In this work, I aim to explore the influence of different state of charge (SOC) intervals on the capacity degradation and power scheduling of battery energy storage systems. I formulate an operational optimization model for a multi-battery energy storage system based on SOC interval division, with the objective of minimizing the total cost of a hybrid power supply system. The model is a mixed-integer nonlinear programming (MINLP) problem. By solving this model, the optimal operation strategy for the hybrid system is obtained. I demonstrate the effectiveness of the proposed approach using a case study of a photovoltaic (PV) and battery hybrid power system. The results show that: (1) Compared with full-interval operation, the multi-battery energy storage system with divided SOC intervals significantly reduces the total cost of the hybrid system—specifically, the total cost is 21.45% lower than that of a full-interval battery storage system. (2) The number of SOC operation intervals is not simply maximized; it must be determined by a trade-off between operation and maintenance costs and capacity degradation costs for specific application scenarios.
1. Introduction
In power supply systems, energy storage systems serve as a bridge between generation and load, effectively enabling demand-side management, peak shaving, valley filling, and improving the utilization of renewable energy sources such as wind and solar power. Among various storage technologies, electrochemical batteries have become a research hotspot. However, during the use of battery energy storage systems (BESS), capacity degradation is inevitable and significantly affects performance and operation. The degradation process is influenced not only by charge/discharge cycles, depth of discharge, and temperature but also by the SOC range in which the battery operates. Many studies have shown that batteries operating at high or low SOC levels, high temperatures, high rates, and deep depths of discharge degrade faster. For instance, it has been reported that the capacity fade rate of lithium-ion batteries in the SOC interval 70%–100% is much higher than that in 0%–30% or 40%–70% intervals. The full SOC interval (0%–100%) exhibits a degradation rate approximately 29.79 times that of partial intervals. This indicates that the SOC operation interval has a substantial impact on battery aging.
To mitigate capacity degradation, one can limit the SOC operation range, but doing so may require a larger battery capacity to meet the same energy demand. Therefore, we can treat energy storage units operating in different SOC intervals as separate types of batteries. This perspective transforms the scheduling optimization problem into a multi-type battery energy storage system optimization problem. In this paper, I propose an operational optimization model based on SOC interval division to investigate the effects of SOC intervals on power scheduling and economic performance of the hybrid power supply system.
2. Problem Statement
As shown in the conceptual diagram (Figure 1), the hybrid power supply system consists of a photovoltaic (PV) system, a battery energy storage system, users, and the grid. The PV generated power can be supplied to users, stored in the battery energy storage system, or sold to the grid. The battery energy storage system is composed of n battery units, each operating in a different SOC interval.
The operational optimization problem is described as follows: (1) The PV power generation curve is known; (2) The user load curve is known; (3) The composition and capacity configuration of the battery energy storage system are known; (4) Other operating parameters and economic parameters (e.g., charge/discharge efficiency, electricity price) are known. The objective is to minimize the total cost of the hybrid system while accounting for battery capacity degradation characteristics and different SOC interval features.
I define the set of battery units as N = {n | n is a battery unit with a different SOC interval, n = 1, 2, …, N} and the set of time intervals as K = {k | k is a time interval, k = 1, 2, …, K}.
3. Model Formulation
3.1 Objective Function
The objective is to minimize the total cost of the hybrid power supply system:
$$ \min C_{total} = C_{bat} + C_{oper} $$
where:
- Cbat is the battery loss cost (a hidden cost caused by degradation):
$$ C_{bat} = \sum_{n \in N} \frac{1 – H_{n,K}}{H_{n,ini} – H_{n,end}} E_{n,rated} \mu_{n,b} $$
- Coper is the operation and maintenance cost (operating cost minus electricity revenue):
$$ C_{oper} = \sum_{n \in N} f_{n,inv} E_{n,rated} \mu_{n,b} – \sum_{k \in K} \mu_{sel} E_{k,OUT} $$
Here, Hn,K is the health state of storage unit n at the end of time interval K; Hn,ini and Hn,end are the initial and end-of-life health states; En,rated is the rated capacity; μn,b is the unit price of available capacity; fn,inv is the operation cost coefficient; μsel is the grid purchase price; Ek,OUT is the amount of electricity sold to the grid at time interval k.
