The relentless pursuit of efficiency and performance in modern aircraft has driven a significant shift towards More Electric Aircraft (MEA) architectures. In this context, energy storage systems have evolved from purely backup roles to becoming integral components for primary power delivery. Among available technologies, the lithium ion battery stands out due to its high specific energy, excellent cycle life, and superior power density. The performance, weight, and volume of the li ion battery pack directly influence critical aircraft metrics such as range, payload, and operational cost. Traditional design methods, heavily reliant on extensive physical testing and conservative safety margins, are often time-consuming, costly, and result in suboptimal, overweight systems.
To address these challenges, model-based design and optimization have emerged as powerful tools. By developing high-fidelity digital models of the li ion battery, engineers can virtually explore a vast design space, predict performance under diverse operational scenarios, and optimize the pack configuration before committing to costly prototyping. While various modeling approaches exist, electrochemical models are particularly valuable as they are rooted in the fundamental physical and chemical processes governing battery behavior. This intrinsic link to physics allows for more accurate predictions, especially under dynamic loads, and provides deeper insight into internal states that are not directly measurable, such as lithium concentration gradients.
One such electrochemical model is the Single-Particle Plus (SP+) model. It builds upon the classic Single-Particle (SP) model but addresses its primary shortcoming: poor accuracy at moderate to high discharge rates. The SP model simplifies the complex porous electrodes of a li ion battery by assuming each electrode can be represented by a single, spherical active material particle. While computationally efficient, it neglects crucial effects like electrolyte concentration gradients, leading to voltage prediction errors under strenuous conditions. The SP+ model elegantly incorporates simplified yet accurate descriptions of these phenomena, making it an ideal candidate for aviation applications where loads can vary significantly. Its balance of accuracy, computational simplicity, and physical interpretability makes it perfectly suited for system-level simulation and optimization of complex li ion battery packs.
Foundations of the SP+ Electrochemical Model
The development of the SP+ model begins with the principles of the standard Single-Particle model. The core assumption is that the reaction current density at the current collector interface for each electrode (positive, p, and negative, n) can be approximated as uniform. This current density \( j_i \) is given by:
$$ j_i \approx \frac{I R_i}{3 F (1 – \varepsilon_i – \varepsilon_{f,i}) l_i A}, \quad i = p, n $$
where \( I \) is the applied current, \( R_i \) is the radius of the active material particle, \( F \) is Faraday’s constant, \( \varepsilon_i \) is the porosity, \( \varepsilon_{f,i} \) is the filler volume fraction, \( l_i \) is the electrode thickness, and \( A \) is the electrode plate area.
The diffusion of lithium within the solid particles is governed by Fick’s second law in spherical coordinates:
$$ \frac{\partial c_s^i}{\partial t} = D_{s,i} \frac{1}{r^2} \frac{\partial}{\partial r} \left( r^2 \frac{\partial c_s^i}{\partial r} \right) $$
with boundary conditions:
$$ \left. \frac{\partial c_s^i}{\partial r} \right|_{r=0} = 0 \quad \text{and} \quad \left. \frac{\partial c_s^i}{\partial r} \right|_{r=R_i} = -\frac{j_i}{D_{s,i}} $$
Here, \( c_s^i \) is the solid-phase lithium concentration, \( D_{s,i} \) is the solid-phase diffusion coefficient, and \( r \) is the radial coordinate.
The cell’s open-circuit potential \( E \) is determined by the difference between the positive and negative electrode equilibrium potentials, which are functions of the surface lithium concentrations (\( c_{s, surf}^i \)):
$$ E = E_p \left( \frac{c_{s, surf}^p}{c_{s, max}^p} \right) – E_n \left( \frac{c_{s, surf}^n}{c_{s, max}^n} \right) $$
The activation overpotential \( \eta_{act} \), arising from charge transfer kinetics, is derived from the Butler-Volmer equation and can be expressed as:
$$ \eta_{act} = \frac{2RT}{F} \left[ \ln\left( \sqrt{m_n^2 + 1} + m_n \right) + \ln\left( \sqrt{m_p^2 + 1} + m_p \right) \right] $$
where \( m_i = \frac{0.5 j_i}{k_i (c_{s, max}^i – c_{s, surf}^i)^{0.5} (c_{s, surf}^i)^{0.5} (c_e)^{0.5}} \), \( k_i \) is the reaction rate constant, \( c_e \) is the electrolyte concentration, \( R \) is the gas constant, and \( T \) is the temperature.
