In recent years, the rapid advancement of energy storage technologies has fundamentally reshaped power grid architectures, evolving from traditional “generation-transmission-distribution-consumption” models to more integrated “generation-transmission-distribution-consumption-storage” systems. My research focuses on a critical application within this paradigm: the deployment of battery energy storage systems in user-side settings, particularly large industrial parks. These parks are characterized by high, regular electricity demand and stringent power quality requirements. The widespread implementation of two-part tariffs and time-of-use pricing makes reducing peak demand and shifting consumption away from high-price periods a primary avenue for cutting operational electricity costs. Herein lies the significant potential for battery energy storage systems. By providing rapid response capabilities, these systems can effectively perform peak shaving and valley filling, directly contributing to the economic and operational efficiency of industrial consumers.
However, the economic viability of deploying battery energy storage systems at scale remains a central challenge. The substantial initial investment and, crucially, the degradation of battery life during operation are major cost factors. In many existing economic dispatch models for industrial parks, the life loss of the battery energy storage system is often oversimplified or entirely neglected. Some approaches estimate costs based on theoretical calendar life, which can severely underestimate degradation under frequent charge-discharge cycles. Others assume a linear relationship between cost and depth of discharge, lacking the precision needed for accurate economic modeling. A more nuanced approach uses methods like rainflow counting to assess cycle life, but this often results in complex, non-linear objective functions that hinder computational speed for dispatch optimization. Therefore, my work aims to bridge this gap by developing a dispatch model that accurately accounts for battery life degradation costs while maintaining computational tractability for practical, day-ahead scheduling.
Furthermore, large-scale industrial applications typically require a battery energy storage system composed of multiple individual storage units or devices. In real-world operation, these units may exhibit heterogeneity due to differences in age, operating environment, manufacturing batch, or initial state of health. Treating the entire battery energy storage system as a single, monolithic entity in dispatch models overlooks this reality. It reduces scheduling flexibility and can lead to situations where individual units are subjected to stressful operating conditions—like deep discharges or frequent cycling—accelerating their degradation disproportionately. To address this, my model independently characterizes each device within the battery energy storage system. This granular approach allows for more precise control, enhances operational flexibility, and facilitates coordinated operation to extend the overall system’s useful life. Given the smaller scale of user-side installations compared to grid-side projects, modeling individual devices is computationally feasible and offers significant practical benefits.
The core of my proposed methodology involves three key innovations. First, I establish a battery life degradation cost model based on depth of discharge (DOD), which accurately captures the non-linear relationship between cycling stress and capacity fade. To ensure the model remains suitable for linear programming solvers and maintains fast solving speeds, I employ a piecewise linearization technique on this cost function. Second, I formulate the dispatch problem with independent decision variables for each unit in the multi-device battery energy storage system. Third, I introduce a State of Charge (SOC) health evaluation function into the overall dispatch objective. This secondary objective works in tandem with the primary economic goal to discourage excessive SOC fluctuations and promote balanced usage among devices, thereby indirectly prolonging battery life. The final model is cast as a Mixed-Integer Linear Programming (MILP) problem, which can be solved efficiently using commercial solvers.
Modeling Battery Life Degradation for Economic Dispatch
Accurately quantifying the daily operational cost attributable to battery wear is paramount for a realistic economic dispatch. The lifetime of a battery energy storage system, particularly lithium iron phosphate (LiFePO4) batteries common in stationary storage, is predominantly limited by cycle life rather than calendar life when used for frequent energy arbitrage. Cycle life is strongly dependent on the depth of discharge experienced during each cycle.
I adopt a semi-empirical model to relate the number of cycles to failure (N_cal) to the depth of discharge (DOD). Experimental data for a typical LiFePO4 battery shows the non-linear relationship summarized in the following table.
| Depth of Discharge (DOD) | Cycle Life (N_cal) |
|---|---|
| 0.4 | 7,200 |
| 0.6 | 6,000 |
| 0.8 | 5,200 |
| 1.0 | 4,700 |
This relationship can be fitted with a power-law function:
$$ N_{\text{cal}} = N_0 \cdot (\text{DOD})^{-a} $$
where \( N_0 \) is the cycle life at 100% DOD (4,700 cycles), and \( a \) is a fitted constant. For the data above, \( a \approx 0.5 \), yielding:
$$ N_{\text{cal}} = 4700 \cdot (\text{DOD})^{-0.5} $$
This clearly illustrates that battery life decreases in a non-linear fashion as the DOD increases.
