Integrating cell energy storage systems into power supply networks is pivotal for demand-side management, load smoothing, and enhancing the utilization of intermittent renewable sources like photovoltaic (PV) generation. The operational economics and longevity of these systems are critically influenced by the rate of battery capacity degradation. Empirical studies consistently demonstrate that the degradation trajectory is not uniform but is heavily dependent on the operational State-of-Charge (SOC) range. Batteries cycled within high or low SOC extremes degrade significantly faster than those operated within moderate SOC windows. This work presents a comprehensive operational optimization framework for a hybrid power supply system employing a multi-cell energy storage system, where distinct battery units are constrained to operate within specific, partitioned SOC intervals. The primary objective is to minimize the total annualized cost, strategically balancing the capital cost associated with capacity degradation against operational expenses. A mixed-integer nonlinear programming (MINLP) model is formulated and solved to determine the optimal power dispatch. A case study of a PV-battery hybrid system demonstrates the substantial economic advantage of the partitioned multi-cell energy storage system approach compared to a conventional single-battery system operating across the full SOC range.
The core challenge in optimizing a cell energy storage system lies in accurately modeling the trade-off between its usable energy throughput (which favors wider SOC swings) and its service life (which benefits from restricted, moderate SOC cycling). Traditional models often treat capacity fade as a simple function of cumulative energy throughput or cycles, neglecting the profound impact of the SOC operating window. This work addresses this gap by conceptualizing a storage system composed of multiple, discrete battery units, each dedicated to a predefined SOC band (e.g., Low: 0%-20%, Medium: 20%-60%, High: 60%-100%). This architecture allows the system scheduler to intelligently route power through the cell energy storage system unit whose SOC band offers the most favorable degradation characteristics for the required service, thereby extending the aggregate system life and reducing long-term costs.
1. System Architecture and Problem Statement
The hybrid power supply system under consideration comprises four main components: a photovoltaic (PV) generation source, an electrical load representing user demand, the main power grid, and the proposed multi-cell energy storage system. The PV system can supply power directly to the load, charge the cell energy storage system, or sell excess power to the grid. The cell energy storage system, in turn, can discharge to meet the load demand or be charged from the PV surplus or the grid. The grid acts as a balancing entity, allowing for the purchase of electricity when local generation and storage are insufficient and the sale of surplus electricity.
The multi-cell energy storage system is comprised of \( N \) distinct battery units. Each unit \( n \in N \) is characterized by its designated SOC operating interval \([SOC_{n,min}, SOC_{n,max}]\), its rated energy capacity \( E_{n}^{rated} \) (kWh), and its rated power \( P_{n}^{rated} \) (kW). The fundamental hypothesis is that the capacity degradation rate for a cell energy storage system unit is a function of its specific SOC cycle interval. The optimization problem is defined over a scheduling horizon \( K \), discretized into time intervals \( k \) (e.g., one hour). The goal is to determine, for each time interval, the optimal power flows between all system components to satisfy the load while minimizing the total annual cost.
The total cost \( C_{total} \) is the sum of the cell energy storage system degradation cost \( C_{bat} \) and the net operational cost \( C_{oper} \).
$$ \min C_{total} = C_{bat} + C_{oper} $$
The degradation cost monetizes the capacity loss of each battery unit over the scheduling period. The operational cost includes routine maintenance costs for the cell energy storage system and the net cost of electricity exchanged with the grid (purchase cost minus sale revenue).
2. Mathematical Optimization Model
2.1 Degradation Cost Model
The degradation cost for the multi-cell energy storage system is calculated based on the capacity fade experienced by each unit. Let \( H_{n,k} \) represent the State-of-Health (SOH) of unit \( n \) at time \( k \), where \( H=1 \) denotes a brand-new cell and \( H=0.8 \) typically indicates the end-of-life (EOL) threshold. The degradation cost for unit \( n \) is proportional to its lost capacity relative to its total usable capacity over its lifetime, multiplied by its unit capital cost.
$$ C_{bat} = \sum_{n \in N} \frac{1 – H_{n,K}}{H_{n,init} – H_{n,EOL}} \cdot E_{n}^{rated} \cdot \mu_{n}^{cap} $$
Here, \( H_{n,init} \) is the initial SOH (typically 1.0), \( H_{n,EOL} \) is the SOH at end-of-life (e.g., 0.8), \( E_{n}^{rated} \) is the rated capacity, and \( \mu_{n}^{cap} \) is the capital cost per kWh of usable capacity.
