In my research, I have developed a novel two-layer multi-objective optimization strategy for battery energy storage systems (BESS) participating in grid frequency regulation. The increasing penetration of renewable energy sources like wind and solar power has led to a decline in system inertia and insufficient frequency regulation capacity. Battery energy storage systems, with their millisecond-level response speed and high control precision, are ideally suited to address these challenges, particularly for high-frequency regulation signals. However, the high cost of BESS necessitates efficient utilization. My proposed strategy specifically tackles the issues of varying frequency modulation costs among different BESS types and the imbalance in the state of charge (SOC) during operation. The core of my approach is a two-layer framework: the frequency modulation instruction decomposition layer and the BESS frequency modulation instruction optimization layer.
Methodology: A Two-Layer Optimization Framework
My strategy is built upon a two-layer architecture to separately handle signal decomposition and power allocation. The first layer, the frequency modulation instruction decomposition layer, leverages the fast response characteristics of BESS. I used Empirical Mode Decomposition (EMD) to separate the automatic generation control (AGC) signal into high-frequency and low-frequency components. The original AGC signal, \(P_{AGC}(t)\), is decomposed as follows:
$$ P_{AGC}(t) = \sum_{i=1}^{n} IMF_i(t) + r_n(t) $$
Where \(IMF_i(t)\) are the intrinsic mode functions and \(r_n(t)\) is the residual. The high-frequency components, specifically \(IMF_1\) and \(IMF_2\), are assigned to the BESS because they require the fastest response. The high-frequency AGC command for the BESS, \(P_{AGC,B}(t)\), is then given by:
$$ P_{AGC,B}(t) = \sum_{i=1}^{k} IMF_{AGC,i}(t) $$
Here, \(k\) is the filtering order, which I set to 2 in my simulation. This ensures that the BESS only handles the rapidly fluctuating part of the regulation signal, which is where their performance advantage is most pronounced.
Modeling the BESS Optimization Layer
The second layer of my strategy is a multi-objective optimization model designed to allocate the high-frequency AGC command among multiple BESS. This model simultaneously considers the economic performance and the SOC balance of the battery energy storage system. I defined two primary objective functions.
Economic Objective with a Resistance Coefficient
The first objective is to minimize the total regulation cost minus the regulation revenue. The cost for a single battery energy storage system \(i\) at time \(t\) includes energy loss cost \(C_{loss,i}(t)\) and lifetime degradation cost \(C_{life,i}(t)\):
$$ C_{loss,i}(t) = m \times T \times \left[ p_{B,i}^d(t) \times \left( \frac{1}{\eta_i^d} – 1 \right) + p_{B,i}^c(t) \times (1 – \eta_i^c) \right] $$
$$ C_{life,i}(t) = \frac{T \times P_{B,i}^{rate}}{2 N_{z,i} \times E_{B,i}^{rate}} \times \left( p_{B,i}^c(t) \times \eta_i^c \times c_{p,i} + \frac{p_{B,i}^d(t) \times c_{p,i}}{\eta_i^d} \right) $$
To prevent the most economical BESS from being overused, I introduced a resistance coefficient \(K_i(t)\). This coefficient is related to the remaining frequency regulation capacity of each BESS. For charging and discharging, the resistance coefficients are:
$$ K_{c,i}(t) = E_{B,i}^{rate} \times \frac{ SOC_i(t-1) – SOC_{i,min} }{T} $$
$$ K_{d,i}(t) = E_{B,i}^{rate} \times \frac{ SOC_{i,max} – SOC_i(t-1) }{T} $$
After normalization, the final resistance coefficient is \(K_i(t) = V_1 \times K_{c,i}(t)’ + V_2 \times K_{d,i}(t)’\). The total cost objective function, incorporating this coefficient, is:
$$ \min C_1(t) = \sum_{i=1}^{n} K_i(t) \times C_{B,i}(t) – B_{total}(t) $$
The total revenue \(B_{total}(t)\) is derived from regulation mileage and capacity payments:
$$ B_{total}(t) = \sum_{i=1}^{n} B_{mile,i}(t) + B_{cap,i}(t) $$
$$ B_{mile,i}(t) = \sum_{t=1}^{T} B_i \times P_{mp,t,i} \times R_{mileage} \times A_{t,i} $$
$$ B_{cap,i}(t) = \sum_{t=1}^{T} B_i \times P_{cp,t,i} \times A_{t,i} $$
The AGC performance index \(A_{t,i}\) is a weighted average of correlation \(k_1\), precision \(k_2\), and delay \(k_3\): \(A = \beta_1 k_1 + \beta_2 k_2 + \beta_3 k_3\).
