With the large-scale integration of renewable energy sources into the power grid, issues such as reduced equivalent rotational inertia during frequency control and insufficient frequency regulation reserve capacity of traditional thermal power units have become prominent. Energy storage systems, particularly those based on battery technology, offer a promising solution due to their fast response and flexible regulation capabilities. Among these, the energy storage cell serves as the fundamental unit, and its coordinated operation is crucial for effective grid support. In this context, I explore a multi-objective cooperative control strategy that leverages the unique characteristics of individual energy storage cells to enhance primary frequency regulation. This strategy not only addresses frequency stability but also optimizes the operational safety and efficiency of the battery system, ensuring sustainable integration into modern power grids.
The core idea revolves around grouping energy storage cells based on their State of Charge (SOC) to form distinct storage units, each with tailored control parameters. By doing so, I can manage the heterogeneity among energy storage cells, which often arises from manufacturing tolerances, aging, or environmental conditions. This grouping allows for a more granular approach to power allocation during frequency disturbances, ensuring that each energy storage cell operates within its safe limits while contributing to grid stability. The strategy integrates virtual inertia and virtual droop control mechanisms, adapting them to the SOC levels of each unit to maximize the utilization of the energy storage cell capacity. Furthermore, I incorporate a multi-objective optimization framework that minimizes frequency deviations and energy losses, leading to a balanced trade-off between performance and battery health. Through detailed modeling and simulation, I demonstrate the effectiveness of this approach in various load disturbance scenarios, highlighting its superiority over conventional methods. This work underscores the importance of considering the individual attributes of each energy storage cell in large-scale deployments, paving the way for more resilient and efficient power systems.
To establish a foundation, I first construct a system model where multiple energy storage units, each comprising grouped energy storage cells, participate in primary frequency regulation. The power system dynamics can be represented by the swing equation, which describes the frequency deviation $\Delta f$ in response to power imbalances. The equation is given by:
$$ \dot{\Delta f} = -\frac{D}{M} \Delta f + \frac{1}{M} \left( \Delta P_G + \Delta P_E – \Delta P_L \right) + \frac{1}{T_g} \Delta P_c, $$
where $D$ is the load damping coefficient, $M$ represents the system inertia, $\Delta P_G$ is the power output deviation from traditional generators, $\Delta P_E$ is the total power output from the energy storage system, $\Delta P_L$ is the load disturbance, and $\Delta P_c$ is the control input from the governor. For primary frequency regulation, the traditional generator response is typically modeled as:
$$ \Delta P_G = -\frac{1}{R} G_g(s) \Delta f, $$
with $R$ being the generator frequency factor and $G_g(s)$ representing the governor transfer function. The energy storage system’s output $\Delta P_E$ is the sum of contributions from all storage units, each consisting of multiple energy storage cells. If I denote the number of units as $n$, then:
$$ \Delta P_E = \sum_{i=1}^{n} \Delta P_i, $$
where $\Delta P_i$ is the power output from the $i$-th unit. This modular approach allows me to manage each energy storage cell group independently based on its SOC and capacity, enhancing flexibility and control precision. The parameters involved in this model are summarized in the following table to provide a clear reference:
| Parameter/Variable | Description | Unit |
|---|---|---|
| $\Delta f$ | Frequency deviation | Hz |
| $\Delta P_G$ | Traditional generator power deviation | p.u. |
| $\Delta P_E$ | Total energy storage system power output | p.u. |
| $\Delta P_L$ | Load disturbance | p.u. |
| $H$ | Inertia time constant | s |
| $D$ | Load damping coefficient | Hz/p.u. |
| $T_g$ | Governor time constant | s |
| $K_r$ | Reheat coefficient | – |
| $T_r$ | Reheat time constant | s |
| $T_t$ | Turbine time constant | s |
| $\Delta P_c$ | Controller input | p.u. |
| $R$ | Generator frequency factor | p.u./Hz |
Each energy storage cell within a unit is characterized by its SOC, which evolves over time based on the power exchanged. The SOC dynamics for the $i$-th unit can be expressed as:
$$ \text{SOC}_i(t + \Delta t) = \text{SOC}_i(t) + \frac{\int_{t}^{t+\Delta t} \Delta P_i^* \, dt}{E_{N,i}}, $$
where $E_{N,i}$ is the rated capacity of the unit, and $\Delta P_i^*$ is the power output or input at that instant. This equation highlights the direct impact of power transactions on the state of each energy storage cell, necessitating careful management to prevent over-charging or over-discharging. To ensure safe operation, I analyze the relationship between the open-circuit voltage (OCV) and SOC for a typical energy storage cell, as shown in various studies. The OCV-SOC curve exhibits a flat region between SOC values of 0.2 and 0.8, where voltage changes are minimal, making it the optimal operating range. Outside this region, the voltage gradient increases significantly, posing risks to the energy storage cell health and system stability. Therefore, I constrain the SOC of each energy storage cell within this range, i.e., $0.2 \leq \text{SOC} \leq 0.8$, to maintain efficient and secure performance.
The output power of each energy storage unit is determined by a combination of virtual inertia and virtual droop control, tailored to the unit’s SOC and capacity. For the $i$-th unit, the control law is:
$$ \Delta P_i = K_{I,i} \frac{d\Delta f}{dt} + K_{D,i} \Delta f, $$
where $K_{I,i}$ and $K_{D,i}$ are the virtual inertia and droop coefficients, respectively. These coefficients are adjusted based on the unit’s available energy reserve, which depends on the SOC and capacity of the energy storage cells in that unit. Specifically, the power allocation proportion $k_i$ for the $i$-th unit is calculated as:
$$ k_i = \frac{C_i \cdot \text{SOC}_i(t)}{\sum_{j=1}^{n} C_j \cdot \text{SOC}_j(t)}, $$
where $C_i$ is the capacity of the unit. Then, the reference power output for the unit is:
$$ P_{i,t}^* = k_i \left( \Delta P_{\text{Req}} – \sum_{j=1}^{n} P_{j,t} \right), $$
with $\Delta P_{\text{Req}} = -K \cdot \Delta f_t$ representing the required power to counteract the frequency deviation, and $K$ being the grid frequency characteristic constant. The control coefficients are then scaled as:
$$ K_{I,i} = \frac{P_{i,t}^*}{P_{i,t,\text{max}}^*} K_{I,\text{max}}, \quad K_{D,i} = \frac{P_{i,t}^*}{P_{i,t,\text{max}}^*} K_{D,\text{max}}, $$
where $P_{i,t,\text{max}}^*$ is the maximum power output among all units, and $K_{I,\text{max}}$ and $K_{D,\text{max}}$ are the maximum allowable control coefficients. This adaptive strategy ensures that units with higher SOC and capacity contribute more to frequency regulation, while those with lower reserves are protected from excessive depletion. However, this approach does not account for energy losses within each energy storage cell, which can affect overall efficiency. Therefore, I extend the strategy to include a multi-objective optimization that minimizes both frequency deviations and energy losses.