The capacity fade model is crucial. The degradation rate \( D_{n,k} \) (capacity loss fraction per cycle) for unit \( n \) at cycle \( k \) is modeled using a power-law relationship dependent on the equivalent full cycle count \( x_{n,k} \) and parameters specific to its SOC interval:
$$ D_{n,k} = A_n \cdot \left( \frac{x_{n,k}}{100} \right)^{b_n} $$
The equivalent cycle count \( x_{n,k} \) accounts for the partial charging/discharging activity:
$$ x_{n,k} = \frac{\eta_{n}^{c} P_{n,k}^{SB} + P_{n,k}^{BD} / \eta_{n}^{d}}{2 E_{n}^{rated}} \cdot t_k $$
where \( \eta_{n}^{c}, \eta_{n}^{d} \) are charge/discharge efficiencies, \( P_{n,k}^{SB} \) is power from PV to storage, \( P_{n,k}^{BD} \) is power from storage to demand, and \( t_k \) is the duration of interval \( k \). The factor of 2 in the denominator converts the energy throughput to an equivalent full cycle depth.
The coefficient \( A_n \) is not a constant but a function of the average SOC \( \overline{SOC}_n \) and the SOC range \( \Delta SOC_n \) for the unit’s operating window:
$$ A_n = A_{1,n} \overline{SOC}_n (1 + A_{2,n} \Delta SOC_n + A_{3,n} (\Delta SOC_n)^2) $$
Parameters \( A_{1,n}, A_{2,n}, A_{3,n}, b_n \) are obtained by fitting the model to experimental cycle life data for batteries cycled within the specific SOC interval corresponding to unit \( n \). The SOH update is then:
$$ H_{n,k} = 1 – \sum_{i=1}^{k} D_{n,i} $$
with the constraint \( 0.8 \leq H_{n,k} \leq 1.0 \).
2.2 Operational Cost and System Constraints
The operational cost \( C_{oper} \) includes a fixed maintenance cost proportional to the installed capacity and the net cost of grid electricity exchange.
$$ C_{oper} = \sum_{n \in N} f_n^{inv} E_{n}^{rated} \mu_{n}^{cap} – \sum_{k \in K} \mu^{sell} E_{k}^{OUT} $$
Here, \( f_n^{inv} \) is an annualized maintenance cost coefficient, \( \mu^{sell} \) is the price for selling electricity to the grid, and \( E_{k}^{OUT} \) is the energy sold in interval \( k \). The cost of purchasing electricity from the grid, \( \sum \mu^{buy} E_{k}^{IN} \), would be added if grid purchases were part of the strategy.
The model is governed by the following key constraints:
1. Power Balance at the PV Source:
$$ P_{k}^{S} t_k = P_{k}^{SD} t_k + \sum_{n \in N} P_{n,k}^{SB} t_k + E_{k}^{OUT} \quad \forall k $$
2. Power Balance at the Load:
$$ P_{k}^{D} t_k = P_{k}^{SD} t_k + \sum_{n \in N} P_{n,k}^{BD} t_k \quad \forall k $$
3. Cell Energy Storage System Dynamics:
$$ E_{n,k}^{B} = E_{n,k-1}^{B} + \left( \eta_{n}^{c} P_{n,k}^{SB} – \frac{P_{n,k}^{BD}}{\eta_{n}^{d}} \right) t_k \quad \forall n, k $$
$$ E_{n,0}^{B} = E_{n,K}^{B} \quad \forall n $$
The final constraint ensures the cell energy storage system returns to its initial energy level over the scheduling cycle (e.g., 24 hours).
4. SOC Operating Limits:
$$ SOC_{n,min} \leq SOC_{n,k} = \frac{E_{n,k}^{B}}{E_{n}^{rated}} \leq SOC_{n,max} \quad \forall n, k $$
5. Charge/Discharge Logic and Power Limits:
To prevent simultaneous charging and discharging, binary variable \( z_{k}^{c} \) is introduced.
$$ 0 \leq \sum_{n \in N} P_{n,k}^{SB} \leq z_{k}^{c} P_{n}^{rated} \quad \forall k $$
$$ 0 \leq \sum_{n \in N} P_{n,k}^{BD} \leq (1 – z_{k}^{c}) P_{n}^{rated} \quad \forall k $$
$$ z_{k}^{c} \in \{0,1\} $$
Additionally, charge/discharge power is limited by the unit’s power rating and its usable energy window:
$$ P_{n,k}^{SB} t_k \leq \min \left( P_{n}^{rated}, (SOC_{n,max} – SOC_{n,min}) E_{n}^{rated} \right) $$
$$ P_{n,k}^{BD} t_k \leq \min \left( P_{n}^{rated}, (SOC_{n,max} – SOC_{n,min}) E_{n}^{rated} \right) $$
This set of equations forms an MINLP model, where the decision variables include all power flows, energy states, the binary charge/discharge indicators, and the degradation states. The solution provides an hourly schedule that minimizes total cost while respecting the physical and degradation-related constraints of the multi-cell energy storage system.
