In recent years, the integration of renewable energy sources into power systems has gained significant attention due to environmental concerns and the need for sustainable development. Energy storage systems, particularly those based on electrochemical technologies, play a crucial role in balancing supply and demand, smoothing load profiles, and enhancing grid stability. Among these, battery energy storage systems (BESS) have emerged as a promising solution, but their performance and economic viability are heavily influenced by capacity degradation over time. The state of charge (SOC) operating range is a critical factor affecting the degradation characteristics of energy storage cells. This study focuses on optimizing the operation of a multi-battery energy storage system by dividing SOC intervals to minimize total costs in a hybrid power supply system. The model developed here considers the distinct capacity degradation behaviors of energy storage cells across different SOC ranges, formulating the problem as a mixed-integer nonlinear programming (MINLP) model. Through a case study of a photovoltaic (PV)-battery hybrid system, I demonstrate the effectiveness of this approach in reducing costs and prolonging the lifespan of energy storage cells.
The hybrid power supply system comprises a PV generation system, multiple energy storage cells, user loads, and the grid. The PV system generates electricity that can be directly supplied to users, stored in the energy storage cells, or sold to the grid. The battery energy storage system consists of multiple energy storage cells, each operating within a specific SOC interval, such as low (L), medium (M), high (H), or full (T) ranges. By partitioning the SOC operating ranges, I aim to leverage the varying degradation rates of energy storage cells to achieve economic benefits. The optimization objective is to minimize the total cost, which includes battery degradation costs and operational maintenance costs. The constraints encompass power balance, battery dynamics, capacity degradation models, and operational limits for the energy storage cells.
To formalize the problem, I define sets for the energy storage cells and time intervals. Let $N = \{ n \mid n = 1, 2, \dots, N \}$ represent the set of energy storage cells with different SOC intervals, and $K = \{ k \mid k = 1, 2, \dots, K \}$ denote the set of time intervals. The objective function is formulated as follows:
$$ \min C_{\text{total}} = C_{\text{bat}} + C_{\text{oper}} $$
where $C_{\text{bat}}$ is the battery degradation cost, and $C_{\text{oper}}$ is the operational maintenance cost. The battery degradation cost accounts for the capacity loss of energy storage cells over time and is expressed as:
$$ C_{\text{bat}} = \sum_{n \in N} \frac{1 – H_{n,K}}{H_{n,\text{ini}} – H_{n,\text{end}}} E_{n,\text{rated}} \mu_{n,b} $$
Here, $H_{n,K}$ is the health state of energy storage cell $n$ at the end of the time horizon, $H_{n,\text{ini}}$ and $H_{n,\text{end}}$ are the initial and end-of-life health states, $E_{n,\text{rated}}$ is the rated capacity, and $\mu_{n,b}$ is the unit capacity cost. The operational maintenance cost includes operational expenses and revenue from selling electricity:
$$ C_{\text{oper}} = \sum_{n \in N} f_{n,\text{inv}} E_{n,\text{rated}} \mu_{n,b} – \sum_{k \in K} \mu_{\text{sel}} E_{k,\text{OUT}} $$
where $f_{n,\text{inv}}$ is a coefficient for operational costs, $\mu_{\text{sel}}$ is the electricity selling price, and $E_{k,\text{OUT}}$ is the electricity sold to the grid in time interval $k$.
The constraints ensure the physical and operational feasibility of the system. The power balance for the PV system is given by:
$$ P_{k,S} t_k = P_{k,SD} t_k + \sum_{n \in N} P_{n,k,SB} t_k + E_{k,\text{OUT}} $$
where $P_{k,S}$ is the PV power output, $P_{k,SD}$ is the power supplied to users, $P_{n,k,SB}$ is the power charged to energy storage cell $n$, and $t_k$ is the duration of time interval $k$. The user demand balance is:
$$ P_{k,D} t_k = P_{k,SD} t_k + \sum_{n \in N} P_{n,k,BD} t_k $$
with $P_{k,D}$ being the user demand and $P_{n,k,BD}$ the power discharged from energy storage cell $n$ to users.
