In modern power systems, the increasing penetration of renewable energy sources has significantly altered the dynamic characteristics of frequency stability. Traditional synchronous generators, which provide inherent rotational inertia, are being replaced by converter-interfaced resources. Among these, the energy storage battery system (BESS) has emerged as a critical element for providing synthetic inertia. However, a key challenge that is often overlooked is the long-term performance degradation of these batteries. Frequent charge and discharge cycles, driven by continuous small-signal disturbances in the grid, accelerate the aging process of the energy storage battery. This power degradation directly reduces the maximum inertia support capability of the BESS. If this degradation is not considered in the evaluation of grid inertia, the assessed upper boundary of the inertia stability region (ISR) can be significantly overestimated. This overestimation poses a severe risk: the actual available inertia might fall below the required lower boundary, leading to frequency instability, such as excessive Rate of Change of Frequency (RoCoF) or frequency deviation.
In this paper, I propose a comprehensive method for assessing the Inertia Stability Region (ISR) of a power grid that explicitly accounts for the long-term aging effects of the energy storage battery. My approach integrates a battery peak power degradation model with a grid frequency response model to dynamically update the boundaries of the ISR. The core contribution is the development of a framework that prevents the overestimation of grid inertia reserves, thereby ensuring operational security. The following sections detail the methodology, from battery aging quantification to the final ISR evaluation and the definition of an inertia stability margin index.
Methodology for Long-Term Peak Power Degradation of Energy Storage Batteries
To accurately assess the inertia support capability of a BESS over its lifetime, it is essential to model the gradual decay of its peak power. The overall process for this assessment is structured into three key steps, summarized in Table I.
| Step | Description | Input | Output |
|---|---|---|---|
| 1 | Simulate the energy storage battery response to grid frequency disturbances to track its State of Energy (SOE). | Grid frequency disturbance, BESS control parameters | Temporal SOE data sequence |
| 2 | Decompose the SOE data using the Rain-flow counting algorithm to extract cycle information. | SOE time series data | Cycle depth (\(d_i\)), cycle count (\(c_i\)), mean SOE (\(M_{SOE}\)) |
| 3 | Apply the long-term peak power degradation model to the extracted cycle information. | Cycle information, operational time | Total peak power degradation (\(P_{total}\)) |
In Step 1, I model the energy storage battery as a virtual inertia source. Its active power output, \(P_{BESS}(t)\), is controlled to counteract the RoCoF, \(df(t)/dt\). This relationship is controlled by an adaptive inertia response coefficient, \(K_B(\mu)\), which is a function of the battery’s SOE (\(\mu\)). The power output is given by:
$$P_{BESS}(t) = -K_B(\mu) \frac{df(t)}{dt}$$
This coefficient is carefully designed to prevent the SOE from being quickly depleted or saturated. The charging coefficient, \(K_c\), and discharging coefficient, \(K_d\), are defined adaptively based on the current SOE, with parameters fitting a sigmoid-like function:
$$
\begin{cases}
K_d = \frac{K_{Bmax}}{1 + 100 K_{Bmax} e^{1 – n\frac{\mu – \mu_{min}}{\mu_{low} – \mu_{min}}}} \\
K_c = \frac{K_{Bmax}}{1 + 100 K_{Bmax} e^{1 – n\frac{\mu_{max} – \mu}{\mu_{max} – \mu_{high}}}}
\end{cases}
$$
where \(K_{Bmax}\) is the maximum coefficient, \(\mu_{min}\) and \(\mu_{max}\) are the SOE safety limits, and \(\mu_{low}\) and \(\mu_{high}\) are the buffer values near these limits. The evolution of the SOE over a time step \(\Delta t\) is then determined by the energy balance equation:
$$\mu(t + \Delta t) = \mu(t) – \frac{P_{BESS}(t) \Delta t}{E_{BESS}}$$
The aging of the energy storage battery is attributed to two primary mechanisms: calendar aging (time-dependent) and cycle aging (usage-dependent). The empirical model I employ to estimate the total peak power degradation, \(P_{total}\), is presented in Table II. This model is derived from fitting experimental data from lithium-ion battery aging tests.
| Degradation Type | Equation |
|---|---|
| Calendar Aging | \(P_{d\_cal} = 0.0033 \cdot M_{SOE}^{0.4513} \cdot t\) |
| Cycle Aging | \(P_{d\_cyc} = \sum_{i=1}^{N} 1.1725 \times 10^{-6} \cdot d_i^{0.7891} \cdot c_i\) |
| Total Degradation | \(P_{total} = P_{d\_cal} + P_{d\_cyc}\) |
In these equations, \(t\) is the operation time, \(M_{SOE}\) is the mean SOE, and \(d_i\) and \(c_i\) are the depth of cycle and number of cycles from the Rain-flow count, respectively. This model allows me to quantitatively link the operational history of the energy storage battery to its reduced capacity to provide inertia support.
