⚡ electrical engineering

Making Aggregations Reliable: Realizability Guarantees for Battery Fleets with Heterogeneous Power and Energy Limits

This paper introduces a realizable composite battery model for heterogeneous fleets that guarantees element-level feasibility through simple linear conditions and a priority-based controller, ensuring computational scalability and robustness while avoiding the intractability of micro-level mixed-integer formulations.

Original authors: Mazen Elsaadany, Mads R. Almassalkhi

Published 2026-06-02

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Mazen Elsaadany, Mads R. Almassalkhi

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are the conductor of a massive orchestra, but instead of violins and drums, your musicians are thousands of individual batteries. Some are small and fast (like a sprinter), while others are large and slow (like a marathon runner). Your goal is to tell this entire fleet when to charge up and when to release energy to help power the city grid.

The problem is that if you try to write a specific instruction for every single one of the 1,000+ batteries, the math becomes so complicated that your computer crashes before it can finish. This is like trying to conduct an orchestra by whispering a unique note to every single musician individually; it's too slow and messy.

So, most people try to treat the whole fleet as one giant super-battery. They say, "Okay, the whole group can charge at 100 megawatts." But here's the catch: just because the group has enough capacity doesn't mean the individual batteries can handle the specific instructions you give them.

The "Impossible" Schedule

If you tell the "super-battery" to charge and discharge at the same time (which math sometimes suggests is efficient), you run into a physical impossibility. You can't tell a single battery to charge and discharge simultaneously; it's like telling a person to eat dinner and diet at the exact same moment. If you ignore this rule in your math, you might send a schedule to the fleet that looks perfect on paper but causes the actual batteries to break their rules or fail to deliver power when needed.

The Paper's Solution: The "Priority Stack"

This paper introduces a new way to manage the fleet that guarantees the instructions sent to the big "super-battery" can actually be carried out by the individual batteries. They call this a Realizable Composite Battery model.

Think of it like a priority line at a busy coffee shop:

The Rule: The shop has a rule: "We will only serve coffee to people who are thirsty (low battery) and only serve tea to people who are full (high battery)."
The Conductor's Job: Instead of telling every single person what to do, the conductor (the computer) just tells the line, "We need 50 coffees and 20 teas right now."
The Disaggregation: The "Priority Stack Controller" (the barista) automatically sorts the line. The people with the emptiest cups get the coffee first. The people with the fullest cups get the tea first.

Because the barista follows a strict, simple rule (lowest energy gets charged first, highest energy gets discharged first), the paper proves that as long as the conductor stays within certain safe limits, the barista will always be able to fulfill the order without anyone breaking the rules.

The "Safety Buffer"

To make sure no single battery gets into trouble (like running out of juice completely or overflowing), the paper suggests adding a "safety buffer."

Imagine the coffee shop has a rule: "We will never let the line get so full that the last person in line can't get a drink, and we won't let the line get so empty that the first person has nowhere to stand."

The paper calculates exactly how much "wiggle room" is needed. If the batteries are all very similar, the buffer is small. If they are very different, the buffer is larger. By keeping the "super-battery" slightly away from its absolute maximum and minimum limits, the system guarantees that every individual battery stays safe.

What the Experiments Showed

The authors tested this idea using a computer simulation of a power grid:

Speed: When they had 1,000 batteries, the old method (trying to calculate every single battery) took a long time to solve. The new method took the same amount of time whether there were 10 batteries or 1,000. It was like switching from counting every grain of sand to weighing the whole bucket.
Accuracy: When they made the "conductor's" instructions more frequent (checking the batteries every minute instead of every 10 minutes), the new method's results became almost identical to the perfect, but slow, old method.
Reliability: Unlike other methods that sometimes gave instructions the batteries couldn't follow, this method always produced a schedule that the batteries could physically execute.

The Catch (The "Uniform Ratio" Assumption)

The paper notes that this magic trick works best if the batteries are somewhat similar in how fast they can charge relative to how much energy they hold (like all cars having a similar gas tank size relative to their engine power).

If the fleet has a mix of very different types of batteries, the system has to be a bit more conservative. It's like the coffee shop having to ignore some of the customers with very large cups to make sure the line moves smoothly. The paper shows that even with this "conservatism," the system still works and finds a good solution, just not quite as cheap as the theoretical perfect solution.

In a Nutshell

This paper provides a mathematical "rulebook" that allows grid operators to treat a chaotic crowd of thousands of different batteries as a single, manageable unit. It guarantees that the instructions given to the group will never be impossible for the individuals to follow, making the whole process faster, safer, and more reliable.

