We propose a comprehensive framework for the optimal control of retired battery energy storage systems (BESS) by incorporating a multi-factor capacity degradation model. The problem of economic viability in different application scenarios is addressed through a novel cost-benefit analysis that accounts for battery aging caused by depth of discharge (DOD), state of charge (SOC), and calendar self-discharge. The lifetime of retired batteries is estimated using a semi-empirical capacity fade model combined with the rainflow counting algorithm. The battery energy storage system is applied to both the user side and the grid side. On the user side, the objective is to maximize total net revenue over the project period, including electricity price arbitrage and reduction of demand charges under time-of-use tariffs. On the grid side, the battery energy storage system is used for peak load shaving, with a multi-objective optimization balancing peak-to-valley load ratio and battery degradation cost. Case studies demonstrate that the proposed method effectively extends battery life while improving economic returns. Sensitivity analysis reveals that the optimal operating strategy depends on battery cost and electricity price differentials.
The rapid growth of electric vehicles has led to a massive volume of retired power batteries. When their capacity degrades to around 80% of initial value, they are no longer suitable for automotive use but can be repurposed in stationary battery energy storage systems. Such cascade utilization reduces resource waste and lowers the upfront cost of energy storage. However, accurate estimation of battery life and operational costs is critical for economic feasibility. Traditional methods often simplify capacity fade by considering only DOD, neglecting the influence of SOC and calendar aging. Moreover, many existing studies treat operation and maintenance costs as fixed percentages, ignoring the cost of capacity degradation during operation. To bridge this gap, we develop a refined model that integrates multiple degradation factors and introduces a capacity degradation cost into the operating expenses of battery energy storage systems.
The remaining paper is organized as follows. Section 2 presents the multi-factor capacity degradation model for retired batteries. Section 3 details the cost-benefit model for battery energy storage systems. Section 4 describes the user-side optimization framework, while Section 5 covers the grid-side multi-objective optimization. Simulation results and sensitivity analysis are provided in Section 6. Finally, conclusions are drawn in Section 7.
2. Capacity Degradation Model for Retired Batteries
Aging of lithium-ion batteries in battery energy storage systems occurs through two mechanisms: calendar aging and cycle aging. Calendar aging is caused by self-discharge and side reactions during idle periods, while cycle aging results from charge-discharge operations. Both are influenced by temperature, charge rate, DOD, and SOC. In stationary applications, temperature and charge rate can be controlled within narrow ranges, so we focus on the effects of DOD, average SOC, and time-dependent self-discharge. Based on the semi-empirical model proposed in [12], the capacity loss ratio \( L \) (fraction of initial capacity lost) for a battery is expressed as:
$$
L = (1 – \alpha) e^{- \beta N} + \alpha \left( 1 – e^{- f_d(\text{DOD}, \sigma, t)} \right)
$$
where \(\alpha\) accounts for the initial rapid capacity fade due to SEI formation, \(\beta\) is a factor influenced by operating conditions, \(N\) is the number of cycles, and \(f_d\) is a function capturing DOD, SOC level (\(\sigma\)), and time \(t\). For retired batteries starting at 80% capacity, the SEI formation phase is already complete, so we simplify the model to:
$$
L = L_0 + (1 – L_0) \left( 1 – e^{- f_d(\text{DOD}, \sigma, t)} \right)
$$
with \(L_0\) being the initial capacity loss (0.2 for 80% initial capacity). The function \(f_d\) is decomposed as:
$$
f_d(\text{DOD}, \sigma, t) = f_{d1}(\text{DOD}) + f_{d2}(\sigma) + f_{d3}(t)
$$
$$
f_{d1}(\text{DOD}) = k_{D1} \text{DOD}^{k_{D2}} + k_{D3}
$$
$$
f_{d2}(\sigma) = k_{\sigma} e^{ (\sigma – \sigma_{\text{ref}}) }
$$
$$
f_{d3}(t) = k_t t
$$
The parameters \(k_{D1}, k_{D2}, k_{D3}, k_{\sigma}, k_t\) are empirical constants determined from cycle aging tests. \(\sigma_{\text{ref}}\) is the reference SOC (typically 0.5). The rainflow counting algorithm is applied to extract equivalent cycles from an irregular SOC profile. Each cycle is characterized by its DOD (peak-to-valley difference) and average SOC. Substituting these values into the degradation model yields the accumulated capacity loss over a given period. The battery end-of-life is defined when total capacity loss reaches 40% (i.e., remaining capacity 60% of initial). Table I summarizes the key parameters used in our simulations.