Energy losses in an energy storage cell primarily arise from internal resistance, contact resistance, and line resistance. The equivalent circuit of a battery cell can be modeled as a voltage source $U_{\text{OCV}}$ in series with an equivalent resistance $R_{\text{eq}}$, which encompasses these resistive components. The terminal voltage $U$ and current $I$ relate as:
$$ U = U_{\text{OCV}} – I R_{\text{eq}} \quad \text{(during discharge)}, $$
$$ U = U_{\text{OCV}} + I R_{\text{eq}} \quad \text{(during charge)}. $$
The energy loss for the $i$-th unit over a time period $T^*$ is given by:
$$ E_{i,\text{loss}} = \int_{t}^{t+T^*} I_i^2 R_{i,\text{eq}} \, dt, $$
where $I_i$ is the current of the unit, and $R_{i,\text{eq}}$ is its equivalent resistance. To incorporate loss minimization into the control strategy, I define adjustment variables $\Delta P_{i,t}$ that modify the power outputs from the initial allocation. The adjusted power for each unit becomes:
$$ P_{i,t}’ = P_{i,t}^* + \Delta P_{i,t}. $$
The multi-objective optimization problem aims to find the optimal adjustments that minimize frequency deviations and total energy losses. The objective functions are formulated as follows. First, to minimize the frequency deviation fluctuations, I consider the rate of change of frequency (RoCoF) squared error:
$$ f_1 = \min \sum_{i=1}^{n} \left( \Delta f_i(k) – \Delta f_i(k-1) \right)^2, $$
where $\Delta f_i(k)$ represents the frequency deviation at time step $k$ for the system with the $i$-th unit’s contribution. Second, to minimize the total energy loss across all energy storage cells, the objective function is:
$$ f_2 = \min \sum_{i=1}^{n} E_{i,\text{loss}}. $$
These objectives are subject to several constraints that ensure operational feasibility and safety. The power balance constraint requires that the sum of adjusted power outputs matches the total required power:
$$ \sum_{i=1}^{n} P_{i,t}’ = P_{E,t}, $$
where $P_{E,t}$ is the total power command for the energy storage system, positive for discharge and negative for charge. The power-current relationship for each energy storage cell must adhere to its limits:
$$ \begin{cases}
P_i = (U_{i,\text{ocv}} – I_i R_{i,\text{eq}}) I_i & \text{for } P_i > 0 \text{ (discharge)} \\
-P_i = (U_{i,\text{ocv}} + I_i R_{i,\text{eq}}) I_i & \text{for } P_i < 0 \text{ (charge)} \\
0 < I_i < I_{i,\text{max}}
\end{cases}, $$
with $I_{i,\text{max}}$ being the maximum current for the $i$-th unit. Additionally, the SOC of each energy storage cell must remain within the safe range:
$$ \text{SOC}_{\text{min}} < S_i(t) < \text{SOC}_{\text{max}}, $$
where I set $\text{SOC}_{\text{min}} = 0.2$ and $\text{SOC}_{\text{max}} = 0.8$ based on the OCV-SOC analysis. The overall optimization model is summarized as:
$$ \begin{aligned}
&\min \; f_s(\chi), \quad s = 1, 2 \\
&\text{subject to:} \\
&\quad h_k(\chi) = 0, \quad k = 1, 2, 3 \\
&\quad g_l(\chi) \leq 0, \quad l = 1, 2
\end{aligned} $$
where $\chi$ denotes the decision vector comprising the adjustment variables $\Delta P_{i,t}$, $h_k(\chi)$ represents equality constraints (e.g., power balance), and $g_l(\chi)$ represents inequality constraints (e.g., SOC and current limits). To solve this multi-objective problem, I employ the NSGA-II algorithm, which is effective for handling non-convex and discontinuous Pareto fronts. The algorithm iteratively generates a population of solutions, evaluates their constraint violations, and performs non-dominated sorting to identify the Pareto optimal set. The constraint violation for a solution $\chi$ is computed as:
$$ \text{CV}(\chi) = \sum_{k=1}^{3} |h_k(\chi)| + \sum_{l=1}^{2} \langle g_l(\chi) \rangle, $$
where $\langle g_l(\chi) \rangle$ is zero if $g_l(\chi) \leq 0$, and equal to $g_l(\chi)$ otherwise. After obtaining the Pareto set, I select the best compromise solution using a satisfaction function $\mu$ defined for each objective:
$$ \mu = \frac{1}{m} \sum_{s=1}^{m} \mu_s, $$
with $m=2$ being the number of objectives, and $\mu_s$ ranging from 0 (completely dissatisfied) to 1 (completely satisfied). This approach ensures a balanced trade-off between frequency regulation performance and energy storage cell efficiency.
