Dynamic Average Consensus with Privacy Guarantees and Its Application to Battery Energy Storage Systems

This paper proposes a privacy-preserving dynamic average consensus algorithm that utilizes randomly generated sinusoidal masking signals to protect reference data from eavesdroppers while maintaining convergence, and demonstrates its practical effectiveness through state-of-charge balancing in battery energy storage systems.

The Gist

Imagine a group of friends trying to decide on the average price of a pizza they all want to order together. They are sitting in a circle, and to make a fair decision, they need to share their individual budget numbers with their neighbors.

The Problem:
In the digital world, this is like a network of smart batteries (like those in a solar power station) trying to balance their energy levels. They talk to each other to figure out the "average" energy level so they can all work together efficiently.

However, there's a catch: Privacy. If they just shout their numbers out, a sneaky eavesdropper (a hacker or a spy) listening in could figure out exactly how much energy each specific battery has. This is dangerous. If a hacker knows exactly how much power a specific battery has, they might know when a factory is running, when a hospital is charging, or they could manipulate the system to cause a blackout.

The Old Solution (and why it failed):
Previous methods tried to hide the numbers by adding "noise" (static) to the conversation, like whispering while someone else is shouting. But this made the group slow to reach a decision, or the final average was slightly wrong. It was like trying to solve a math problem while wearing foggy glasses.

The New Solution (The "Sinusoidal Mask"):
This paper proposes a clever new trick called Dynamic Average Consensus with Privacy. Here is how it works, using a simple analogy:

The Analogy: The "Secret Rhythm" Game

Imagine the friends are trying to agree on the average height of the group, but they don't want to reveal their own height.

1. The Setup: Before they start talking about heights, each friend secretly picks a unique, random rhythm (a sine wave) and whispers it to their immediate neighbors.

   • In the paper: Each battery generates a random "masking signal" (a wave) and shares it securely with neighbors.

2. The Mask: Each friend adds up all the rhythms they received from neighbors and subtracts the rhythms they sent out. This creates a Composite Mask.

   • The Magic: Because everyone is doing this, the masks cancel each other out perfectly across the whole group. The total sum of all masks is zero. It's like everyone wearing a heavy coat that weighs them down, but since everyone wears one, the group's average weight hasn't changed at all.

3. The Conversation: Now, when they talk about their "height," they don't say their real height. They say: "My height PLUS my secret mask."

   • In the paper: The battery sends its "masked" energy level (Real Energy + Mask) to neighbors.

4. The Result:

   • For the Group: Since the masks cancel out (sum to zero), the group can still calculate the true average perfectly and quickly. The system works just as fast as if they told the truth.
   • For the Spy: The spy hears the conversation. They see the "Height + Mask." But they don't know the specific rhythm (mask) any single person is using. It's like trying to guess a person's height when they are wearing a coat that changes shape every second. The spy can't figure out the real height, nor can they figure out how fast the person is growing (the derivative).

Why is this a big deal?

• Speed vs. Security: Usually, you have to choose between being fast or being secure. This method gives you both. The batteries reach a consensus (agreement) just as fast as before.
• Protecting the "Secret Sauce": It doesn't just hide the current number; it hides how that number is changing. This is crucial. If a hacker knows how fast a battery is draining, they can predict when it will die. This method hides that too.
• Real World Use: The authors tested this on a Battery Energy Storage System (BESS). Imagine a neighborhood with solar panels and big batteries. They need to share power so no one runs out. This algorithm lets them share power perfectly without letting a hacker know exactly how much juice is in any single battery.

The Bottom Line

Think of this algorithm as a group of dancers moving in perfect sync.

• They need to move together to the beat (reach consensus).
• But they don't want the audience to know which dancer is leading or how fast they are spinning (privacy).
• So, they all wear glowing, shifting costumes (the masking signals).
• To the audience (the hacker), the dancers look like a blur of light; you can't tell who is who or what they are doing individually.
• But to the dancers themselves, they can still see each other clearly, move in perfect unison, and hit the exact right beat.

This paper proves that you can have a perfectly synchronized, high-speed team that keeps its individual secrets safe from prying eyes.

Technical Summary

Here is a detailed technical summary of the paper "Dynamic Average Consensus with Privacy Guarantees and Its Application to Battery Energy Storage Systems" by Maithripala, Qiu, and Lin.

1. Problem Statement

Context: Dynamic Average Consensus (DAC) is a fundamental algorithm in multi-agent systems used to track the average of time-varying reference signals. It is critical for applications like distributed economic dispatch and battery energy storage system (BESS) coordination.
The Challenge: Standard DAC algorithms require agents to exchange local state information with neighbors. This direct sharing exposes sensitive data (specifically, the time-varying reference signals $z_i(t)$ and their derivatives $\dot{z}_i(t)$) to external eavesdroppers.
Limitations of Existing Solutions:

• Static Consensus: Most privacy-preserving methods (e.g., differential privacy, state decomposition) are designed for static consensus and do not apply well to dynamic signals.
• Existing DAC Privacy: Previous attempts to protect DAC (e.g., modifying the graph Laplacian) often degrade convergence rates.
• Prior Masking Methods: A previous masking approach by the authors (adding constant random values) protected the reference signal but failed to protect its derivative (rate of change), which is often equally sensitive in control applications.