SOC Balance Objective
The second objective is to minimize the SOC imbalance among all BESS. This is crucial for the sustainability and health of the battery energy storage system. The objective function is:
$$ \min C_2(t) = \sum_{i=1}^{n} \left| SOC_{pc,i}(t) \right| $$
Where \(SOC_{pc,i}(t)\) is the SOC deviation of a single BESS from the average SOC of all BESS at the previous time step, \(SOC_J(t-1)\):
$$ SOC_{pc,i}(t) = SOC_i(t-1) + \Delta SOC_i(t) – SOC_J(t-1) $$
$$ \Delta SOC_i(t) = \frac{T}{E_{B,i}^{rate}} \left( p_{B,i}^c(t) \times \eta_i^c + \frac{p_{B,i}^d(t)}{\eta_i^d} \right) $$
$$ SOC_J(t-1) = \frac{1}{n} \times \sum_{i=1}^{n} SOC_i(t-1) $$
Constraints and Solution Method
These objectives are subject to power balance, rated power, and SOC constraints:
$$ P_{AGC,B}(t) = \sum_{i=1}^{n} P_{B,i}(t) $$
$$ -P_{B,i}^{max}(t) \leq P_{B,i}(t) \leq P_{B,i}^{max}(t) $$
$$ SOC_{min,i}(t) \leq SOC_i(t) \leq SOC_{max,i}(t) $$
To solve this multi-objective optimization problem, I utilized the NSGA-II algorithm to generate a Pareto front of optimal solutions for the battery energy storage system power commands.
Simulation Setup and Verification
I validated my strategy using a simulation model of a regional power grid containing three BESS of different types: sodium-sulfur (BESS1), lithium-ion (BESS2), and lead-carbon (BESS3). The key parameters for these battery energy storage systems are summarized in the table below.
| Parameter | BESS1 (NaS) | BESS2 (Li-ion) | BESS3 (Lead-Carbon) |
|---|---|---|---|
| Rated Power / MW | 4×4 | 6×4 | 4×4 |
| Rated Capacity / MWh | 4×4 | 6×4 | 4×4 |
| Unit Power Cost / RMB·kW⁻¹ | 2500 | 1500 | 1500 |
| Float Life / years | 12 | 10 | 10 |
| Cycle Life / times | 3000 | 2000 | 2500 |
| SOC Upper Limit | 0.9 | 0.9 | 0.9 |
| SOC Lower Limit | 0.1 | 0.1 | 0.1 |
| Charge/Discharge Efficiency | 0.85 | 0.95 | 0.90 |
| Initial SOC | 0.8 | 0.5 | 0.2 |
I compared my proposed two-layer optimization strategy (Strategy A) against a dynamic proportional allocation strategy (Strategy B), which distributes the regulation command based on the remaining capacity of each battery energy storage system.
Analysis of Results
EMD-Based Command Decomposition
First, I applied EMD to a typical AGC signal. The decomposition revealed that the first two IMFs, IMF1 and IMF2, contained the highest frequency content. These two components were summed to create the high-frequency command for the BESS. This step is fundamental to my strategy, as it ensures that the fast-responding battery energy storage systems are tasked with the portion of the regulation signal for which they are best suited.
