Dynamic Optimal Power Flow in Microgrids using the Alternating Direction Method of Multipliers

Smart devices, storage and other distributed technologies have the potential to greatly improve the utilisation of network infrastructure and renewable generation. Decentralised control of these technologies overcomes many scalability and privacy concerns, but in general still requires the underlying problem to be convex in order to guarantee convergence to a global optimum. Considering that AC power flows are non-convex in nature, and the operation of household devices often requires discrete decisions, there has been uncertainty surrounding the use of distributed methods in a realistic setting. This paper extends prior work on the alternating direction method of multipliers (ADMM) for solving the dynamic optimal power flow (D-OPF) problem. We utilise more realistic line and load models, and introduce a two-stage approach to managing discrete decisions and uncertainty. Our experiments on a suburb-sized microgrid show that this approach provides near optimal results, in a time that is fast enough for receding horizon control. This work brings distributed control of smart-grid technologies closer to reality.

💡 Research Summary

This paper tackles the emerging challenge of operating power systems that are increasingly populated with smart devices, distributed generation, storage, and electric vehicles. Traditional market operations solve unit‑commitment and optimal power flow (OPF) problems assuming inflexible loads and a centralized architecture. The authors argue that such assumptions break down when household‑level assets, which often involve discrete decisions (e.g., appliance start times) and stochastic behavior (e.g., solar output), must be coordinated.

To address this, the authors formulate a dynamic optimal power flow (D‑OPF) problem over a receding‑horizon of n time steps. The network is abstracted into components, terminals, and connections. Each component (bus, line, generator, shiftable load, battery) carries its own decision variables, cost function, and local constraints. Terminals carry four time‑indexed variables – real power p, reactive power q, voltage magnitude v, and phase angle θ – and connections enforce power balance and voltage/angle equality. This formulation deliberately duplicates variables at each terminal to enable decomposition.

The core solution method is the Alternating Direction Method of Multipliers (ADMM). The augmented Lagrangian penalizes connection constraints, and ADMM iteratively optimizes two disjoint sets of components (Phase 1 and Phase 2) while updating dual variables that correspond to locational marginal prices. By placing all buses in Phase 2 and all other components in Phase 1, intra‑phase connections disappear, allowing each component to solve a small, independent sub‑problem. Buses admit an analytical solution; other components are solved with Gurobi (for mixed‑integer linear/quadratically constrained formulations) or Ipopt (for general nonlinear sub‑problems).

A key contribution is a two‑stage handling of discrete decisions and uncertainty. In Stage 1 the authors solve a continuous relaxation of the D‑OPF, obtaining dual variables that capture price signals. In Stage 2 these price signals are fixed, and a mixed‑integer program determines the binary start‑up variables of shiftable loads and the on/off status of batteries or other discrete devices. This sequential approach preserves the convergence properties of ADMM while still delivering high‑quality integer solutions.

The methodology is evaluated on a modified 70‑bus, 11 kV distribution network that mimics a suburban microgrid. The original static PQ loads are replaced by detailed household models comprising an uncontrollable background draw, two shiftable appliances, optional photovoltaic generation, and optional battery storage. All houses are directly attached to the 11 kV buses (low‑voltage details are abstracted). The implementation is written in C++ with CasADi for automatic differentiation, Gurobi for MILP/QCQP, and Ipopt for nonlinear sub‑problems. Although the ADMM loop is executed sequentially, the authors time the longest component per iteration to estimate the wall‑clock time of a fully distributed deployment.

Results show that the algorithm converges within a few dozen seconds on the test hardware (dual 6‑core AMD Opteron servers). The total operational cost is within 1–2 % of a centralized, globally optimal solution obtained by a monolithic solver, confirming near‑optimality. Voltage and line‑thermal limits are respected throughout the horizon, and the method is fast enough to be embedded in a receding‑horizon control (model predictive control) framework, enabling real‑time coordination of distributed resources.

The paper’s contributions can be summarized as:

1. Extending ADMM‑based D‑OPF to incorporate full AC line physics and realistic household models, moving beyond the linearized or convex approximations common in prior work.
2. Introducing a two‑stage pricing and integer‑decision mechanism that handles discrete appliance scheduling and uncertainty while retaining ADMM convergence guarantees.
3. Demonstrating on a realistic suburban microgrid that the approach yields near‑optimal solutions with computational times suitable for online control.

Limitations include reliance on convex relaxations for the AC power‑flow equations (which may be inaccurate under heavily loaded or low‑voltage conditions) and the need for a centralized initialization of dual variables. Future work could explore tighter non‑convex ADMM convergence theory, hierarchical multi‑level decomposition (e.g., distribution‑level and feeder‑level coordination), and integration of stochastic forecasts for renewable generation.

Overall, the study brings distributed optimization of smart‑grid resources a step closer to practical deployment by reconciling realistic network physics, discrete household decisions, and the scalability offered by ADMM.