Approximating CCCV charging using SOC-dependent tapered charging power constraints in long-term microgrid planning

1. Problem Statement

Traditional long-term microgrid planning models typically assume constant power charging for Battery Energy Storage Systems (BESS). This simplification overlooks critical physical realities of lithium-ion batteries, specifically:

• Internal Resistance Rise: As the State of Charge (SOC) increases, internal resistance rises, leading to significant $I^2R$ losses and thermal stress.
• CCCV Protocols: Real-world batteries utilize Constant Current–Constant Voltage (CCCV) charging, where charging power tapers off at high SOC levels to mitigate these losses.
• Planning vs. Operation Gap: While Battery Management Systems (BMS) and inverters can emulate this tapering behavior at the pack level, long-term planning models often ignore it.
• Consequences: Ignoring tapering leads to overestimated charging flexibility, resulting in undersized BESS investments, underestimated costs, and potential reliability issues (load shedding) when systems face dynamic, fast-charging demands.

2. Methodology

The authors propose a tractable, scalable approach to integrate CCCV-like behavior into long-term optimization models without resorting to complex non-linear physics.

A. Proof-of-Concept Simulation (MATLAB/Simulink)

• Objective: Quantify the efficiency gains and energy loss reductions of tapering under varying internal resistance ($R_s$).
• Setup: A 100Ah Li-Ion pack (40 cells in series) was simulated under two strategies:
  1. Constant Current (CC): Fixed 1C rate.
  2. Tapered Charging: Current steps down (1C $\to$ 0.5C $\to$ 0.2C $\to$ 0.1C) as SOC crosses 80%, 90%, and 95%.
• Findings: Tapering significantly reduced energy losses and improved efficiency, particularly in batteries with higher internal resistance (e.g., second-life batteries). For $R_s = 0.01\Omega$, tapering reduced energy loss by 49 kWh compared to constant charging.

B. Optimization Framework

The study developed a multi-year microgrid planning model using Pyomo and GUROBI to minimize total system cost (Capital + Degradation + Load Shedding penalties).

• Baseline Model: Assumes constant power charging limits.
• Proposed Model (SOC-Dependent TCP): Introduces Tapered Charging Power (TCP) constraints.
  • Mechanism: The SOC range is divided into discrete bands (e.g., 0–80%, 80–90%, etc.).
  • Constraints: Binary variables activate specific SOC bands. As the battery enters higher SOC bands, the maximum allowable charging power ($P_{chg}$) is dynamically reduced using a tapering factor ($\beta_b$).
  • Formulation: This approximates the CCCV curve using linear constraints and Big-M constants, making it solvable within Mixed-Integer Linear Programming (MILP) frameworks.

3. Key Contributions

1. Novel Constraint Formulation: Proposed a scalable method to approximate CCCV charging behavior in long-term planning using SOC-dependent power limits, bridging the gap between cell-level physics and system-level optimization.
2. Quantified Trade-offs: Demonstrated that while tapering improves efficiency and reduces thermal stress, it reduces charging speed, necessitating larger battery capacities to meet the same energy delivery requirements.
3. Reliability Analysis: Showed that neglecting tapering in the planning phase leads to system designs that fail under operational constraints, causing significant load shedding when fast-charging demands occur.

4. Case Study Results

The model was tested on a synthetic profile featuring a solar-load mismatch (high PV midday, high load evening) requiring rapid BESS charging. Four scenarios were compared:

| Case | Tapering | Sizing | Key Outcome |
| :--- | :--- | :--- | : |
| Case 1 | Enabled | Free | Optimal with Tapering: Required larger BESS (13.75 MWh) and PV (3.53 MW). Total Cost: $12.82M. No load shedding. |
| Case 2 | Disabled | Free | Optimal without Tapering: Smaller BESS (11.0 MWh) and larger PV. Total Cost: $12.32M. No load shedding (but assumes unrealistic charging speed). |
| Case 3 | Disabled | Fixed to Case 1 | Isolated Test: Using Case 1's oversized hardware without tapering constraints yielded the same cost, confirming the model doesn't penalize fast charging unless explicitly modeled. |
| Case 4 | Enabled | Fixed to Case 2 | Stress Test: Enforcing tapering on the undersized Case 2 system resulted in 14,238 MWh of Load Shedding, proving reliability risks. |

Key Findings:

• Oversizing: Accounting for tapering increases BESS sizing by ~25% (from 11 MWh to 13.75 MWh) to compensate for reduced charging rates at high SOC.
• Cost Impact: The "realistic" cost (Case 1) is ~4% higher than the "idealized" cost (Case 2) due to the need for larger storage capacity.
• Reliability Risk: Systems sized without tapering constraints (Case 2) become unreliable when actual tapering behavior is enforced during operation (Case 4).

5. Significance and Conclusion

This paper highlights that tapering is a critical modeling factor for accurate long-term microgrid planning.

• Investment Accuracy: Planners must account for reduced charging flexibility at high SOC to avoid undersizing storage, which leads to higher operational costs and reliability failures.
• Second-Life Batteries: The benefits of tapering are even more pronounced for batteries with higher internal resistance (e.g., second-life EV batteries), making this approach vital for circular economy applications.
• Future Work: The authors plan to linearize the model further for better scalability, incorporate C-rate-dependent degradation models, and apply the framework to high-demand use cases like EV fleet charging and frequency regulation.

In summary, the study provides a rigorous mathematical framework to ensure that microgrid investments are robust against the physical limitations of battery chemistry, preventing costly reliability failures in renewable-heavy grids.