Inertia Stability Region (ISR) Evaluation Considering Battery Aging
My overall framework for evaluating the ISR while considering battery aging is designed to be dynamic and iterative. The flow of the evaluation is as follows: First, I estimate the power disturbance, \(\Delta P\), using a load forecasting algorithm. This disturbance is then used in a System Frequency Response (SFR) model to calculate the system frequency dynamics. The SFR model, which integrates synchronous generators, renewable energy sources, and BESS, is central to this analysis. Its core dynamic equations relate the RoCoF and frequency deviation to the disturbance and total system inertia.
The frequency response of the grid can be represented in a simplified form after applying the inverse Laplace transform to the SFR transfer function:
$$
\begin{cases}
\frac{df}{dt} = – \frac{R \Delta P}{DR+1} \alpha \omega_n e^{-\zeta \omega_n t} \sin(\omega_r t + \phi) \\
\Delta f_d = \frac{R \Delta P}{DR+1} (1 + \sqrt{1-\zeta^2} \alpha \omega_n e^{-\zeta \omega_n t_d})
\end{cases}
$$
where \(R\) is the droop, \(D\) is the damping coefficient, \(\omega_n\) is the natural frequency, and \(\zeta\) is the damping ratio. The damping ratio and natural frequency are functions of the total system inertia, \(H\), defined as:
$$H = H_g + H_{re} + H_{BESS}$$
1. Calculation of the ISR Upper Boundary: This boundary, \(E^{high}_{sys}\), represents the maximum total inertial power available in the system. It is determined by the rated capacities of all generators, with the contribution from the BESS being scaled down by the aging degradation factor, \(P_{total}\). The formula for this boundary is a key part of my method, as it directly incorporates the aging effect:
$$E^{high}_{sys} = H_g S^{SG}_N + H_{re} S^{re}_N + (1 – P_{total}) H_{BESS} S^{BESS}_N$$
Here, \(S^{SG}_N\), \(S^{re}_N\), and \(S^{BESS}_N\) represent the rated capacities of the synchronous generators, renewable sources, and the energy storage battery, respectively. As \(P_{total}\) increases over time, the upper boundary of the ISR declines.
2. Calculation of the ISR Lower Boundary: The lower boundary, \(E^{low}_{sys}\), represents the minimum amount of inertia required to keep the system within safe frequency limits after a disturbance. I formulate this as an optimization problem that minimizes the total system inertia constant \(H\) while respecting stability constraints:
$$
\begin{aligned}
& \text{min } H \\
& \text{s.t.} \\
& \quad -R_{max} \leq \frac{df}{dt} \leq R_{max} \\
& \quad -\Delta f_{max} \leq \Delta f_d \leq \Delta f_{max}
\end{aligned}
$$
After solving for the minimum inertia constant, \(H_{min}\), the lower boundary \(E^{low}_{sys}\) is computed by weighting the contributions from each source based on their current output, \(P_{SG}\), \(P_{re}\), and \(P_{BESS}\).
3. Real-time Inertia Calculation: The actual total inertia of the grid at any moment, \(E_{sys}\), is calculated based on the instantaneous power output of each source:
$$E_{sys} = H_g P_{SG} + H_{re} P_{re} + H_{BESS} P_{BESS}$$
To evaluate the security of the system, I define the Inertia Stability Margin, \(r_{sys}\), which quantifies the relative distance of the current inertia from the critical lower boundary:
$$r_{sys} = \frac{E_{sys} – E^{low}_{sys}}{E^{low}_{sys}} \times 100\%$$
A positive value of \(r_{sys}\) indicates a stable state. A negative value indicates a deficit of inertia, signaling a risk of frequency instability and the need for corrective actions like load shedding or increasing virtual inertia from other sources.
Simulation and Results Analysis
I validated my proposed method using a MATLAB/Simulink platform, simulating the SFR model with parameters representing a typical power system with high penetration of energy storage battery and renewable energy. The key system parameters are listed in Table III.
| Component | Parameter | Value |
|---|---|---|
| Synchronous Generator (x5) | Total Rated Capacity (\(S^{SG}_N\)) | 50 MW |
| Inertia Constant, \(H_g\) | 5 s | |
| Renewable Energy Source | Rated Capacity (\(S^{re}_N\)) | 40 MW |
| Virtual Inertia, \(H_{re}\) | 4.8 s | |
| Energy Storage Battery (BESS) | Rated Capacity (\(S^{BESS}_N\)) | 30 MW |
| Virtual Inertia, \(H_{BESS}\) | 4.8 s |
To simulate long-term operation, I used real-world 24-hour power disturbance data as the input to the SFR model. This data was applied cyclically to drive the energy storage battery model and generate its SOE profile over many years.
