Operator-Level Disaggregation in Energy Systems

• Operator-level disaggregation is a method that decomposes aggregate energy measurements into individual agent contributions while preserving privacy and scalability.
• The approach leverages alternate projections and secure multiparty computation to derive agent-level allocations without exposing private constraints.
• Analysis shows that polyhedral cuts and distributed computations enable efficient resource management in high-dimensional systems like smart grids and microgrids.

Operator-level disaggregation refers to a set of computational and algorithmic methods for decomposing aggregate measurements or system-level resource allocations into component-level (operator, appliance, agent, or nodal) contributions, with an emphasis on preserving privacy, scalability, and practical utility for large-scale systems such as smart grids, microgrids, and other multi-agent resource management contexts. In recent literature, the term “operator-level disaggregation” most often appears in the context of energy systems, especially for disaggregating aggregate electrical consumption or flexibility to assign and monitor individual profiles without exposing private constraints or behaviors.

1. Privacy-Preserving Disaggregation Algorithm

The principal methodology for operator-level disaggregation in large multi-agent systems is a distributed algorithm that jointly solves a global, possibly nonconvex, resource allocation (“master”) problem and a subordinate disaggregation subproblem. The process begins with the central operator solving for an aggregate allocation vector $p$ (e.g., a time series of power setpoints) subject to system-wide constraints. Subsequently, the disaggregation phase seeks agent-level profiles $x_n$ such that each $x_n$ obeys private, agent-specific constraints and collectively sum to the aggregate $p$ at each time index.

The computational engine underlying this approach is an alternate projections method (APM). APM iterates between:

• A projection $\mathcal{P}_X$ onto the product constraint set $X = \prod_n X_n$ (agent-local constraints: demand, individual bounds, etc.), computed locally by each agent.
• A projection $\mathcal{P}_Y$ onto the affine set $Y = \{ y \mid \sum_n y_n = p \}$ (enforcing the aggregate allocation at every time period).

A typical APM iteration can be written: $x^{(k+1)} = \mathcal{P}_X(y^{(k)})$

$y^{(k+1)} = \mathcal{P}_Y(x^{(k+1)})$

If the problem admits a feasible disaggregation, this process converges to a common point. If infeasibility is encountered (i.e., no $x_n$ exist that satisfy constraints and sum to $p$), the method converges to a periodic orbit (2-cycle), and an aggregate inequality (polyhedral cut) is generated according to Hoffman’s theorem. This cut is then added to the master problem as an additional constraint, and the process is repeated.

A representative cut takes the form: $\sum_{t \in T_0} p_t \leq A_{T_0}$ where $T_0$ and $A_{T_0}$ are selected based on the analysis of the fixed-point residuals.

2. Resource Allocation Problem Formulation

The resource allocation problem solved by the operator is formulated as: $$\begin{array}{ll} \text{minimize} & f(p) \ \text{subject to} & p \in P \ & x_n \in X_n \quad \forall n \ & \sum_n x_{n,t} = p_t \quad \forall t \in T \end{array}$$ where $f(p)$ is a (possibly nonconvex) global objective, $P$ describes system-level allocation constraints (such as power generation or grid import/export limits), each $X_n$ encodes private agent-level requirements and bounds (e.g., device-level flexibility, time-coupling, etc.), and $x_n$ represents the individual load or generation profile for agent $n$. The operator never receives $x_n$ or the detailed structure of $X_n$ (preserving privacy). Instead, only the sum constraint and the aggregate $p$ are communicated.

3. Privacy Mechanisms

Privacy preservation is accomplished via two mechanisms:

• Secure Multiparty Computation (SMC):

Each agent computes its own projection locally and exchanges random shares of its iterate with other agents. The aggregation protocol is a la Von Neumann, which is implemented locally by each agent, preserving her privacy needs. Globally, only the aggregate sum $S = \sum_n x_n$ is revealed; no agent’s specific $x_n$ is ever transmitted. The operator uses a dedicated SMC protocol (Algorithm SMCA) to compute and collect only the sum, ensuring non-disclosure of individual data.