| Parameter | Value |
|---|---|
| \(k_{D1}\) | 0.085 |
| \(k_{D2}\) | 1.2 |
| \(k_{D3}\) | 0.012 |
| \(k_{\sigma}\) | 0.015 |
| \(k_t\) (per day) | 5e-5 |
| \(\sigma_{\text{ref}}\) | 0.5 |
| Initial capacity loss \(L_0\) | 0.2 |
3. Cost-Benefit Model for Battery Energy Storage System
The total cost of a battery energy storage system consists of investment cost, operation and maintenance (O&M) cost, and capacity degradation cost. The investment cost includes the retired battery pack, power conversion system (PCS), and battery management system (BMS):
$$
C_{\text{inv}} = C_E E_{\text{BESS}} + C_P P_{\text{BESS}} + C_B E_{\text{BESS}}
$$
where \(C_E\) is the unit capacity cost of retired batteries (0.6 yuan/Wh), \(C_P\) is the unit power cost of PCS (0.2 yuan/W), and \(C_B\) is the unit capacity cost of BMS (0.1 yuan/Wh). \(E_{\text{BESS}}\) and \(P_{\text{BESS}}\) are the rated energy and power of the battery energy storage system. The O&M cost is proportional to the battery capacity and increases as the battery degrades. We use the empirical relation from [28]:
$$
K_M = 0.05165 \times (1 – L)^{-6}
$$
and the annual O&M cost is \(C_M = K_M \times E_{\text{BESS}}\). The capacity degradation cost reflects the loss of battery value due to aging during operation:
$$
C_O = \frac{C_E E_{\text{BESS}} \times \Delta L}{L_R}
$$
where \(\Delta L\) is the capacity loss incurred during a specific operation period, and \(L_R\) is the total allowable capacity loss from start to end of life (0.2 for retired batteries). This cost is subtracted from the revenue when evaluating net profit.
On the user side, the battery energy storage system generates revenue through two mechanisms: (1) price arbitrage by charging during low-price hours and discharging during peak-price hours, and (2) reducing the maximum demand charge under the two-part tariff for industrial users. The arbitrage revenue over the project lifetime \(\tau\) is:
$$
I_{\text{arb}} = \sum_{i=1}^{\tau} \sum_{d=1}^{365} \sum_{t=1}^T \left( 1 – \frac{L_{i,d}}{1-L_0} \right) \left( p_t P_{\text{dis}}(t) – p_t P_{\text{ch}}(t) \right) \Delta t
$$
where \(L_{i,d}\) is the cumulative capacity loss at day \(d\) of year \(i\), and \(p_t\) is the time-of-use electricity price. The demand charge reduction revenue is:
$$
I_{\text{peak}} = \sum_{i=1}^{\tau} \sum_{m=1}^{12} p_{\text{peak}} \Delta P_{\text{peak}, i,m}
$$
where \(\Delta P_{\text{peak},i,m}\) is the reduction in monthly peak load due to the battery energy storage system, and \(p_{\text{peak}}\) is the unit demand charge (51.2 yuan/kW). The objective on the user side is to maximize net profit:
$$
\max \, I = I_{\text{arb}} + I_{\text{peak}} – C_{\text{inv}} – \sum_{i=1}^{\tau} C_{M,i} – \sum_{i=1}^{\tau} C_{O,i}
$$
On the grid side, the battery energy storage system is used to shave the net load peak and fill the valley. The performance is measured by the peak-to-valley load ratio \(K = \max(P_L)/\min(P_L)\). The optimization aims to minimize both \(K\) and the cumulative capacity degradation cost \(C_O\) simultaneously, forming a multi-objective problem:
$$
\min \, (K, C_O)
$$
subject to SOC constraints, daily energy balance, and power limits:
$$
\text{SOC}_{\min} \leq \text{SOC}(t) \leq \text{SOC}_{\max}
$$
$$
\text{SOC}(t=0) = \text{SOC}(t=24)
$$
$$
0 \leq P_{\text{ch}}(t) \leq P_{\text{BESS}}, \quad 0 \leq P_{\text{dis}}(t) \leq P_{\text{BESS}}
$$
4. User-Side Optimization and Simulation
We simulate a user-side battery energy storage system with rating 1.5 MW / 6 MWh, SOC limits 0.1 and 0.9, and a project lifetime of 10 years. The load profile is a typical industrial day (data from [19]). Three scenarios are compared:
- Scenario 1: Ignore capacity degradation and its cost. Objective is to maximize net profit assuming no battery fade.
- Scenario 2: Consider capacity degradation but ignore degradation cost. Objective still maximizes net profit, but battery aging reduces effective capacity.
- Scenario 3: Consider both capacity degradation and degradation cost (full model).