Objective: Develop a privacy-preserving DAC algorithm that:

1. Protects both the reference signals ($z_i(t)$) and their derivatives ($\dot{z}_i(t)$) from external eavesdroppers.
2. Maintains the original convergence rate and steady-state accuracy of conventional DAC.
3. Allows for tunable privacy levels.

2. Methodology

The authors propose a Sinusoidal Masking Strategy integrated into the standard DAC framework.

A. Algorithmic Design

The core idea is to mask the reference signal using a composite signal constructed from pairwise exchanges between neighbors.

1. Initialization: Each agent $i$ generates a set of sinusoidal signals $s_{ij}(t) = A_m \sin(\omega_{ij} t)$ for each neighbor $j$. Here, $A_m$ is a predefined amplitude, and $\omega_{ij}$ is a random frequency unique to the pair.
2. Mask Construction: Agent $i$ constructs a local masking signal $m_i(t)$ by summing the differences between signals received from neighbors and signals sent to them:
   $$m_i(t) = \sum_{j \in \mathcal{N}_i} (s_{ji}(t) - s_{ij}(t))$$
   Crucially, this construction ensures that the sum of all masking signals across the network is zero ($\sum m_i(t) = 0$) and the sum of their derivatives is also zero ($\sum \dot{m}_i(t) = 0$) for all time $t$.
3. Masked Update Rule: Instead of using the raw reference $z_i(t)$, agents use the masked signal $z_{m,i}(t) = z_i(t) + m_i(t)$ in the DAC update law:
   $$\dot{\hat{z}}_{a,i}(t) = \dot{z}_{m,i}(t) - \beta \sum_{j=1}^N a_{ij} (\hat{z}_{a,i}(t) - \hat{z}_{a,j}(t))$$
   where $\hat{z}_{a,i}$ is the estimate of the average, and $\beta$ is a tuning gain.

B. Theoretical Analysis

• Convergence: The authors prove (Theorem 1) that because the sum of the masking signals and their derivatives is zero, the network average of the masked signals equals the true average. The algorithm retains the exponential convergence rate of the standard DAC, governed by the graph Laplacian's second smallest eigenvalue ($\lambda_2$) and the gain $\beta$.
• Privacy Guarantee: The authors prove (Theorem 2) that an external eavesdropper observing all communication links and estimator states cannot uniquely reconstruct $z_i(t)$ or $\dot{z}_i(t)$.
  • Proof Logic: For any observed trajectory, there exist infinitely many distinct reference trajectories ($z'(t) = z(t) + \delta(t)$) and corresponding masks ($m'(t) = m(t) - \delta(t)$) that produce the exact same estimator output. Since $\delta(t)$ can be any zero-sum signal, the true signal is mathematically indistinguishable from infinite alternatives.

3. Application: Battery Energy Storage Systems (BESS)

The algorithm is applied to State-of-Charge (SoC) balancing in a networked BESS.

• Goal: Balance the SoC of $N$ battery units while tracking a total desired power profile $p^*(t)$.
• Implementation:
  • The "reference signal" for the DAC is the local energy state ($x_i(t)$) of each battery.
  • The DAC estimates the average energy state ($x_a(t)$) and the average desired power ($p_a(t)$).
  • A distributed power allocation law uses these estimates to determine individual charging/discharging currents.
• Privacy Benefit: This prevents external attackers from inferring the specific SoC or power output of individual battery units, which could reveal operational strategies or battery health status.

4. Key Contributions

1. Dual-Variable Privacy: Unlike previous methods, this scheme protects both the signal value and its derivative (rate of change), which is critical for dynamic control systems.
2. Preserved Performance: The masking mechanism does not alter the system's Laplacian matrix, thereby preserving the original convergence speed and steady-state accuracy of the standard DAC.
3. Tunable Privacy: The privacy level is adjustable via the signal amplitude parameter ($A_m$). Larger amplitudes increase the difficulty of reconstruction for an attacker without affecting convergence.
4. Low Complexity: The method relies on simple sinusoidal signal generation and addition, avoiding the heavy computational overhead of cryptography or complex state decomposition.

5. Simulation Results

The authors validated the approach using a 6-unit BESS in a ring topology.

• Performance: The system achieved accurate SoC balancing and precise tracking of a time-varying total power demand ($p^*(t) = 4200 \sin(t) + 4200$ W).
• Privacy Validation:
  • An external eavesdropper attempted to reconstruct the true battery states using a standard observer.
  • Result: The reconstructed signals deviated significantly from the true trajectories.
  • Metric: The Root-Mean-Square Error (RMSE) between the attacker's estimate and the true signal increased linearly with the masking amplitude $A_m$, confirming that privacy can be tuned.

6. Significance

This work bridges a critical gap in distributed control by enabling secure dynamic consensus without sacrificing performance. It is particularly significant for:

• Critical Infrastructure: Protecting the operational data of microgrids and BESS from market manipulation or cyber-physical attacks.
• Scalability: Offering a computationally lightweight solution suitable for large-scale networks where encryption is impractical.
• Theoretical Rigor: Providing a formal proof that derivative privacy is achievable in dynamic consensus, expanding the scope of secure multi-agent control beyond static settings.