• Aggregate Polyhedral Cuts:

When APM fails to achieve consensus, the aggregate (not individual) information from the 2-cycle or periodic orbit is used to construct a violated aggregate cut, which is then added to the feasible set of the master problem. Since the cut is defined only in terms of the sum profile (not any individuals’ data or constraints), individual privacy is not breached.

4. Aggregation and Disaggregation Workflow

The process can be led stepwise:

• The operator first solves a master problem $\min f(p)$ over the current approximation of feasible aggregates $P^{(s)} \subseteq P$, initially with no polyhedral cuts.
• For the computed $p$, agents execute the local APM-based disaggregation:
  • Each agent computes $\mathcal{P}_{X_n}(y_n^{(k)})$ on its own, using only its private $X_n$.
  • Aggregation to project onto $Y$ (sum constraint) is run via SMC, never revealing $x_n$.
• If the sum constraint can be met within tolerance, agent allocations are feasible, and an allocation $\{x_n\}$ is found.
• If not, a new aggregate cut is derived, uploaded to the master problem, and the process repeats.

This design ensures feasibility (identifies when aggregate allocation is not compatible with individual constraints) and privacy (neither $x_n$ nor $X_n$ are exposed to the operator or other agents at any step).

5. Scalability and Computational Properties

Theoretical and numerical analysis demonstrate:

• The number of APM iterations required for consensus or cycle detection is upper-bounded by a factor growing linearly with agent number $N$ (Theorem “BoundCvgRate”: if $$\|x^{(k)}-x^\infty\|_2 \leq \|x^{(0)} - x^\infty\|_2 \cdot (N T)^k$$ for contraction factor $N T < 1$).
• The master problem (number of aggregate cuts) does not grow rapidly with the number of agents; numerical results show the quantity added is nearly constant, suggesting the method can be used to solve the allocation problem in high dimension, while addressing privacy issues.
• Local computations per agent are independently parallelizable, with per-iteration cost $\mathcal{O}(T)$. Aggregate projection via SMC requires only a single sum per time index.
• For applied microgrid dispatch problems, the burden added by additional agents (e.g., households) to the total projection count is sublinear—leading to overall high practical efficiency even as system size increases.

6. Practical Implications

The approach yields several practical benefits for microgrid and energy system operators:

• Privacy Preservation: Consumer-level data such as appliance usage, personal constraints, or behavioral patterns remain undisclosed. All computations are performed via sums and aggregate inequalities.
• Efficient Resource Management: The operator enforces a global objective (e.g., cost minimization, system-level fairness, integration of renewable and storage resources) with guaranteed agent-level feasibility.
• Realistic Model Handling: The algorithm supports nonconvex, mixed-integer agent constraints (e.g., start-up costs, minimum up-time), so on/off loads and other operational complexities are naturally accommodated.
• Distributed and Scalable: By offloading computation to agents and leveraging SMC, the method is practical for real-world systems with thousands of agents and limited network bandwidth.
• Minimal Overhead: The number of required master problem iterations (aggregate cuts) is nearly unaffected by system size, making the method robust for growing systems.

A plausible implication is that such operator-level disaggregation algorithms can serve as a foundational layer for large-scale, privacy-sensitive resource allocation in distributed energy systems, with potential applicability to other domains requiring privacy-preserving aggregation and allocation.

7. Context and Significance

The privacy-preserving operator-level disaggregation paradigm contrasts with traditional approaches that require either disclosure of detailed agent constraints and behaviors (e.g., supervisory or centralized scheduling) or heavy communication overhead. Unlike approaches that approximate aggregate behavior by overlaying synthetic flexibility characterizations, the method here constructs feasibility cuts based directly on inability to find feasible disaggregation and provides strong privacy guarantees.

The framework is also adaptable to settings beyond microgrids—such as distributed scheduling, federated optimization, and other networked resource allocation problems—where agent privacy, feasibility, and scalability are critical requirements.

In summary, operator-level disaggregation, as implemented via distributed alternate projections, polyhedral cuts, and SMC, forms a mathematically rigorous and scalable foundation for integrating flexible loads and users into the operation of modern energy systems, achieving privacy, feasibility, and efficiency in high-dimensional, agent-rich environments (Jacquot et al., 2019).