The advancement of energy storage battery systems is pivotal for realizing the global “dual carbon” objectives, facilitating rapid frequency regulation, smoothing the output fluctuations of renewable energy sources, and enhancing overall power system stability and reliability. By the end of 2022, the cumulative installed capacity of power storage projects in China reached 59.8 GW, with new-type storage accounting for 13.1 GW. Within this segment, energy storage battery technology dominates, representing approximately 97% of the market share. Projections indicate that the installed capacity of new-type storage will exceed 50 GW by 2025.
However, this rapid growth is accompanied by significant safety concerns. Over the past decade, 32 major safety incidents have been reported at large-scale energy storage power stations globally. This underscores an urgent need to significantly improve the innovation capability, safety, reliability, and longevity of new energy storage technologies. Consequently, the reliability assessment of energy storage battery systems has become a critical research focus, encompassing studies on individual battery cells and the overall system.
Traditional reliability assessment methods for energy storage battery systems often rely on simplistic models that fail to capture the complex, multi-dimensional nature of performance degradation and the intricate interconnections within a battery pack. This study addresses these limitations by proposing a comprehensive framework for assessing the operational reliability of energy storage battery systems, explicitly considering multi-dimensional performance degradation.
1. Limitations of Traditional Reliability Assessment Methods
Conventional approaches to evaluating the reliability of an energy storage battery system often suffer from two major shortcomings. First, they typically consider performance degradation along a single dimension, such as capacity fade, while neglecting other critical indicators like voltage characteristics. Second, they inadequately model the system’s reliability by oversimplifying the impact of series-parallel connections among numerous battery cells, often assuming all cells are identical.
The traditional reliability \( R \) of a single energy storage battery cell is defined as the probability that its performance metrics simultaneously exceed minimum required thresholds:
$$ R = \Pr\{ Q \geq Q_{\text{req}}, d \geq d_{\text{req}} \} $$
where \( Q \) and \( d \) represent the cell’s capacity and a defined voltage characteristic, respectively, and \( Q_{\text{req}} \) and \( d_{\text{req}} \) are their corresponding minimum required values.
For a system comprising \( N \) identical cells, the system reliability \( R_s \) is frequently calculated as:
$$ R_s = R^N $$
This exponential relationship makes the system reliability extremely sensitive to the estimated single-cell reliability \( R \). Furthermore, this model ignores the inherent uncertainty in each cell’s state of health and the system’s topology, leading to potentially inaccurate assessments—either overly optimistic or pessimistic.
2. A Novel Framework for Multi-Dimensional Reliability Assessment
The proposed methodology enhances the reliability assessment of energy storage battery systems by introducing two core innovations: a Gaussian Process-based model for voltage characteristic distribution and a Multi-Dimensional Universal Generating Function (UGF) framework for system-level evaluation.
2.1 Gaussian Process-Based Voltage Characteristic Distribution
Voltage characteristics, derived from differential voltage (DV) analysis, provide profound insights into the internal electrochemical state and degradation mechanisms of an energy storage battery. A key voltage feature, defined as the distance between two inflection points on the discharge DV curve, exhibits a consistent decreasing trend with cycling, correlating strongly with capacity fade.
To accurately capture the distribution of this voltage characteristic \( d^* \) for a cell during a specific cycle, we employ Gaussian Process (GP) regression. Let \( \mathcal{D} = \{ \mathbf{X_Q}, \mathbf{Y_{DV}} \} \) be the dataset from a discharge cycle, where \( \mathbf{X_Q} = [x_{q1}, x_{q2}, …, x_{qn}]^T \) is the vector of discharged capacity sampling points and \( \mathbf{Y_{DV}} = [y_{dv1}, y_{dv2}, …, y_{dvn}]^T \) is the corresponding vector of differential voltage values.
The GP model assumes a joint Gaussian prior distribution over the observed values and the predicted function value \( f_{DV}^* \) at a new input point \( x_{q*} \):
$$
\begin{bmatrix}
\mathbf{Y_{DV}} \\
f_{DV}^*
\end{bmatrix}
\sim \mathcal{N}\left(
\mathbf{0},
\begin{bmatrix}
\mathbf{K} & \mathbf{K}_*^T \\
\mathbf{K}_* & k_{**}
\end{bmatrix}
\right)
$$
Here, \( \mathbf{K} \) is the covariance matrix computed between all observed points using a kernel function \( k(x_i, x_j) \), such as the squared exponential kernel:
$$ k(x_i, x_j) = \sigma_f^2 \exp\left(-\frac{(x_i – x_j)^2}{2l^2}\right) $$
\( \mathbf{K}_* \) is the covariance vector between the new point and all observed points, and \( k_{**} \) is the variance at the new point.