In a dispatch context, the battery energy storage system may not undergo a single, full-depth cycle per day. It may experience multiple partial cycles. To account for this, I use the concept of equivalent full cycles. The wear incurred by a discharge event at a certain DOD is converted to an equivalent cycle count at 100% DOD. The daily wear accumulation, expressed as an equivalent cycle count factor \( E_{eq} \), is calculated as:
$$ E_{eq} = \sum_{t=1}^{T} \left| \frac{1}{(\text{DOD}_t)^{-a}} – \frac{1}{(\text{DOD}_{t-1})^{-a}} \right| $$
where \( \text{DOD}_t \) is the depth of discharge at time interval \( t \), and \( T \) is the total number of intervals in the scheduling horizon (e.g., 24 hours). This formulation effectively “counts” the incremental wear caused by changes in DOD throughout the day, focusing on discharge phases as the primary cause of degradation.
The anticipated operational life of the battery energy storage system in years (\( T_{\text{cal}} \)) is then:
$$ T_{\text{cal}} = \frac{N_0}{365 \cdot E_{eq}} $$
The daily amortized cost of the battery investment (\( C_b \)) must consider the capital recovery factor, not simply divide the initial cost by \( T_{\text{cal}} \). The formulation is:
$$ C_b = \left( C_{\text{bin}} \cdot \frac{r(1+r)^{T_{\text{cal}}}}{(1+r)^{T_{\text{cal}}} – 1} + C_{\text{bw}} \right) / 365 $$
where \( C_{\text{bin}} \) is the total initial investment cost for the battery energy storage system, \( r \) is the annual discount rate, and \( C_{\text{bw}} \) is the annual maintenance cost. The initial cost is calculated based on the rated power and energy capacity of each device \( n \) in the system:
$$ C_{\text{bin}} = \sum_{n=1}^{N} (C_{\bar{P}} \cdot \bar{P}_n + C_{\bar{E}} \cdot \bar{E}_n) $$
$$ C_{\text{bw}} = \sum_{n=1}^{N} (C_w \cdot \bar{P}_n) $$
Here, \( \bar{P}_n \) and \( \bar{E}_n \) are the rated power and capacity of device \( n \), \( C_{\bar{P}} \) and \( C_{\bar{E}} \) are per-unit power and energy costs, \( C_w \) is the annual maintenance cost per unit power, and \( N \) is the total number of devices.
The variable \( E_{eq} \), and consequently \( C_b \), is a non-linear function of the decision variables \( \text{DOD}_t \). To incorporate this into a linear optimization framework, I apply piecewise linearization to the cost function. The daily battery degradation cost is approximated as:
$$ C^f_b = \sum_{t=1}^{T} (K_d \cdot \Delta \text{DOD}_t + B_d) $$
where \( \Delta \text{DOD}_t \) represents the change in depth of discharge in period \( t \), and \( K_d \) and \( B_d \) are parameters obtained from the piecewise linear segments of the original non-linear cost function derived from Equations (3) and (5). This transformation is crucial for maintaining the model as a MILP, ensuring it can be solved rapidly for practical applications like day-ahead scheduling for an industrial park’s battery energy storage system.
Optimization Model for Industrial Park Dispatch
My optimization model schedules the operations of the industrial park’s resources, including power purchase from the grid, on-site renewable generation (e.g., wind turbines), and the multi-device battery energy storage system. The objective is to minimize total daily cost while respecting technical constraints.
Objective Function
The dispatch aims to balance primary economic costs with secondary battery health objectives. The composite objective function is:
$$ \min \, F = \min \, ( C_1 + \lambda \cdot F_2 ) $$
where \( C_1 \) is the total economic cost, \( F_2 \) is the SOC health evaluation function, and \( \lambda \) is a weighting coefficient that balances their influence. The value of \( \lambda \) is tuned so that the contribution of \( \lambda F_2 \) is proportionally much smaller than \( C_1 \), ensuring economic efficiency remains the primary driver.