3. Case Study: PV-Battery Hybrid System Analysis
A practical case study is designed to evaluate the proposed model. The system consists of a PV array, a residential load profile, grid connection, and the cell energy storage system. Typical diurnal profiles for PV generation and load demand are used. The cell energy storage system is configured in two primary scenarios for comparison:
- Scenario B-T (Baseline): A single battery unit operating across the full SOC range [0, 1.0].
- Scenario B-LMH (Proposed): A multi-cell energy storage system with three distinct units: Unit L (Low: [0, 0.2]), Unit M (Medium: [0.2, 0.6]), Unit H (High: [0.6, 1.0]).
To ensure a fair comparison, the total usable energy capacity is kept identical across scenarios. Each unit in B-LMH has a rated capacity of 500 kWh. For B-T, a single unit with appropriate capacity is used. All batteries are assumed to be Lithium-ion NMC type with charge/discharge efficiency \( \eta^c = \eta^d = 0.95 \). The unit capital cost \( \mu^{cap} \) is set at 1595.6 $/kWh. The maintenance coefficient \( f^{inv} \) is 0.01 per year. The grid sell price \( \mu^{sell} \) is based on a typical feed-in tariff. The model is implemented and solved for a one-year operational period, discretized into hourly intervals, using the GAMS platform with the SCIP solver.
3.1 Degradation Parameters from Empirical Data
The degradation model parameters \( A_{1,n}, A_{2,n}, A_{3,n}, b_n \) are derived by fitting the power-law model to experimental cycle-life data for batteries cycled within specific SOC windows. The table below summarizes the fitted parameters for the four SOC interval types.
| Cell Unit | SOC Interval | Avg. SOC \( \overline{SOC}_n \) | SOC Range \( \Delta SOC_n \) | \( A_{1,n} \) | \( A_{2,n} \) | \( A_{3,n} \) | \( b_n \) | R² |
|---|---|---|---|---|---|---|---|---|
| L | [0, 0.2] | 0.1 | 0.2 | 4.11858 | 10.84936 | 50.24678 | 0.82851 | 0.98 |
| M | [0.2, 0.6] | 0.4 | 0.4 | 1.80043 | 3.60227 | 7.51058 | 1.02531 | 0.99 |
| H | [0.6, 1.0] | 0.8 | 0.4 | 1.21842 | 1.60217 | 2.50542 | 1.49401 | 0.96 |
| T | [0, 1.0] | 0.5 | 1.0 | 3.45210 | 0.85432 | 0.00000 | 1.88975 | 0.97 |
The parameters clearly show the dependency of degradation on the SOC window. The high interval (H) has a larger exponent \( b_n \), indicating accelerated fade after more cycles, while the full-interval (T) model shows the most aggressive overall degradation characteristic.
3.2 Optimization Results and Economic Comparison
The optimized annual costs for the two main scenarios are presented below. The results highlight the fundamental trade-off between degradation expense and operational/maintenance cost.
| Scenario | Degradation Cost (\( C_{bat} \)) | Operational Cost (\( C_{oper} \)) | Total Cost (\( C_{total} \)) | System Capacity Fade |
|---|---|---|---|---|
| B-T (Full Interval) | $381,500 | $7,932 | $389,432 | 8.8% |
| B-LMH (Partitioned) | $281,312 | $24,592 | $305,903 | L:1.6%, M:3.8%, H:1.6% |
The results are striking. The proposed multi-cell energy storage system (B-LMH) reduces the total annual cost by **21.45%** compared to the conventional single-unit system (B-T). This saving is primarily driven by a drastic reduction in degradation cost (26.3% lower), which outweighs the increase in operational cost. The B-T system suffers from severe capacity fade (8.8%), making its degradation cost dominate the total expense (97.96%). In contrast, the partitioned system effectively contains fade within each unit, especially protecting the High and Low interval units. The operational cost for B-LMH is higher because maintaining three separate units with tighter SOC constraints incurs slightly higher cycling losses and requires more sophisticated scheduling, but this is a worthwhile trade-off.