The optimization is solved using particle swarm optimization (PSO) with 100 particles and 100 iterations. Decision variables are the 24-hour charging/discharging power. Table II summarizes the results.
| Item | Scenario 1 | Scenario 2 | Scenario 3 |
|---|---|---|---|
| Net profit (10^4 yuan) | 1491.29 | 1272.33 | 954.61 |
| Arbitrage revenue (10^4 yuan) | 1725.36 | 1394.16 | 1315.31 |
| Demand charge reduction (10^4 yuan) | 217.98 | 333.81 | 400.26 |
| Investment cost (10^4 yuan) | 450 | 450 | 450 |
| O&M cost (10^4 yuan) | 2.04 | 5.63 | 4.55 |
| Degradation cost (10^4 yuan) | 0 | 0 | 306.41 |
| Battery life (years) | 10 | 10 | 10 |
| Total capacity loss (%) | 0 | 19.73 | 17.02 |
Table III shows the equivalent cycles extracted from the optimal schedule of each scenario using the rainflow counting method.
| Scenario | Cycle | DOD | Average SOC |
|---|---|---|---|
| 1 | 1 | 0.0736 | 0.8632 |
| 2 | 0.1833 | 0.5194 | |
| 3 | 0.0192 | 0.1132 | |
| 4 | 0.8 | 0.5 | |
| 2 | 1 | 0.0534 | 0.5887 |
| 2 | 0.7994 | 0.5003 | |
| 3 | 1 | 0.8 | 0.5 |
In Scenario 1, ignoring degradation leads to excessive cycling (four cycles per day) and reduces the actual battery life to 8.32 years. If we recalculate the profit considering actual fade, the net profit drops to 961.08×10^4 yuan, a 35.55% decrease. Scenario 2, which accounts for degradation but not its cost, yields a net profit of 1272.33×10^4 yuan, but if the degradation cost were deducted, the real profit would be only 917.21×10^4 yuan. Scenario 3, using the full model, reduces daily cycles to one (0.8 DOD) and achieves a net profit of 954.61×10^4 yuan, which is higher than the adjusted profit of Scenario 2. This demonstrates that incorporating capacity degradation cost leads to a more conservative but ultimately more economical operation of the battery energy storage system.
Sensitivity analysis is performed on battery unit cost and peak-valley electricity price differential. The dynamic payback period is calculated using a discount rate of 6%. As shown in Figure 6 (not shown here), when the battery price drops to 0.3 yuan/Wh, the net profit of Scenario 2 increases to 1353.41×10^4 yuan with a payback period of 1.80 years, while Scenario 3 yields 1181.58×10^4 yuan and 1.83 years. Conversely, when the peak-valley differential decreases to 0.776 yuan/kWh, Scenario 2’s net profit falls to 1011.74×10^4 yuan (payback 4.36 years) and Scenario 3 to 681.15×10^4 yuan (payback 4.33 years). Interestingly, when battery costs are low and price differential is high, Scenario 2 (without degradation cost) can outperform Scenario 3 because the revenue from additional cycling outweighs the degradation cost. This highlights the need for adaptive operating strategies based on market conditions.
5. Grid-Side Multi-Objective Optimization
For the grid-side application, we consider a battery energy storage system with 0.8 MW / 1.6 MWh connected to a load profile with 2.4 MW photovoltaic generation. The original net load has a peak of 3.715 MW and valley of 0.6463 MW, yielding a ratio \(K=5.75\). Two scenarios are compared:
- Scenario A: Minimize only the peak-to-valley ratio \(K\) without considering degradation cost.
- Scenario B: Multi-objective optimization to minimize both \(K\) and degradation cost \(C_O\).
The multi-objective particle swarm optimization (MOPSO) is used, and the optimal trade-off is selected via TOPSIS with entropy weights. Table IV shows the results.
| Item | Scenario A | Scenario B |
|---|---|---|
| Peak-to-valley ratio \(K\) | 2.75 | 2.99 |
| Effective peak (MW) | 3.0648 | 3.1635 |
| Effective valley (MW) | 1.116 | 1.0587 |
| Battery life (years) | 8.35 | 10 |
| Total capacity loss (%) | 20 | 15.11 |
| Degradation cost (10^4 yuan) | 96 | 72.53 |
In Scenario A, the battery operates two cycles per day (DOD 0.5828 and 0.8), achieving a lower peak-to-valley ratio but exhausting the battery in 8.35 years with a degradation cost of 96×10^4 yuan. Scenario B reduces cycling to a single deep cycle (DOD 0.7225), resulting in a slightly higher ratio (2.99) but extending the battery life to 10 years and lowering degradation cost to 72.53×10^4 yuan. Thus, by sacrificing some load leveling performance, the battery energy storage system can achieve a longer operational lifetime and better economic efficiency.
6. Conclusion
We have presented a comprehensive optimal control framework for retired battery energy storage systems that incorporates multi-factor capacity degradation, including DOD, SOC, and calendar aging. The rainflow counting method allows accurate estimation of cycle-by-cycle degradation. By introducing a capacity degradation cost into the operation objective, the proposed model automatically balances revenue and battery wear. The user-side case study shows that ignoring degradation leads to overestimation of profitability, while considering it yields a more realistic and often more profitable operating schedule. Sensitivity analysis further reveals that when battery costs are low and price spreads are high, a more aggressive cycling strategy can be beneficial. For grid-side peak shaving, a multi-objective approach enables a trade-off between load leveling performance and battery longevity. The proposed method provides a valuable tool for the economic and sustainable deployment of retired batteries in stationary storage.