Technical Summary: Making Aggregations Reliable for Heterogeneous Battery Fleets

Problem Statement
Aggregated Battery Energy Storage Systems (BESS) enable large fleets of heterogeneous battery elements to participate in wholesale electricity markets and system-level optimization. However, explicitly scheduling and dispatching each individual element in a large fleet is computationally impractical. While existing aggregate models often rely on convex relaxations to reduce complexity, they typically ignore the non-convex complementarity constraint (preventing simultaneous charging and discharging) at the element level. Consequently, these models can produce dispatch solutions that appear feasible at the aggregate level but are physically infeasible when implemented on individual batteries, potentially violating energy limits or complementarity constraints. There is a lack of aggregate models that simultaneously scale to large, heterogeneous fleets and guarantee realizable dispatch under these constraints.

Methodology
The paper proposes a Realizable Composite Battery (RCB) model designed for heterogeneous BESS fleets. The core methodology involves:

Composite Modeling: The fleet is modeled as a single composite battery operating on a coarse scheduling time step ( $\Delta t$ ), aggregating the power and energy states of $N$ individual elements.
Priority Stack Controller (PSC): A disaggregation strategy is introduced where individual elements are sorted by their normalized State of Energy (SOE). Elements with lower normalized SOE are prioritized for charging, while those with higher normalized SOE are prioritized for discharging. This controller operates at a finer control time step ( $\delta t$ ).
Sufficient Conditions for Realizability: The authors derive simple linear conditions on the aggregate charging ( $P_c$ ) and discharging ( $P_d$ ) trajectories that guarantee the PSC can successfully disaggregate the command without violating individual constraints. These conditions rely on three structural assumptions:
- Symmetric power limits (equal max charge/discharge power per element).
- Uniform power-to-energy ratios across all elements.
- Homogeneous charging/discharging efficiencies.
- (Additionally, the initial spread of normalized SOE must be bounded).
The derived conditions include:
- A "cutting plane" constraint on $P_c + P_d$ to prevent simultaneous charging and discharging of the same element.
- A "buffer" constraint on the aggregate SOE ( $E$ ), ensuring it stays within $[\tilde{\epsilon}, 1-\tilde{\epsilon}]$ of the total capacity, where $\tilde{\epsilon}$ is a function of the control time step and power-to-energy ratios. This buffer accommodates the dispersion of individual SOEs caused by the priority-based control.

Key Contributions

Extension to Heterogeneity: The paper extends the composite battery framework to heterogeneous elements, deriving a linear aggregate model that captures varying power and energy limits.
Guaranteed Realizability: It generalizes the priority-based disaggregation strategy and provides sufficient linear conditions under which any feasible aggregate dispatch is guaranteed to be realizable at the element level, respecting power limits, energy limits, and complementarity.
Characterization of Feasible Region: The authors characterize the aggregate feasible region, demonstrating that accuracy improves as the control granularity ( $\delta t$ ) is refined, while maintaining computational tractability.

Numerical Results
The proposed model was evaluated using a Unit Commitment (UC) problem with fleet sizes ranging from $N=10$ to $N=1000$ .

Feasibility: Unlike relaxed micro-models (which ignore complementarity) that produced non-realizable schedules, the RCB model produced feasible dispatch solutions in all cases.
Scalability: The solve time for the RCB model was effectively independent of the fleet size (remaining near 2 seconds), whereas the Mixed-Integer Linear Programming (MILP) micro-model solve time increased sharply with $N$ (reaching ~52 seconds for $N=1000$ in a 2-hour horizon and scaling poorly in a 24-hour horizon).
Optimality: With coarse control granularity ( $\delta t = 10$ min), the RCB model was conservative, yielding higher costs due to tightened constraints. However, as $\delta t$ was refined (down to 1 min), the RCB objective cost converged to the optimal MILP benchmark while maintaining near-constant solve times.
Robustness: The study examined relaxing the uniform power-to-energy ratio assumption. By artificially limiting power or energy capacities to enforce a uniform ratio, the authors showed that realizability could be restored, though this introduced a trade-off between conservatism (reduced usable capacity) and performance cost.

Significance and Claims
The paper claims to address a critical gap in the literature by providing an aggregate modeling framework that scales to large heterogeneous fleets while guaranteeing that dispatch decisions are physically realizable. The significance lies in the ability to integrate battery fleets into system-level optimization (like Unit Commitment) without the computational burden of micro-modeling or the risk of infeasible dispatches inherent in standard convex relaxations. The authors note that while the current framework relies on specific structural assumptions (particularly uniform power-to-energy ratios), the proposed linear conditions and priority-based controller offer a tractable path toward reliable aggregation, with future work needed to relax these assumptions and extend the framework to market participation and network-constrained settings.