The ohmic overpotential \( \eta_{ohm} \), primarily from the Solid-Electrolyte Interphase (SEI) layer, is:
$$ \eta_{ohm} = R_{SEI,n} F j_n – R_{SEI,p} F j_p $$
The terminal voltage \( V \) in the simple SP model is then:
$$ V = E – \eta_{act} – \eta_{ohm} $$
Enhancements in the SP+ Model: Simplified Dynamics
The SP+ model introduces key simplifications to capture missing dynamics without prohibitive computational cost.
1. Solid-Phase Diffusion with a Three-Parameter Approximation: Instead of solving the full partial differential equation, the lithium concentration profile within a particle is approximated by a parabolic function characterized by three parameters: the volume-averaged concentration \( c_{s,a}^i(t) \), the surface concentration \( c_{s,s}^i(t) \), and the volume-averaged concentration flux \( q_{a}^i(t) \). The governing equations simplify to:
$$ c_{s,a}^i(t) = c_{s,0}^i – \int_0^t \frac{3 j_i}{R_i} dt $$
$$ \frac{d}{dt} q_{a}^i(t) + \frac{30 D_{s,i}}{R_i^2} q_{a}^i(t) + \frac{45 j_i}{2 R_i^2} = 0 $$
$$ c_{s,s}^i(t) = c_{s,a}^i(t) + \frac{[8 D_{s,i} q_{a}^i(t) – j_i] R_i}{35 D_{s,i}} $$
For constant current (CC) conditions, an analytical solution exists for \( q_{a}^i(t) \):
$$ q_{a}^i(t) = \frac{3 j_i}{4 D_{s,i}} \left[ \exp\left( -\frac{t}{\tau_s^i} \right) – 1 \right], \quad \tau_s^i = \frac{R_i^2}{30 D_{s,i}} $$
Here, \( \tau_s^i \) is the solid-phase diffusion time constant, a crucial parameter characterizing the diffusion speed.
2. Simplified Liquid-Phase Diffusion: Under CC operation, the electrolyte concentration distribution reaches a steady-state profile. The concentration in the electrodes approximates a parabola, and in the separator, a straight line. This allows for the calculation of the concentration overpotential \( \eta_e \) between the electrodes:
$$ \eta_e = 2 \frac{RT}{F} (1 – t_+) (\ln c_n – \ln c_p) $$
where \( t_+ \) is the transference number, and \( c_n \), \( c_p \) are the electrolyte concentrations at the negative and positive current collectors, respectively.
3. Parameter Reduction and Physical Interpretation: The SP+ model consolidates numerous physical parameters into a smaller, more manageable set, each retaining a clear physical meaning. This is vital for parameter identification and model scalability. The terminal voltage equation becomes:
$$ V = E – \eta_{act} – \eta_e – \eta_{ohm} $$
The final set of parameters for an SP+ li ion battery model is summarized in the table below. This parameter set effectively captures the essential electrochemical behavior needed for accurate aviation li ion battery pack simulation.
| Parameter | Physical Meaning | Related Internal State |
|---|---|---|
| \( y_0 \) | Initial lithiation fraction in the positive electrode (-) | Battery State of Charge (SOC) |
| \( Q_p \) (A·s) | Positive electrode capacity | Total amount of active material in the positive electrode |
| \( Q_n \) (A·s) | Negative electrode capacity | Total amount of active material in the negative electrode |
| \( y_{ofs} \) (-) | Positive/negative electrode capacity offset | Relative alignment of electrode design |
| \( \tau_s^p \) (s) | Positive electrode solid-phase diffusion time constant | Characteristic speed of solid diffusion in the cathode |
| \( \tau_s^n \) (s) | Negative electrode solid-phase diffusion time constant | Characteristic speed of solid diffusion in the anode |
| \( P_{con} \) (mol·m⁻³·A⁻¹) | Liquid-phase diffusion proportionality coefficient | Magnitude of electrolyte concentration gradients |
| \( \tau_e \) (s) | Liquid-phase diffusion time constant | Dynamic characteristic of electrolyte diffusion |
| \( c_0 \) (mol·m⁻³) | Initial electrolyte lithium-ion concentration | Typically around 1000 mol/m³ (1 mol/L) |
| \( P_{act} \) (m⁻¹·⁵·mol⁰·⁵·s) | Activation polarization coefficient | Characteristic of charge transfer reaction kinetics |
| \( R_{ohm} \) (Ω) | Ohmic internal resistance | Combined ohmic resistance (SEI, contacts, etc.) |
Modeling and Validation of Li-ion Battery Packs
The performance of a multi-cell li ion battery pack is not a simple linear sum of its individual cells. Inconsistencies in voltage, internal resistance, capacity, and initial SOC among cells lead to imbalanced currents in parallel strings and potential over-charge/discharge in series connections. Therefore, a robust pack model must account for cell-to-cell variations.