1. Economic Cost Function (\( C_1 \)):
This encompasses all monetized costs:
$$ C_1 = C_e + C_d + C_r + C^f_b $$
- Energy Charge (\( C_e \)): Cost of electricity purchased from the main grid at time-of-use rates.
$$ C_e = \sum_{t=1}^{T} m(t) \cdot P_N(t) $$
where \( m(t) \) is the electricity price at time \( t \), and \( P_N(t) \) is the power drawn from the grid. - Demand Charge (\( C_d \)): A monthly fee based on the peak power demand (\( P_{\text{peak}} \)) observed during the billing cycle. For a daily model, we consider the impact on the daily apportioned cost or directly minimize the peak power. A common formulation is:
$$ C_d = a \cdot P_{\text{peak}} $$
where \( a \) is the demand charge rate, and \( P_{\text{peak}} = \max_{t} (P_N(t)) \). This can be linearized by introducing an auxiliary variable constrained to be greater than or equal to \( P_N(t) \) for all \( t \). - Renewable Energy Cost (\( C_r \)): Includes the cost of using renewable power (often lower than grid power) and a penalty for curtailment to promote full utilization.
$$ C_r = \sum_{t=1}^{T} \left[ C_{\text{NE}} \cdot P_{\text{NE}}^{\text{use}}(t) + C_{\text{pun}} \cdot ( P_{\text{NE}}(t) – P_{\text{NE}}^{\text{use}}(t) ) \right] $$
where \( C_{\text{NE}} \) is the cost per kWh of renewable energy, \( P_{\text{NE}}(t) \) is the available renewable power, \( P_{\text{NE}}^{\text{use}}(t) \) is the portion actually used, and \( C_{\text{pun}} \) is the penalty for curtailment. - Battery Degradation Cost (\( C^f_b \)): The piecewise linearized cost from Equation (6), representing the daily wear cost of the battery energy storage system.
2. SOC Health Evaluation Function (\( F_2 \)):
This function promotes operating practices that benefit the long-term health of the individual devices within the battery energy storage system. It has two components:
$$ F_2 = F_{\text{ave}} + F_{\text{wav}} $$
- SOC Coordination Index (\( F_{\text{ave}} \)): Penalizes deviations of individual device SOC from the average SOC of all devices. This encourages balanced usage, preventing any single unit from being consistently over- or under-utilized.
$$ F_{\text{ave}} = \sum_{t=1}^{T} \sum_{n=1}^{N} | S^{t}_{\text{OC,ave}} – S^{t, n}_{\text{OC}} | $$
where \( S^{t}_{\text{OC,ave}} = \frac{1}{N} \sum_{n=1}^{N} S^{t, n}_{\text{OC}} \) is the average SOC at time \( t \), and \( S^{t, n}_{\text{OC}} \) is the SOC of device \( n \) at time \( t \). - SOC Fluctuation Index (\( F_{\text{wav}} \)): Penalizes large deviations of a device’s SOC from its own daily average. This discourages excessive and frequent deep cycling for individual devices.
$$ F_{\text{wav}} = \sum_{t=1}^{T} \sum_{n=1}^{N} | S^{n}_{\text{OC,ave}} – S^{t, n}_{\text{OC}} | $$
where \( S^{n}_{\text{OC,ave}} = \frac{1}{T} \sum_{t=1}^{T} S^{t, n}_{\text{OC}} \) is the daily average SOC for device \( n \).
Both \( F_{\text{ave}} \) and \( F_{\text{wav}} \) use absolute values, which can be linearized using standard techniques (introducing auxiliary variables and constraints).
Constraints
The model is subject to the following operational constraints for each time period \( t \):
1. Power Balance Constraint:
$$ P_N(t) + P_{\text{NE}}^{\text{use}}(t) + P_{\text{dis\_bat}}^{\text{total}}(t) = P_{\text{cha\_bat}}^{\text{total}}(t) + P_L(t) $$
where \( P_{\text{dis\_bat}}^{\text{total}}(t) = \sum_{n=1}^{N} P_{\text{dis\_bat}}^{n}(t) \) is the total discharge power from all devices in the battery energy storage system, \( P_{\text{cha\_bat}}^{\text{total}}(t) = \sum_{n=1}^{N} P_{\text{cha\_bat}}^{n}(t) \) is the total charge power, and \( P_L(t) \) is the industrial park load. A device cannot charge and discharge simultaneously, so for each device \( n \) and time \( t \), binary variables are used to enforce \( P_{\text{cha\_bat}}^{n}(t) \cdot P_{\text{dis\_bat}}^{n}(t) = 0 \).
2. Renewable Generation Constraint:
$$ 0 \leq P_{\text{NE}}^{\text{use}}(t) \leq P_{\text{NE}}(t) \leq P^{\text{max}}_{\text{NE}} $$
where \( P^{\text{max}}_{\text{NE}} \) is the installed capacity.