3.2 Constraints
3.2.1 Supply and Demand Balance
Power balance for PV system:
$$ P_{k,S} t_k = P_{k,SD} t_k + \sum_{n \in N} P_{n,k,SB} t_k + E_{k,OUT} $$
Load demand satisfaction:
$$ P_{k,D} t_k = P_{k,SD} t_k + \sum_{n \in N} P_{n,k,BD} t_k $$
where Pk,S is PV power generation, Pk,SD is power supplied to load directly from PV, Pn,k,SB is power charged to battery unit n, Pk,D is load demand, Pn,k,BD is power discharged from battery unit n to load, and tk is the duration of interval k.
3.2.2 Battery Energy Storage System
Energy balance for each battery unit:
$$ E_{n,k,B} = E_{n,k-1,B} + \eta_{n,c} P_{n,k,SB} – \frac{P_{n,k,BD}}{\eta_{n,d}} $$
Energy cyclic condition:
$$ E_{n,0} = E_{n,K} $$
Capacity degradation model based on SOC interval division:
$$ D_{n,k} = A_n \left( \frac{x_{n,k}}{100} \right)^{b_n} $$
Equivalent cycle number:
$$ x_{n,k} = \frac{\eta_{n,c} P_{n,k,SB} + \frac{P_{n,k,BD}}{\eta_{n,d}}}{2 E_{n,rated}} t_k $$
Pre-exponential factor An as function of average SOC and SOC range:
$$ A_n = A_{1,n} S_{n,m} (1 + A_{2,n} \Delta S_n + A_{3,n} \Delta S_n^2) $$
Substituting gives:
$$ D_{n,k} = A_{1,n} S_{n,m} (1 + A_{2,n} \Delta S_n + A_{3,n} \Delta S_n^2) \left( \frac{x_{n,k}}{100} \right)^{b_n} $$
Health state constraint:
$$ H_{n,k} = 1 – D_{n,k} $$
$$ 0.8 \leq H_{n,k} \leq 1.0 $$
SOC limits:
$$ S_{n,min} \leq S_{n,k} \leq S_{n,max} $$
$$ S_{n,k} = \frac{E_{n,k,B}}{E_{n,rated}} $$
Charge/discharge status constraints (binary variable):
$$ 0 \leq \sum_{n \in N} P_{n,k,SB} \leq z_{k,c} P_{n,rated} $$
$$ 0 \leq \sum_{n \in N} P_{n,k,BD} \leq (1 – z_{k,c}) P_{n,rated} $$
Power limits:
$$ P_{n,k,SB} t_k \leq \min \left[ P_{n,rated}, (S_{n,max} – S_{n,min}) E_{n,rated} \right] $$
$$ P_{n,k,BD} t_k \leq \min \left[ P_{n,rated}, (S_{n,max} – S_{n,min}) E_{n,rated} \right] $$
The overall model is a MINLP with variables.
4. Case Study
4.1 Basic Data
I use a PV-battery hybrid power supply system as a case study. The PV generation and load power curves are shown in Figure 2 (not included here). The battery energy storage system consists of lithium-ion batteries (ternary manganese oxide composite) of the same batch. Based on SOC intervals, the battery units are classified into four types: Low (L) [0, 0.2], Medium (M) [0.2, 0.6], High (H) [0.6, 1.0], and Full (T) [0, 1.0]. The charge/discharge efficiency is 0.95, rated power is 1/3 of rated capacity, unit capacity cost is 1595.6 CNY/(kWh). Operation cost coefficient is 0.01. The study period is one year, with hourly time intervals.
The battery cycling data for different SOC intervals are given in Table 1 (based on literature).
| Cycle Number | Remaining Capacity (%) – L | Remaining Capacity (%) – M | Remaining Capacity (%) – H | Remaining Capacity (%) – T |
|---|---|---|---|---|
| 0 | 100.00 | 100.00 | 100.00 | 100.00 |
| 100 | 98.73 | 97.01 | 95.02 | 96.16 |
| 200 | 95.86 | 93.98 | 92.94 | 92.32 |
| 300 | 94.36 | 92.02 | 89.30 | 89.63 |
| 400 | 93.05 | 89.87 | 86.16 | 84.78 |
| 500 | 91.92 | 87.07 | 80.48 | 71.82 |
| 600 | 90.87 | 82.66 | 68.72 | 50.23 |
Using these data, I fitted the model parameters for each SOC interval. The results are shown in Table 2.
| Battery Unit | Average SOC Sn,m | SOC Range ΔSn | A1,n | A2,n | A3,n | bn | R2 |
|---|---|---|---|---|---|---|---|
| L | 0.1 | 0.2 | 4.11858 | 10.84936 | 50.24678 | 0.82851 | 0.98 |
| M | 0.4 | 0.4 | 1.80043 | 3.60227 | 7.51058 | 1.02531 | 0.99 |
| H | 0.8 | 0.4 | 1.21842 | 1.60217 | 2.50542 | 1.49401 | 0.96 |
All R2 values are above 0.95, indicating good fit.
