The increasing integration of renewable energy sources like wind and solar into power grids presents significant challenges to system frequency stability and power quality. These variable resources lack the inherent inertia and governor response of traditional synchronous generators, making frequency regulation more difficult. To address this, battery energy storage systems (BESS) have emerged as a highly effective solution due to their rapid response and precise power control capabilities. This study focuses on the optimal sizing of an energy storage battery system specifically for providing secondary frequency regulation (SFR) services. We propose a comprehensive capacity optimization model that maximizes net economic benefit over the system’s lifetime and employ the Antlion Optimization (ALO) algorithm to solve it.
The core problem is determining the optimal rated power and energy capacity for an energy storage battery participating in SFR. Oversizing leads to unnecessary capital costs, while undersizing reduces potential revenue and regulatory performance. Our methodology involves two integrated models: a techno-economic evaluation model and an operational simulation model.
1. Techno-Economic Evaluation Model for Energy Storage Battery
This model calculates the net present value of the energy storage battery project over its lifetime, considering all costs and revenues.
1.1 Battery Lifetime Estimation
The lifespan of the lithium-ion energy storage battery is a critical cost factor. We estimate it based on equivalent charge-discharge cycles using the rainflow counting algorithm to process the State-of-Charge (SOC) profile obtained from operation simulation. The equivalent full cycles per year \(N_d\) is calculated as:
$$N_d = K_2 \times \sum_{i=1}^{K_1} \frac{4000}{N(\text{DOD}_d(i))}$$
where \(K_1\) is the number of half-cycles counted by the rainflow algorithm, \(K_2\) is the scaling factor to one year, and \(N(\text{DOD}_d)\) is the cycle life at a given depth-of-discharge (DOD). The battery lifetime \(T_{life}\) (in years) is then:
$$T_{life} = \frac{4000}{N_d}$$
1.2 Cost Model
The total lifetime cost \(C\) includes investment \(C_{inv}\), operation & maintenance \(C_{OM}\), recycling \(C_{rc}\), and other penalty costs \(C_{ot}\).
$$C = C_{inv} + C_{OM} + C_{rc} + C_{ot}$$
The detailed formulas are:
Investment Cost:
$$C_{inv} = K_{PCS} P_N + \sum_{k=1}^{n} K_{BESS} E_N (1+r)^{-[kT_L/(n+1)]}$$
O&M Cost:
$$C_{OM} = K_{POM} P_N \frac{(1+r)^{T_L}-1}{r(1+r)^{T_L}} + \sum_{t=1}^{T_L} K_{EOM} W (1+r)^{-t}$$
Recycling Cost:
$$C_{rc} = K_{Prc} P_N (1+r)^{-T_L} + \sum_{j=1}^{n+1} K_{Erc} E_N (1+r)^{-jT_L/(n+1)}$$
Other Costs (for deficit/surplus energy):
$$C_{ot} = \sum_{t=1}^{T_L} (K_{lack}E_{lack} + K_{loss}E_{loss})(1+r)^{-t}$$
| Parameter | Symbol | Value | Unit |
|---|---|---|---|
| Discount Rate | \(r\) | 0.06 | – |
| Project Lifetime | \(T_L\) | 20 | year |
| PCS Specific Cost | \(K_{PCS}\) | 900,000 | ¥/MW |
| BESS Specific Cost | \(K_{BESS}\) | 1,600,000 | ¥/(MWh) |
| PCS O&M Cost | \(K_{POM}\) | 68,400 | ¥/MW |
| BESS O&M Cost | \(K_{EOM}\) | 68.3 | ¥/(MWh) |
1.3 Revenue Model
The total revenue \(B\) comes from frequency regulation energy \(B_{ele}\), reserve capacity \(B_{cap}\), and environmental benefits \(B_{env}\).
$$B = B_{ele} + B_{cap} + B_{env}$$
Frequency Regulation Energy Revenue:
$$B_{ele} = \sum_{t=1}^{T_L} K_{DAY} K_{ele} W_s (1+r)^{-t}$$
Reserve Capacity Revenue:
$$B_{cap} = \sum_{t=1}^{T_L} \frac{K_{DAY}}{2} \left[ \sum_{i=1}^{N} (K_u P_m^{(up)}(i) + K_d P_m^{(dn)}(i)) \right] (1+r)^{-t}$$
where \(P_m^{(up)} = P_N – P_m\) and \(P_m^{(dn)} = P_m – (-P_N)\).
Environmental Benefit:
$$B_{env} = \sum_{t=1}^{T_L} \left[ \sum_{i} (K_{ai} M_{ai}) W^+ \right] K_{DAY} (1+r)^{-t}$$
1.4 Objective Function
The goal is to maximize the net present benefit \(P\) over the project lifetime by optimizing the rated power \(P_N\) and energy capacity \(E_N\) of the energy storage battery.
$$\max f(P) = B – C$$
$$\text{subject to: } P_N, E_N > 0$$
2. Operational Simulation Model for Energy Storage Battery
This model simulates the real-time operation of the energy storage battery to provide the SOC profile, charge/discharge energy, and power outputs needed for the evaluation model. The input is a sequence of Area Control Error (ACE) signals.