3. Battery Energy Storage System Constraints (for each device \( n \)):
- Power Limits:
$$ 0 \leq P_{\text{cha\_bat}}^{n}(t) \leq u_{\text{cha}}^{n}(t) \cdot \bar{P}_n $$
$$ 0 \leq P_{\text{dis\_bat}}^{n}(t) \leq u_{\text{dis}}^{n}(t) \cdot \bar{P}_n $$
$$ u_{\text{cha}}^{n}(t) + u_{\text{dis}}^{n}(t) \leq 1 $$
where \( u_{\text{cha}}^{n}(t) \) and \( u_{\text{dis}}^{n}(t) \) are binary variables indicating charge and discharge modes. - State of Charge Dynamics:
$$ S^{t, n}_{\text{OC}} = S^{t-1, n}_{\text{OC}} + \left( \frac{\eta_{\text{cha}} \cdot P_{\text{cha\_bat}}^{n}(t)}{\bar{E}_n} – \frac{ P_{\text{dis\_bat}}^{n}(t) }{\eta_{\text{dis}} \cdot \bar{E}_n} \right) \cdot \Delta t $$
where \( \eta_{\text{cha}} \) and \( \eta_{\text{dis}} \) are charge and discharge efficiencies, and \( \Delta t \) is the time interval length (e.g., 1 hour). - SOC Boundaries:
$$ \text{SOC}^{\text{min}} \leq S^{t, n}_{\text{OC}} \leq \text{SOC}^{\text{max}} $$
Typically, \( \text{SOC}^{\text{min}} \) might be 0.1-0.2 and \( \text{SOC}^{\text{max}} \) 0.9-0.95 to prevent stress. - Depth of Discharge: The DOD for device \( n \) at time \( t \) is defined relative to its maximum capacity: \( \text{DOD}_t^n = 1 – S^{t, n}_{\text{OC}} \). The system-wide DOD change used in the cost model is aggregated appropriately from individual device behaviors, though the piecewise cost function can also be applied per device for finer accuracy.
- Initial and Final SOC: Often set to a specific value (e.g., 0.5) to ensure the battery energy storage system is ready for the next day’s schedule: \( S^{0, n}_{\text{OC}} = S^{T, n}_{\text{OC}} = S^{\text{setpoint}} \).
Case Study and Numerical Analysis
To validate the proposed model, I developed a case study based on a typical industrial park’s summer load profile. The park has a peak load of approximately 9 MW and hosts a 0.5 MW wind turbine. The battery energy storage system initially comprises two devices with different characteristics, reflecting a scenario of system expansion: Device 1 is rated at 250 kW / 500 kWh, and Device 2 is rated at 250 kW / 450 kWh. The cost parameters are: \( C_{\bar{P}} = 600 \) USD/kW, \( C_{\bar{E}} = 1800 \) USD/kWh, \( C_w = 97 \) USD/(kW·year). The time-of-use electricity tariff has three periods: peak (08:00-12:00, 17:00-21:00) at 0.107 USD/kWh, off-peak (12:00-17:00, 21:00-24:00) at 0.064 USD/kWh, and valley (00:00-08:00) at 0.031 USD/kWh. The demand charge rate is 12 USD/kW-month, apportioned daily. The wind energy cost \( C_{\text{NE}} \) is 0.04 USD/kWh, and the curtailment penalty \( C_{\text{pun}} \) is set at 0.10 USD/kWh to encourage full consumption. The discount rate \( r \) is 5%. The SOC limits are set at 0.2 and 0.9. The initial SOC for the two devices are set differently at 0.6 and 0.4, with a final SOC target of 0.5 for both, simulating a daily cycle that returns to a moderate, healthy state.
The model was implemented in MATLAB, formulating it as a MILP, and solved using the Gurobi optimizer. I compared the results of my proposed model against three alternative dispatch strategies to isolate the benefits of its key features:
| Dispatch Strategy | Description |
|---|---|
| Strategy A (Proposed) | Full model: Includes battery degradation cost via piecewise linear DOD model, independent device modeling, and SOC health function (\( \lambda \) tuned). |
| Strategy B | Ignores battery life degradation cost (\( C^f_b = 0 \)). |
| Strategy C | Excludes the SOC health evaluation function (\( \lambda = 0 \)). |
| Strategy D | Aggregates the two battery devices into a single, equivalent monolithic battery energy storage system (950 kWh, avg. SOC 0.505). |
The key results for a typical 24-hour schedule are summarized in the following table, showing the breakdown of daily costs.