4.2 Results and Discussion
I compare two configurations: B-LMH (three units operating in L, M, H intervals) and B-T (single unit operating in full interval T). Both total capacities are set to 500 kWh each. The optimization results are summarized in Table 3.
| Battery Storage System | Capacity Degradation (%) | Battery Loss Cost (CNY) | O&M Cost (CNY) | Total Cost (CNY) |
|---|---|---|---|---|
| B-LMH | 1.6 / 3.8 / 1.6 | 281311.5 | 24591.5 | 305902.9 |
| B-T | 8.8 | 381500.0 | 7932.2 | 389432.2 |
The total cost of the hybrid system with B-T is 389432.2 CNY, while with B-LMH it is 305902.9 CNY—a reduction of 21.45%. The battery loss cost dominates the total cost in both cases. For B-T, battery loss accounts for 97.96% of total cost; for B-LMH, it accounts for 91.96%. The multi-battery system sacrifices some O&M cost (higher than B-T) but significantly reduces battery degradation, resulting in lower overall cost.
The power scheduling and SOC profiles of B-LMH are shown in Figure 3 (not shown). The medium-interval energy storage cell (M) is used most heavily due to its favorable balance between capacity range and degradation. The degradation rate curves (Figure 5) show that after one year, the M and H intervals exhibit similar degradation rates, while L is slightly lower. The degradation of the full-interval cell is much higher.
4.3 Economic Impact of Different SOC Interval Compositions
To further investigate the effect of SOC interval composition, I compare various single-interval and multi-interval configurations. All have the same total capacity. Table 4 presents the results.
| BESS Configuration | Total Cost (CNY) | Battery Loss Cost (CNY) | O&M Cost (CNY) | Degradation per Unit (%) |
|---|---|---|---|---|
| B-T | 389432.2 | 381500.8 | 7932.2 | 8.8 |
| B-L | 319960.9 | 280924.0 | 39036.9 | 1.4 |
| B-M | 338584.7 | 318547.7 | 20037.0 | 3.2 |
| B-H | 286356.0 | 264432.8 | 156159.0 | 2.7 |
| B-LM | 326291.1 | 299916.1 | 26375.0 | 1.6/2.9 |
| B-LH | 287719.3 | 260085.7 | 27633.7 | 1.6/2.3 |
| B-MH | 307883.1 | 286903.0 | 20980.1 | 3.8/1.9 |
| B-LMH | 305902.9 | 281311.5 | 24591.5 | 1.6/3.8/1.6 |
Key observations:
- The full-interval T system always yields the highest total cost due to severe degradation.
- Among single-interval systems, B-H has the lowest total cost, followed by B-L and B-M. This is because at the end of one year, the H-interval energy storage cell undergoes fewer equivalent cycles (only about 121 cycles) and its degradation is in the initial stage, where its degradation rate is lower than M and L. The O&M cost of H is also lower due to its higher rated capacity (same as M) but less degradation cost.
- Among two-interval combinations, B-LH achieves the lowest total cost, followed by B-MH, then B-LM. The presence of H-interval cells tends to reduce total cost because they provide high capacity with relatively low degradation early on.
- The three-interval B-LMH system lies in the middle, indicating that more intervals do not necessarily lead to lower cost. The optimal number of SOC intervals depends on the trade-off between the cost of added operation & maintenance and the benefit of reduced degradation.
5. Conclusion
In this work, I have proposed an operational optimization method for a multi-battery energy storage system based on SOC interval division. By considering the distinct capacity degradation characteristics of energy storage cells operating in different SOC ranges, a mixed-integer nonlinear programming model was developed to minimize the total cost of a hybrid PV-battery system. The case study yielded the following conclusions:
- Regardless of the combination, any multi-battery system composed of energy storage cells with divided SOC intervals achieves a lower total cost than a system using a single full-interval energy storage cell.
- The charge/discharge frequencies among different SOC-interval cells are unbalanced. The optimizer prioritizes energy storage cells with lower degradation rates and lower capacity-related operating costs.
- The number of SOC operation intervals is not simply maximized; it is determined by the site-specific trade-off between operation and maintenance costs and capacity degradation costs. The proposed model effectively extends battery lifespan and reduces the annual total cost of the hybrid power supply system.