3.3 Dispatch Strategy and SOC Evolution
The optimization algorithm develops distinct dispatch strategies for the two cell energy storage system configurations. In the B-LMH system, the scheduler preferentially uses the **M (Medium)** unit for the bulk of energy transactions, as it offers a favorable compromise between usable energy range (\( \Delta SOC = 0.4 \)) and moderate degradation rate. The **L (Low)** and **H (High)** units are used more sparingly, primarily for fine-tuning and during periods of very high or low residual load. The SOC trajectories confirm that each unit stays rigorously within its designated bounds: Unit L cycles between 0 and 0.2, Unit M between 0.2 and 0.6, and Unit H between 0.6 and 0.84 (its upper limit was set below 1.0 for longevity).
In the B-T system, the single battery undergoes deep cycles between approximately 0.05 and 0.98 SOC. This wide swing maximizes its immediate utility but triggers the severe degradation characterized by the full-interval parameters. The cumulative degradation effect over one year is significantly larger, validating the core premise that SOC window management is critical for cell energy storage system economics.
3.4 Sensitivity Analysis: Impact of SOC Interval Composition
To further generalize the findings, the cost performance of various other multi-cell energy storage system compositions was analyzed, keeping total usable capacity constant. The results are summarized below.
| Cell Energy Storage System Composition | Total Cost (\( C_{total} \)) | Dominant Cost Component |
|---|---|---|
| Single Unit: B-T (Full [0,1.0]) | $389,432 | Degradation (98%) |
| Single Unit: B-L (Low [0,0.2]) | $319,961 | Degradation (88%) |
| Single Unit: B-M (Medium [0.2,0.6]) | $338,585 | Degradation (94%) |
| Single Unit: B-H (High [0.6,1.0]) | $286,356 | Degradation (92%) |
| Dual Units: B-LM ([0,0.2] & [0.2,0.6]) | $326,291 | Degradation (92%) |
| Dual Units: B-LH ([0,0.2] & [0.6,1.0]) | $287,719 | Degradation (90%) |
| Dual Units: B-MH ([0.2,0.6] & [0.6,1.0]) | $307,883 | Degradation (93%) |
| Triple Units: B-LMH (Proposed) | $305,903 | Degradation (92%) |
The analysis reveals several key insights. First, any partitioning of the SOC range leads to lower total cost than the full-interval B-T system. Second, among single-partition systems, B-H (high interval only) performs best in this specific yearly simulation because its high average SOC correlates with a lower early-life degradation rate for the given number of equivalent cycles. Third, the composition of the multi-cell energy storage system matters: the B-LH dual-unit system achieves the lowest cost among all configurations tested, even lower than the triple-unit B-LMH. This indicates that the optimal number and specification of partitions are not simply “more is better,” but are the result of a nuanced trade-off. Adding a third unit (M) in B-LMH increases operational flexibility but also adds maintenance cost; the optimizer finds that for this particular load/PV profile, the benefit does not fully offset the added expense compared to B-LH. The optimal architecture is therefore scenario-dependent, determined by the specific interplay of degradation curves, cost coefficients, and renewable generation/load patterns.
4. Conclusion and Implications
This study establishes a rigorous operational optimization framework for hybrid power systems employing a multi-cell energy storage system, founded on the principle of SOC interval partitioning. By constraining different battery units to specific SOC windows, the system can strategically leverage the distinct degradation characteristics of each window. The formulated MINLP model successfully coordinates power flows to minimize total cost, explicitly accounting for capacity fade.
The case study demonstrates compelling economic advantages. A partitioned three-unit cell energy storage system reduced total annualized cost by over 21% compared to a traditional single-battery system. The primary mechanism is the drastic suppression of capacity degradation, which is the dominant cost driver in conventional systems. The reduction in degradation cost far outweighs the associated increase in operational complexity and maintenance. Further analysis showed that the optimal multi-cell energy storage system configuration—the number and SOC ranges of the constituent units—is not universal but must be tailored to the specific application context, balancing degradation costs against operational expenses.
This work underscores the importance of incorporating physics-based, SOC-dependent degradation models into the planning and operational scheduling of cell energy storage systems. The proposed multi-cell energy storage system paradigm offers a practical pathway to significantly extend battery service life and improve the levelized cost of storage, enhancing the economic viability of renewable-integrated power systems. Future work could explore dynamic reconfiguration of SOC intervals and integrate real-time degradation estimation for adaptive management of the cell energy storage system.