The posterior predictive distribution for \( f_{DV}^* \) is also Gaussian:
$$ f_{DV}^* | \mathbf{X_Q}, \mathbf{Y_{DV}}, x_{q*} \sim \mathcal{N}\left( \bar{f}_{DV}^*, \mathbb{V}(f_{DV}^*) \right) $$
with mean and variance given by:
$$ \bar{f}_{DV}^* = \mathbf{K}_* \mathbf{K}^{-1} \mathbf{Y_{DV}} $$
$$ \mathbb{V}(f_{DV}^*) = k_{**} – \mathbf{K}_* \mathbf{K}^{-1} \mathbf{K}_*^T $$
This allows us to construct confidence intervals for the entire DV curve. The upper and lower bounds, \( f_{\text{up}} \) and \( f_{\text{low}} \), at \( \pm M \) standard deviations are:
$$ f_{\text{up}} = \bar{f}_{DV}^* + M \sqrt{\mathbb{V}(f_{DV}^*)} $$
$$ f_{\text{low}} = \bar{f}_{DV}^* – M \sqrt{\mathbb{V}(f_{DV}^*)} $$
By finding the first and last inflection points (\( x_{\text{up1}}, x_{\text{up2}} \)) and (\( x_{\text{low1}}, x_{\text{low2}} \)) on these boundary curves using numerical methods like Newton-Raphson, we estimate the bounds for the voltage feature: \( d_{\text{up}} = x_{\text{up2}} – x_{\text{up1}} \) and \( d_{\text{low}} = x_{\text{low2}} – x_{\text{low1}} \). The mean \( \mu_d \) and standard deviation \( \sigma_d \) of the voltage characteristic’s distribution are then:
$$ \mu_d = \frac{1}{2}(d_{\text{low}} + d_{\text{up}}) $$
$$ \sigma_d = \frac{1}{2M}(d_{\text{up}} – d_{\text{low}}) $$
Thus, the voltage characteristic for a cell in a given cycle is modeled as a normally distributed random variable:
$$ d^* \sim \mathcal{N}(\mu_d, \sigma_d^2) $$
This provides a probabilistic description of one critical degradation dimension for the energy storage battery.
2.2 Multi-Dimensional Universal Generating Function (UGF) Framework
To integrate multiple degradation dimensions, we employ the UGF technique. We consider two primary dimensions: the voltage characteristic \( d^* \) and the capacity \( Q^* \). The capacity is also modeled as a normally distributed variable, informed by empirical fade models:
$$ Q^* \sim \mathcal{N}(\mu_Q, \sigma_Q^2) $$
Each performance dimension is discretized into a set of ordered health states, from perfect (State 1) to failed (State \(L\) for voltage, State \(M\) for capacity). For the voltage characteristic of cell \(i\), the probability \( p_{i,j}^d \) of being in state \( j \) is calculated by integrating its probability density function over the corresponding interval \( D_{ij} = [d_{ij}^{\text{lower}}, d_{ij}^{\text{upper}}] \):
$$ p_{i,j}^d = \int_{d_{ij}^{\text{lower}}}^{d_{ij}^{\text{upper}}} \frac{1}{\sqrt{2\pi\sigma_d^2}} \exp\left(-\frac{(x – \mu_d)^2}{2\sigma_d^2}\right) dx $$
The UGF for the voltage characteristic is:
$$ u_i^d(z) = \sum_{j=1}^{L_i^v} p_{i,j}^d z^{j} $$
Similarly, for capacity:
$$ p_{i,k}^Q = \int_{Q_{ik}^{\text{lower}}}^{Q_{ik}^{\text{upper}}} \frac{1}{\sqrt{2\pi\sigma_Q^2}} \exp\left(-\frac{(x – \mu_Q)^2}{2\sigma_Q^2}\right) dx $$
$$ u_i^Q(z) = \sum_{k=1}^{M_i^Q} p_{i,k}^Q z^{k} $$
The multi-dimensional UGF for a single energy storage battery cell \( i \), combining both degradation dimensions, is obtained using a composition operator \( \otimes_{\max} \). This operator multiplies the probabilities of concurrent states and takes the maximum of their state indices (reflecting that a cell’s overall health is limited by its worst-performing dimension):
$$ u_i(z) = u_i^d(z) \otimes_{\max} u_i^Q(z) = \sum_{j=1}^{L_i^v} \sum_{k=1}^{M_i^Q} p_{i,j}^d p_{i,k}^Q z^{\max(j,k)} = \sum_{e=1}^{E_i} p_{i,e} z^{e} $$
where \( E_i = \max(L_i^v, M_i^Q) \) and \( p_{i,e} \) is the aggregated probability of the cell being in overall health state \( e \). The reliability \( R_i \) of the cell, given a minimum required state \( \gamma \), is:
$$ R_i = \Pr(e \leq \gamma) = \sum_{e \leq \gamma} p_{i,e} $$
2.3 System-Level Reliability Considering Series-Parallel Topology
An energy storage battery system is composed of many cells connected in series and parallel. The system’s overall UGF is computed by recursively applying series and parallel composition operators to the UGFs of individual cells or modules.