For the energy storage cells, the state of charge dynamics are governed by:
$$ E_{n,k,B} = E_{n,k-1,B} + \eta_{n,c} P_{n,k,SB} – \frac{P_{n,k,BD}}{\eta_{n,d}} $$
where $E_{n,k,B}$ is the energy stored, $\eta_{n,c}$ and $\eta_{n,d}$ are charging and discharging efficiencies. The SOC is defined as $S_{n,k} = E_{n,k,B} / E_{n,\text{rated}}$, and must satisfy $S_{n,\text{min}} \leq S_{n,k} \leq S_{n,\text{max}}$. The capacity degradation model for energy storage cells is based on a power-law relationship:
$$ D_{n,k} = A_n \left( \frac{x_{n,k}}{100} \right)^{b_n} $$
where $D_{n,k}$ is the degradation rate, $x_{n,k}$ is the equivalent cycle count, and $A_n$ and $b_n$ are parameters. The equivalent cycle count is calculated as:
$$ x_{n,k} = \frac{ \eta_{n,c} P_{n,k,SB} + \frac{P_{n,k,BD}}{\eta_{n,d}} }{2 E_{n,\text{rated}} } t_{n,k} $$
The parameter $A_n$ depends on the SOC interval and is expressed as:
$$ A_n = A_{1,n} S_{n,m} (1 + A_{2,n} \Delta S_n + A_{3,n} \Delta S_n^2) $$
where $S_{n,m}$ is the average SOC, $\Delta S_n$ is the SOC range, and $A_{1,n}$, $A_{2,n}$, $A_{3,n}$ are fitted parameters. The health state $H_{n,k} = 1 – D_{n,k}$ must remain between 0.8 and 1.0. Charging and discharging power constraints include:
$$ 0 \leq \sum_{n \in N} P_{n,k,SB} \leq z_{k,c} P_{n,\text{rated}} $$
$$ 0 \leq \sum_{n \in N} P_{n,k,BD} \leq (1 – z_{k,c}) P_{n,\text{rated}} $$
where $z_{k,c}$ is a binary variable indicating charging state. Additionally, power limits ensure that charging and discharging do not exceed the rated power or usable capacity:
$$ P_{n,k,SB} t_k \leq \min \left[ P_{n,\text{rated}}, (S_{n,\text{max}} – S_{n,\text{min}}) E_{n,\text{rated}} \right] $$
$$ P_{n,k,BD} t_k \leq \min \left[ P_{n,\text{rated}}, (S_{n,\text{max}} – S_{n,\text{min}}) E_{n,\text{rated}} \right] $$
This MINLP model is solved to obtain the optimal power scheduling strategy for the hybrid system.
To validate the model, I consider a case study of a PV-battery hybrid power supply system. The PV generation and user load profiles are based on typical daily curves, with time intervals of one hour over a year. The energy storage cells are lithium-ion batteries with different SOC intervals: low (L: 0-0.2), medium (M: 0.2-0.6), high (H: 0.6-1.0), and full (T: 0-1.0). Each energy storage cell has a rated capacity of 500 kWh, charging and discharging efficiencies of 0.95, and a unit capacity cost of 1595.6 currency units per kWh. The operational cost coefficient $f_{n,\text{inv}}$ is set to 0.01, and the electricity selling price is based on market rates. The capacity degradation parameters are derived from experimental data, as shown in the table below.
| Energy Storage Cell | Average SOC $S_{n,m}$ | SOC Range $\Delta S_n$ | $A_{1,n}$ | $A_{2,n}$ | $A_{3,n}$ | $b_n$ | 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 |
The model is implemented in GAMS using the SCIP solver, with a relative optimality gap of 5%. I compare two scenarios: a multi-battery energy storage system with L, M, and H cells (B-LMH) and a single-battery system with a T cell (B-T). The results indicate that the B-LMH system reduces the total cost by 21.45% compared to B-T, primarily due to lower battery degradation costs. The total cost for B-LMH is 305,902.9 currency units, with battery degradation costing 281,311.5 units and operational maintenance costing 24,591.5 units. In contrast, B-T incurs a total cost of 389,432.2 units, with degradation costs dominating at 381,500.0 units.
The power scheduling and SOC variations for the energy storage cells in B-LMH show that the M cell is utilized most intensively, followed by H and L cells. The equivalent cycle counts and degradation rates align with the scheduling patterns, emphasizing the trade-offs between degradation and operational costs. For instance, the degradation rate for M cells is higher than for L cells but lower than for H cells under certain cycle counts. The SOC trajectories remain within the specified limits, demonstrating effective utilization of each energy storage cell.
| Energy Storage System | Total Cost (units) | Battery Degradation Cost (units) | Operational Maintenance Cost (units) | Degradation Rate (%) |
|---|---|---|---|---|
| B-T | 389,432.2 | 381,500.8 | 7,932.2 | 8.8 |
| B-L | 319,960.9 | 280,924.0 | 39,036.9 | 1.4 |
| B-M | 338,584.7 | 318,547.7 | 20,037.0 | 3.2 |
| B-H | 286,356.0 | 264,432.8 | 156,159.0 | 2.7 |
| B-LM | 326,291.1 | 299,916.1 | 26,375.0 | 1.6/2.9 |
| B-LH | 287,719.3 | 260,085.7 | 27,633.7 | 1.6/2.3 |
| B-MH | 307,883.1 | 286,903.0 | 20,980.1 | 3.8/1.9 |
| B-LMH | 305,902.9 | 281,311.5 | 24,591.5 | 1.6/3.8/1.6 |
Further analysis of different SOC interval compositions reveals that systems with partitioned SOC intervals generally yield lower total costs than the full-interval system. For example, single-interval systems like B-H and B-L show cost reductions due to lower degradation, but multi-interval systems like B-LH achieve the best trade-off. The number of SOC intervals is not necessarily better when increased; it depends on the specific application and the balance between degradation and operational costs. This highlights the importance of optimizing the composition of energy storage cells based on their SOC-dependent degradation characteristics.
In conclusion, the proposed optimization model effectively minimizes the total cost of a hybrid power supply system by leveraging SOC interval division for energy storage cells. The results demonstrate that partitioned multi-battery energy storage systems outperform full-interval systems in terms of cost savings and lifespan extension. Future work could explore dynamic SOC interval adjustments and real-time scheduling to further enhance performance. The integration of advanced degradation models and machine learning techniques may also improve the accuracy of capacity forecasting for energy storage cells.