BESS Power Output and SOC Performance
The power output profiles for the three BESS under both strategies showed clear differences. Under my proposed strategy, in the initial period (0-12 min), only BESS1 and BESS2, which had higher initial SOCs (0.8 and 0.5), were dispatched to discharge. BESS3, with an initial SOC of 0.2, was not called upon. This decision, driven by the SOC balance objective and the resistance coefficient, quickly began to equalize the SOC levels. In contrast, the dynamic proportional allocation primarily dispatched BESS1 because it had the most remaining capacity, which did little to address the SOC imbalance and led to higher costs due to BESS1’s higher unit power cost.
The SOC trajectories vividly illustrate the advantage of my strategy. The standard deviation of the SOCs across the three battery energy storage systems serves as a clear metric for balance. The figure below shows the SOC standard deviation over time.
$$ \text{SOC Standard Deviation} = \sqrt{ \frac{1}{n} \sum_{i=1}^{n} (SOC_i – \overline{SOC})^2 } $$
My strategy rapidly reduced the standard deviation from an initial high value to a low, stable level (sum over the period = 6.18), indicating a well-balanced state among the battery energy storage systems. The dynamic proportional strategy, however, maintained a high standard deviation (sum = 22.91) throughout the simulation, showing persistent SOC imbalance.
Economic Performance Comparison
The economic performance of each BESS and the overall system is summarized in the following table.
| Strategy | BESS | Reg. Cost (RMB) | Reg. Revenue (RMB) | Net Profit (RMB) | Performance Index A |
|---|---|---|---|---|---|
| Proposed | BESS1 | 10,854.63 | 12,864.51 | 2,009.88 | 0.9352 |
| BESS2 | 10,484.37 | 12,322.53 | 1,838.16 | 0.9015 | |
| BESS3 | 9,005.13 | 10,563.52 | 1,558.39 | 0.8956 | |
| Total | 30,344.13 | 35,750.56 | 5,406.43 | ||
| Proportional | BESS1 | 18,091.43 | 20,263.29 | 2,171.86 | 0.9647 |
| BESS2 | 10,343.75 | 12,359.26 | 2,015.51 | 0.8523 | |
| BESS3 | 5,695.44 | 6,246.83 | 551.39 | 0.5124 | |
| Total | 34,130.62 | 38,869.38 | 4,738.76 |
The results demonstrate a critical trade-off. While the dynamic proportional strategy gave BESS1 a higher profit and performance index because it was the primary unit used, it did so at the expense of BESS2 and BESS3, which had low performance and profits. The total system profit was 4,738.76 RMB. In contrast, my proposed strategy resulted in a more balanced distribution of profits among the three battery energy storage systems. BESS1’s profit was lower, but BESS2 and BESS3’s profits were significantly higher. The overall system profit increased by 14.09% to 5,406.43 RMB. My strategy achieves this superior aggregate economic outcome by ensuring a fair and balanced utilization of all battery energy storage systems, preventing the degradation and underutilization of some units while over-burdening others. The high and consistent AGC performance indices for all BESS under my strategy further indicate that the entire fleet operated more reliably.
Conclusion
In my work, I have developed and validated a two-layer multi-objective optimization strategy for battery energy storage systems participating in frequency regulation. My key conclusions are as follows:
1. The proposed strategy, by decoupling the frequency regulation signal into high and low-frequency components, permits the battery energy storage system to focus on the tasks for which they are most efficient, thereby utilizing their fast-response characteristics optimally.
2. By integrating an economic objective function that includes a resistance coefficient and an SOC balance objective, my strategy successfully navigates the trade-off between profitability and system health. The simulation results confirm a 14.09% increase in total net profit compared to a conventional dynamic proportional allocation strategy.
3. The inclusion of SOC balance as an explicit objective leads to a profound improvement in the uniformity of SOC levels across all battery energy storage systems in the fleet. This not only extends the operational lifespan of individual units by preventing deep discharges and overcharges but also ensures that the entire system is more resilient and sustainable over long-term operation. My research provides a robust framework for the efficient and balanced utilization of battery energy storage systems in modern power grids.