For two components with UGFs \( u_1(z) = \sum_a p_{1,a} z^{a} \) and \( u_2(z) = \sum_b p_{2,b} z^{b} \):
- Parallel Connection: The system performance is determined by the better of the two components (e.g., current sharing). The corresponding operator \( \otimes_{\min} \) takes the minimum state index:
$$ u_{\text{par}}(z) = u_1(z) \otimes_{\min} u_2(z) = \sum_a \sum_b p_{1,a} p_{2,b} z^{\min(a,b)} $$ - Series Connection: The system performance is limited by the weaker of the two components (e.g., voltage chain). The operator \( \otimes_{\max} \) is used:
$$ u_{\text{ser}}(z) = u_1(z) \otimes_{\max} u_2(z) = \sum_a \sum_b p_{1,a} p_{2,b} z^{\max(a,b)} $$
By applying these operators according to the system’s physical topology, we obtain the UGF for the entire energy storage battery system:
$$ U_{\text{sys}}(z) = \Omega(u_1(z), u_2(z), …, u_N(z)) = \sum_{h=1}^{H} p_{\text{sys},h} z^{h} $$
where \( \Omega \) represents the recursive composition process and \( p_{\text{sys},h} \) is the probability of the system being in overall state \( h \). The system reliability \( R_s \), given a minimum required system state \( \beta \), is:
$$ R_s = \Pr(h \leq \beta) = \sum_{h \leq \beta} p_{\text{sys},h} $$
This approach accurately aggregates the heterogeneous reliability of individual energy storage battery cells while accounting for the system’s configuration.
3. Case Study and Comparative Analysis
The proposed method was validated using aging data from NASA 18650 lithium-ion energy storage batteries. A system of 144 cells was configured in two ways: 12 parallel strings of 12 series cells each (Series-Parallel, SP), and 12 series modules of 12 parallel cells each (Parallel-Series, PS). The minimum required system state \( \beta \) was set to 9. Reliability was tracked over 1000 charge-discharge cycles and compared against two benchmark methods:
- M1 (Traditional): Uses the exponential model \( R_s = R^N \), ignoring cell heterogeneity and topology.
- M2 (Single-Dimension UGF): Uses the UGF framework but considers only capacity degradation.
- M3 (Proposed Multi-Dimension UGF): The full proposed method considering both voltage characteristic and capacity degradation.
| Assessment Method | Key Characteristics | Cycle at which \( R_s \) begins significant decline | Cycle at which \( R_s \) reaches ~0 |
|---|---|---|---|
| M1 (Traditional) | Assumes identical cells, ignores topology. | ~410 | ~700 |
| M2 (Single-Dimension) | Considers cell heterogeneity and topology, but only capacity fade. | ~780 | ~950 |
| M3 (Proposed) | Considers cell heterogeneity, topology, and multi-dimensional degradation (capacity + voltage). | ~782 | ~902 |
The results clearly demonstrate the shortcomings of the traditional method (M1), which predicts a much earlier and steeper reliability drop, failing to capture the redundancy and averaging effects present in a real energy storage battery pack. Both M2 and M3 show a more gradual and realistic decline, beginning around cycle 780. However, M3 predicts system failure approximately 48 cycles earlier than M2. This is because M3’s \( \otimes_{\max} \) operator correctly models that failure in either dimension (voltage characteristic or capacity) can lead to cell failure, providing a more conservative and comprehensive reliability estimate for the energy storage battery system.
| System Topology | Description | Reliability Trend |
|---|---|---|
| Series-Parallel (SP) | 12 parallel strings, each with 12 series cells. | Higher initial reliability, slower decline. Failure occurs later. |
| Parallel-Series (PS) | 12 series modules, each with 12 parallel cells. | Lower reliability compared to SP for identical cell count. Failure occurs earlier. |
Table 2 highlights a critical finding: system topology significantly impacts reliability, even for systems with the same total number of energy storage battery cells. The SP configuration generally offers higher reliability because a failure in a series string can be partially compensated by current redistribution across parallel strings. This crucial factor is completely overlooked by traditional methods like M1, which would predict identical reliability for both SP and PS configurations.
4. Conclusion
This work presents a novel and robust framework for assessing the operational reliability of energy storage battery systems. By integrating a Gaussian Process model to probabilistically characterize voltage feature degradation and a Multi-Dimensional Universal Generating Function framework to aggregate the effects of multiple degradation mechanisms and system topology, the method overcomes the significant limitations of traditional assessment approaches.
The key conclusions are:
- The proposed multi-dimensional method provides a more scientifically accurate and conservative estimate of energy storage battery system reliability compared to traditional single-failure-threshold or single-dimension models.
- The voltage characteristic derived from differential voltage analysis serves as a valuable complementary dimension to capacity for tracking the health state of an energy storage battery.
- The topological configuration of cells (series-parallel vs. parallel-series) has a substantial impact on system-level reliability, a factor that must be considered during the design and evaluation phases of an energy storage battery system.
This enhanced reliability assessment methodology provides a powerful tool for optimizing the design, maintenance scheduling, and replacement strategies for energy storage battery systems, ultimately contributing to safer and more economically efficient integration of energy storage into the modern power grid.