Signal Decomposition and Allocation: The ACE signal is decomposed using Empirical Mode Decomposition (EMD) into Intrinsic Mode Functions (IMFs) with different frequencies:
$$\sum_{u=1}^{9} \text{imf}_u = \text{EMD}(ACE)$$
A selection variable \(k_h\) (from 1 to 9) determines the allocation. High-frequency components (lower index IMFs) are assigned to the fast-responding energy storage battery, while low-frequency components are assigned to traditional generators.
$$ACE_{BESS} = \sum_{u=1}^{k_h} \text{imf}_u$$
Power Dispatch Logic: At each time step, the power setpoint for the energy storage battery \(P_{BESS,set}\) is derived from \(ACE_{BESS}\), but is constrained by the rated power \(P_N\) and the current SOC to keep it within safe limits [SOC\(_{min}\), SOC\(_{max}\)]. A capacity margin is also considered in the final sizing.
3. Optimization Using Antlion Algorithm (ALO)
The Antlion Optimizer is a nature-inspired metaheuristic algorithm. In this context, each “ant” represents a potential solution vector \((P_N, k_h)\). The “antlions” represent elite solutions. The algorithm iteratively improves the population of solutions by modeling ants moving randomly around antlions within a shrinking hypersphere (the “pit trap”). The fitness of each ant is evaluated using the integrated evaluation and simulation models. The antlion with the best fitness (highest net benefit) becomes the elite, guiding the search. After many iterations, the algorithm converges to an optimal or near-optimal configuration for the energy storage battery.
4. Case Study and Analysis
A case study was conducted using a day-long ACE signal sequence from a 250 MW system. The ALO parameters were set with 100 antlions and 1000 iterations. The optimization variables were bounded: \(P_N\) ∈ (0, 90) MW and \(k_h\) ∈ {1, 2, …, 9}.
The optimal solution found by the ALO was \(P_N^* = 62.58 \text{ MW}\) and \(k_h^* = 3\), meaning the energy storage battery should handle the first three highest-frequency IMFs of the decomposed ACE signal. The corresponding optimal energy capacity was \(E_N^* = 11.00 \text{ MWh}\). To analyze the impact of signal allocation and capacity margin, a comparative study was performed.
| \(k_h\) | With Capacity Margin (Group A) | Without Capacity Margin (Group B) | ||||||
|---|---|---|---|---|---|---|---|---|
| Net Benefit \(P\) (10^8 ¥) | Capacity \(E_N\) (MWh) | Life \(T_{life}\) (Year) | SOC Std. | Net Benefit \(P\) (10^8 ¥) | Capacity \(E_N\) (MWh) | Life \(T_{life}\) (Year) | SOC Std. | |
| 1 | 2.73 | 11.36 | 7.70 | 0.098 | 2.70 | 6.36 | 3.09 | 0.175 |
| 2 | 2.77 | 7.64 | 2.76 | 0.090 | 2.69 | 4.85 | 1.25 | 0.148 |
| 3 | 2.80 | 10.99 | 3.84 | 0.095 | 2.79 | 7.99 | 2.32 | 0.131 |
| 4 | 2.75 | 8.20 | 2.23 | 0.090 | 2.61 | 5.21 | 1.12 | 0.142 |
| 5 | 2.69 | 16.46 | 5.88 | 0.097 | 2.61 | 10.46 | 2.85 | 0.152 |
| 6 | 1.80 | 69.61 | 20.00 | 0.199 | 2.05 | 59.61 | 20.00 | 0.233 |
| 7 | 1.93 | 63.77 | 20.00 | 0.198 | 1.98 | 61.77 | 20.00 | 0.204 |
| 8 | 1.92 | 63.80 | 20.00 | 0.159 | 2.23 | 59.80 | 20.00 | 0.169 |
| 9 | 1.92 | 64.12 | 20.00 | 0.159 | 2.02 | 60.12 | 20.00 | 0.170 |
Key Findings:
1. Optimal Signal Allocation: The maximum net benefit is achieved when the energy storage battery handles only high-frequency ACE components (\(k_h\) = 1-5). Allocating low-frequency signals (\(k_h \ge 6\)) to the battery causes significant charge/discharge imbalance, requiring a much larger \(E_N\) which drastically increases capital costs without proportional revenue gains.
2. Effectiveness of Capacity Margin: For the optimal high-frequency allocation cases (\(k_h\) = 1-5), implementing a capacity margin (Group A) extends the battery’s operational life and increases the net benefit compared to the no-margin case (Group B). This confirms the economic value of properly sizing the energy storage battery with some headroom.
3. Inefficiency in Low-Frequency Service: When the energy storage battery is assigned low-frequency signals (\(k_h \ge 6\)), its capacity is underutilized (evident from low SOC standard deviation). Adding a margin in this scenario only increases cost, reducing net benefit.
5. Conclusion
This study presents a holistic framework for optimizing the capacity of an energy storage battery system providing secondary frequency regulation. The integrated model combines a detailed lifetime techno-economic analysis with a physics-based operational simulation. The Antlion Optimization algorithm proves effective in solving this complex, non-linear problem. The results demonstrate that economic viability is highest when the energy storage battery is dedicated to responding to high-frequency ACE components, leveraging its speed advantage. Furthermore, incorporating a reasonable capacity margin in the sizing process is a financially sound strategy, as it enhances system longevity and overall net benefit. This methodology provides a valuable tool for planners and investors seeking to deploy energy storage battery systems for grid stability in high-renewable penetration scenarios.