approach is particularly valuable for complex power system problems where traditional gradient-based methods may falter due to discontinuities or high dimensionality.
In conclusion, this research presents a comprehensive method for optimizing the capacity of battery energy storage systems in secondary frequency regulation. By integrating lifecycle cost-benefit analysis with signal decomposition and metaheuristic optimization, it offers a practical tool for grid operators and planners. The findings highlight that battery energy storage systems are most economically viable when dedicated to high-frequency ACE components, with carefully calibrated capacity margins. The antlion algorithm serves as a powerful solver, enabling efficient exploration of the solution space. Future work could incorporate battery rate characteristics and real-time market dynamics to further enhance the model. Ultimately, this approach supports the integration of renewable energy by providing reliable frequency regulation, contributing to a more resilient and sustainable power grid.