Our methodology starts with parameter identification for a single aviation-grade lithium cobalt oxide (LCO) li ion battery cell (nominal: 3.7V, 4Ah). Using experimental data from pulse and constant current discharge tests, the SP+ model parameters are extracted. Key parameters, such as diffusion time constants and ohmic resistance, are identified as functions of discharge rate to improve accuracy across the operational range. The identified parameter set forms the basis of a high-fidelity cell model implemented in the Saber simulation environment.
Single-Cell Model Validation: The single-cell SP+ model is validated against experimental data at discharge rates from 1C to 5C. The comparison shows excellent agreement. The table below quantifies the model’s accuracy, demonstrating its capability to predict the terminal voltage of a single li ion battery with high precision, which is the foundational step for reliable pack simulation.
| Discharge Scenario | Absolute Average Error (V) | Relative Average Error (%) |
|---|---|---|
| 1C Constant Current | 0.0181 | 0.49 |
| 2C Constant Current | 0.0177 | 0.49 |
| 3C Constant Current | 0.0275 | 0.76 |
| 4C Constant Current | 0.0203 | 0.58 |
| 5C Constant Current | 0.0166 | 0.48 |
Battery Pack Model Construction and Validation: To construct a pack model, multiple instances of the validated single-cell model are connected in series and/or parallel. Critically, cell inconsistency is simulated by applying a statistical distribution (e.g., using a Monte Carlo method) to key parameters like initial SOC (\(y_0\)) and capacity (\(Q_p, Q_n\)) for each cell in the pack. This creates a more realistic representation of a physical li ion battery pack.
Two pack configurations were modeled and validated:
1. A 3.7V, 60Ah pack (15 cells in parallel).
2. A 28V, 60Ah pack (7 series branches of 15 parallel cells).
The pack models were subjected to constant current discharge tests. The simulation results for the 3.7V pack across multiple C-rates are shown in the table below, confirming the model’s accuracy at the pack level. The 28V pack model also showed strong agreement with experimental data under a 45A load. These validations prove that the SP+ model, combined with statistical variation, can accurately predict the collective behavior of a complex aviation li ion battery pack, making it a trustworthy tool for design exploration.
| Discharge Scenario (3.7V, 60Ah Pack) | Absolute Average Error (V) | Relative Average Error (%) |
|---|---|---|
| 1C Constant Current | 0.0186 | 0.53 |
| 2C Constant Current | 0.0162 | 0.45 |
| 3C Constant Current | 0.0094 | 0.26 |
| 4C Constant Current | 0.0095 | 0.27 |
| 5C Constant Current | 0.0109 | 0.31 |
Case Study: Pack Optimization for a Typical Aviation Power Profile
With a validated modeling framework in place, we demonstrate its application in optimizing an aviation li ion battery pack for a specific duty cycle. The goal is to meet all electrical requirements while minimizing the pack’s mass and volume—two of the most critical metrics in aerospace design.
System Requirements:
- Supply a 24V system with a 1V line drop, meaning the pack must maintain a voltage above (24V + 1V) = 25V at its terminals.
- Load profile: 100A average for 15 minutes, followed by 150A average for 15 minutes (30 minutes total).
- The pack voltage must not fall below 20V at the system level (accounting for the line drop, this means the pack terminal voltage must stay above 21V).
- In a real system, the load current is often voltage-dependent; this relationship is modeled in the simulation.