Battery Peak Power Degradation Assessment Results: The Rain-flow counting of the simulated SOE data revealed that the energy storage battery primarily experiences shallow cycles, with most depths of discharge being less than 5%. This is typical for inertia support, which involves fast but small power bursts. The long-term projection of peak power degradation is shown in Table IV.
| Time (Months) | Peak Power Degradation, \(P_{total}\) (%) |
|---|---|
| 0 | 0.0 |
| 20 | ~0.5 |
| 60 | ~1.3 |
| 120 | ~2.4 |
This confirms that even a moderate level of cyclic activity can lead to a noticeable decline in the energy storage battery‘s peak power capability over a 10-year period.
ISR Evaluation Results: Figure 1 illustrates the core finding of my work: the dynamic shift of the ISR due to battery aging. As the energy storage battery ages from 1 month to 120 months, the upper boundary of the ISR decreases from approximately 525 MWs to about 500 MWs. This narrowing of the safe operating region is a critical factor for grid operators. The plot also shows the ISR lower boundary, which varies with the magnitude of the power disturbance. The gap between the real-time system inertia and the lower boundary defines the inertia stability margin, \(r_{sys}\).
This figure visually represents the core concept of my method. The real-time inertia level is shown in trajectory within this region. The evaluation model provides this continuous boundary, allowing the system to anticipate when the real-time inertia might approach or cross the lower boundary.
Risk Analysis and Validation: The most significant finding from my simulation is the risk of overestimating the ISR. If the energy storage battery aging is ignored, the ISR upper boundary is assessed to be about 2.4% higher than its true value after 120 months. Consider a scenario where a large disturbance causes the ISR lower boundary to rise. Under the “aging-ignored” assessment, the operator might see this lower boundary still below the assumed upper boundary and assume sufficient inertia reserves. However, in reality, the actual (degraded) ISR upper boundary is lower. The ISR lower boundary could surpass the true upper boundary, a state I call an “inertia deficit zone.” This would lead to the system operating with insufficient inertia, causing the RoCoF and frequency deviation (\(\Delta f_d\)) to violate their prescribed limits, a situation confirmed by my detailed time-domain simulations.
To further validate the accuracy of my method, I performed a test where the system’s calculated inertia, \(E_{sys}\), was set exactly equal to the ISR lower boundary. A power disturbance of 0.03 pu was then applied. The resulting RoCoF was 0.29 Hz/s, and the frequency deviation was 0.19 Hz. Both values are just below the safety limits of \(R_{max} = 0.3\) Hz/s and \(\Delta f_{max} = 0.2\) Hz. This precision confirms that my method for determining the lower boundary is accurate and non-conservative; it provides the exact minimum inertia required without being overly restrictive.
Finally, I conducted a sensitivity analysis to demonstrate the flexibility of the ISR lower boundary evaluation. By varying the constraints \(R_{max}\) and \(\Delta f_{max}\), I observed how the required inertia value changes. The results are summarized in Table V.
| \(R_{max}\) (Hz/s) | \(\Delta f_{max}\) (Hz) | ISR Lower Boundary (MWs) |
|---|---|---|
| 0.2 | 0.05 | ~440 |
| 0.3 | 0.10 | ~410 |
| 0.6 | 0.20 | ~380 |
The analysis shows a clear inverse relationship: as the frequency stability requirements become stricter (lower \(R_{max}\) and \(\Delta f_{max}\)), the ISR lower boundary increases. The range of these boundary values, spanning approximately 60 MWs, demonstrates that the ISR model can adapt its assessment to a wide range of operational and security standards.
Conclusion
In conclusion, I have presented a novel method for evaluating the Inertia Stability Region of a power grid that explicitly accounts for the long-term aging of the energy storage battery. The key findings from my work are as follows:
- By integrating a power degradation model with a system frequency response model, my ISR evaluation method establishes a direct link between the aging of the energy storage battery and the dynamic contraction of the grid’s safe inertial operating region. This allows for proactive, rather than reactive, management of inertia reserves.
- The proposed Inertia Stability Margin index, \(r_{sys}\), provides a clear, quantitative measure of the grid’s inertia health. A negative value serves as an unambiguous trigger for corrective actions, such as increasing virtual inertia from other sources or implementing load shedding, thereby preventing frequency instability.
- My simulation results validate the accuracy of the ISR boundary assessment. The lower boundary is determined with precision, ensuring that the system is not evaluated as being overly stable (which would waste resources) or, more importantly, as being stable when it is actually at risk of instability due to the overestimation of the upper boundary caused by energy storage battery degradation.
This work provides a crucial tool for grid operators and planners to ensure the reliable and secure operation of modern, converter-dominated power systems that heavily rely on energy storage battery systems for frequency stability. Future work will focus on extending this model to account for the stochastic nature of renewable generation and developing optimal control strategies that actively balance the trade-off between providing inertia support and preserving the lifespan of the energy storage battery.