To validate the proposed strategy, I conduct simulations using MATLAB/Simulink, considering a power system with a 600 MW thermal generator and a base frequency of 50 Hz. The system parameters are set as: $H = 5.92$, $D = 2.75$, $T_g = 0.1$, $K_r = 25$, $T_r = 10$, $T_t = 0.2$, and $R = 0.05$. The energy storage system comprises three units, each formed by grouping energy storage cells with similar SOC values. The parameters for each unit are detailed in the table below, assuming each energy storage cell has an internal resistance of 10 mΩ:
| Unit | Initial SOC | Capacity (MWh) | Equivalent Resistance (Ω) |
|---|---|---|---|
| Unit 1 | 0.80 | 4.0 | 0.20 |
| Unit 2 | 0.70 | 6.0 | 0.25 |
| Unit 3 | 0.65 | 5.5 | 0.22 |
I simulate a sequence of load disturbances to test the strategy under dynamic conditions. The load profile includes a sudden increase of 0.12 p.u. at 10 seconds, a decrease to -0.06 p.u. at 40 seconds, and another increase to 0.10 p.u. at 70 seconds. I compare four control strategies: (1) no energy storage, (2) power allocation based solely on SOC and capacity without optimization, (3) the proposed multi-objective optimized strategy, and (4) traditional equal power distribution among units. The frequency deviation results show that the proposed strategy significantly reduces both the maximum frequency deviation and the settling time compared to other approaches. Specifically, the RoCoF is minimized, and the steady-state error is maintained within acceptable limits. The power outputs of each unit under the optimized strategy demonstrate adaptive allocation, where units with higher SOC and lower resistance contribute more power, thereby reducing overall losses.
The SOC trajectories of the energy storage cells in each unit reveal that the optimized strategy keeps the SOC values within the safe range and promotes convergence among units, preventing any single energy storage cell from reaching extreme states. In contrast, traditional equal distribution leads to uneven SOC depletion, risking over-discharge for some cells. The energy losses are quantified over different time intervals, as shown in the following table, where the proposed strategy achieves the lowest loss values, indicating improved efficiency:
| Time Interval (s) | Proposed Strategy Loss (p.u.) | Non-optimized Allocation Loss (p.u.) | Equal Distribution Loss (p.u.) |
|---|---|---|---|
| 10–40 | 0.0103 | 0.0113 | 0.0109 |
| 40–70 | 0.0075 | 0.0089 | 0.0087 |
| 70–100 | 0.0101 | 0.0110 | 0.0111 |
These results highlight that the multi-objective optimization reduces energy losses by 5.5% to 15.7% compared to non-optimized methods, directly benefiting the longevity and performance of each energy storage cell. Moreover, the frequency regulation performance is enhanced, with faster recovery and smaller deviations. The adaptive control coefficients $K_{I,i}$ and $K_{D,i}$ vary dynamically based on the power allocation, ensuring that each energy storage cell operates at its optimal point. This granular control is crucial for large-scale deployments where individual cell characteristics can significantly impact system behavior.
In conclusion, the multi-objective cooperative control strategy presented here offers a comprehensive solution for integrating energy storage cells into primary frequency regulation. By grouping cells based on SOC and tailoring control parameters, I address the heterogeneity among energy storage cells, ensuring safe and efficient operation. The optimization framework balances frequency stability and energy loss minimization, leading to superior performance compared to conventional approaches. This strategy not only enhances grid resilience but also extends the lifespan of energy storage cells by avoiding extreme SOC conditions and reducing cumulative stress. Future work could explore real-time implementation with advanced forecasting techniques and expand the model to include other storage technologies. Ultimately, this approach underscores the critical role of detailed energy storage cell management in achieving sustainable and reliable power systems.