Conventional Sizing Calculation:
First, the total energy requirement is calculated:
$$ E_{req} = (100A \times 24V \times 0.25h) + (150A \times 24V \times 0.25h) = 1500 Wh $$
Assuming 90% energy efficiency and starting from 100% SOC, the required pack energy is:
$$ E_{pack\_req} = 1500 Wh / 0.9 \approx 1666.67 Wh $$
Using a cell with a nominal 3.7V platform, the minimum series count (\(N_s\)) to stay above 21V is \( \lceil 21V / 3.7V \rceil = 6\). The maximum series count to stay below a reasonable upper voltage is often \( \lceil 28V / 3.7V \rceil = 7\) (for a 28V bus). We analyze both options.
For a 6-series configuration, the minimum pack capacity is:
$$ C_{pack\_min} = \frac{1666.67 Wh}{(24V – 1V)} \approx 72.46 Ah $$
Accounting for 90% usable capacity: \( 72.46 Ah / 0.9 \approx 80.51 Ah \).
With 4Ah cells, the minimum parallel count (\(N_p\)) is \( \lceil 80.51 / 4 \rceil = 21 \). Initial design: 6S21P (126 cells).
For a 7-series configuration, the minimum pack capacity is:
$$ C_{pack\_min} = \frac{1666.67 Wh}{(25.9V – 1V)} \approx 66.93 Ah $$
Usable capacity: \( 66.93 Ah / 0.9 \approx 74.37 Ah \).
Minimum parallel count: \( \lceil 74.37 / 4 \rceil = 19 \). Initial design: 7S19P (133 cells).
The conventional method suggests two viable but conservative configurations, with the 6S21P appearing slightly better in cell count.
Model-Based Optimization:
Using the Saber SP+ pack model, we simulate the exact load profile, including the voltage-dependent current draw. We start with the conservative designs:
- 6S21P Pack: Simulation confirms it meets requirements with a comfortable voltage margin.
- 7S19P Pack: Simulation also confirms it meets requirements.
The key optimization step is to reduce the parallel count in each configuration until the voltage trajectory approaches, but does not violate, the minimum limit (21V at pack terminals). This minimizes the number of cells, directly reducing weight and volume.
Optimization Results:
- For the 6S configuration: A 6S20P pack (120 cells) was found to be the limit, operating very close to the cutoff voltage at the end of the duty cycle. The 6S21P (126 cells) is therefore selected as the optimal safe configuration for this path.
- For the 7S configuration: A 7S17P pack (119 cells) was found to be viable with a reasonable margin. A 7S16P pack (112 cells) operated too close to the limit.
The final optimized configurations are:
| Configuration | Total Cells | Status |
|---|---|---|
| 6S21P (Conventional) | 126 | Feasible |
| 6S20P (Optimized) | 120 | Marginal, not recommended |
| 7S17P (Optimized) | 119 | Optimal – Safely meets requirement with fewest cells |
| 7S19P (Conventional) | 133 | Feasible but conservative |
This analysis reveals a non-intuitive optimal solution. While the initial calculation favored a 6S topology, the high-fidelity SP+ model simulation, which accounts for the cell’s nonlinear voltage behavior under dynamic load and the system’s voltage drop, showed that a 7S17P configuration actually uses the fewest cells (119). This translates to the lightest and most compact li ion battery pack for this specific aviation application. The model-based approach enabled this discovery, which would be difficult and costly to achieve through trial-and-error physical testing.
Conclusion
The adoption of lithium ion battery technology in aviation is accelerating, moving beyond auxiliary roles into primary power domains. This shift places unprecedented demands on the performance, weight, and reliability of the li ion battery pack. The SP+ electrochemical model has been demonstrated as a powerful enabler for advanced design and optimization in this field. Its foundation in physical principles ensures accurate prediction of cell and pack behavior under realistic, dynamic loads, while its simplified mathematical structure makes it suitable for system-level simulation and iterative design processes.
By transitioning from traditional, conservative sizing methods to a model-based optimization framework, significant gains in system efficiency can be realized. The presented case study clearly shows how high-fidelity simulation can identify an optimal li ion battery pack configuration that minimizes mass and volume while rigorously meeting all electrical and safety constraints. This approach not only leads to better-performing aircraft systems but also substantially reduces development time and cost by front-loading the design process with virtual prototyping and analysis. As aviation continues its electrification journey, such model-based techniques will be indispensable for unlocking the full potential of the li ion battery, ensuring that energy storage systems contribute positively to the overall mission of lighter, more efficient, and more capable aircraft.