| Cost Component (USD) | Strategy A (Proposed) | Strategy B (No Degradation Cost) | Strategy C (No SOC Health) | Strategy D (Aggregated System) |
|---|---|---|---|---|
| Energy Charge (\( C_e \)) | 5,557.5 | 5,549.5 | 5,556.8 | 5,570.0 |
| Demand Charge (\( C_d \)) | 847.0 | 847.0 | 847.0 | 847.0 |
| Renewable Cost & Penalty (\( C_r \)) | 288.1 | 288.1 | 288.1 | 288.1 |
| Battery Degradation Cost (\( C^f_b \)) | 96.1 | 0.0 | 96.8 | 86.6 |
| Total Daily Cost (\( C_1 \)) | 6,788.7 | 6,684.6 | 6,788.7 | 6,791.7 |
| SOC Health Function Value (\( F_2 \)) | 14.1 | 25.3 | 25.3 | N/A (single device) |
Analysis of the SOC profiles and cost data reveals several important insights. First, comparing Strategy A and Strategy B demonstrates the critical impact of accounting for battery life. While Strategy B shows a lower total daily cost because it ignores degradation, this is a misleading short-term view. In reality, the battery energy storage system under Strategy B exhibited more frequent and aggressive charge-discharge cycles, which would lead to faster capacity fade and higher long-term replacement costs. The proposed model (A) internalizes this future cost, leading to slightly more conservative battery usage and a higher but more accurate daily cost. The increase of approximately 104 USD in daily cost represents the “wear tariff” for using the battery energy storage system sustainably.
Second, the comparison between Strategy A and Strategy C highlights the role of the SOC health function. While the total economic cost \( C_1 \) is nearly identical (a difference of only 0.1 USD), the battery degradation cost in Strategy C is slightly higher (96.8 vs. 96.1 USD). More tellingly, the SOC health function value \( F_2 \), which quantifies imbalance and fluctuation, is significantly worse for Strategy C (25.3 vs. 14.1). Examining the SOC trajectories showed that without the health function, the two devices operated with greater SOC divergence and spent more time at the upper and lower SOC bounds. This indicates higher stress on individual units within the battery energy storage system, which would likely translate to accelerated, non-uniform aging over time. The proposed model successfully coordinates the devices, promoting healthier operating patterns without compromising economic efficiency.
Third, Strategy D, which uses an aggregated model, results in a total cost slightly higher than Strategy A (6,791.7 vs. 6,788.7 USD). Although its calculated degradation cost is lower, its energy charge is higher. This is because the aggregated model cannot exploit the flexibility of having two independently controllable devices with different initial states. It makes less optimal decisions regarding when and how much to charge/discharge, leading to marginally higher grid electricity purchases. This confirms that independent modeling of devices within a battery energy storage system provides tangible, albeit modest, economic benefits by enabling more precise and flexible control.
The scheduling behavior under the proposed model clearly shows the battery energy storage system charging during late-night valley periods when electricity is cheapest and discharging during the afternoon peak price period. It also discharges slightly during the morning peak to reduce demand charges. The wind generation is fully consumed in all scenarios due to the curtailment penalty. The SOC of both devices under Strategy A converges smoothly and remains within a balanced, middle range for most of the day, avoiding extreme states.
Conclusions and Future Perspectives
In this work, I have presented a comprehensive economic dispatch model for industrial parks that integrates a realistic cost model for battery life degradation. The model is designed for practical application, employing piecewise linearization to maintain computational efficiency while accurately reflecting the non-linear impact of depth of discharge on the cycle life of a battery energy storage system. A key advancement is the explicit modeling of individual storage devices within a larger system, which enhances scheduling flexibility and allows for health-conscious coordination among units. The inclusion of an SOC health evaluation function as a secondary objective further steers the optimization towards operating patterns that promote longevity, such as reducing SOC fluctuations and maintaining balance between devices.
The case study results validate the model’s effectiveness. They demonstrate that neglecting battery degradation cost leads to an economically shortsighted dispatch strategy that accelerates system wear. They also show that the SOC health function can mitigate stressful operating conditions for individual devices without incurring significant economic penalty. Finally, they confirm that modeling a multi-device battery energy storage system at the unit level, rather than as an aggregated entity, can yield more economical and technically sound dispatch instructions.
For industrial park operators, this model provides a valuable tool for optimizing the use of their battery energy storage system assets, ensuring that economic benefits from energy arbitrage and demand charge management are realized without inadvertently shortening the system’s operational life. Future research directions could involve integrating more detailed, chemistry-specific aging models that account for factors like temperature and C-rate, extending the model to consider uncertainty in renewable generation and load forecasting, and exploring the participation of such coordinated battery energy storage systems in ancillary service markets for additional revenue streams.