Making nonlinear inverter dynamics linear enough to control.

Four threads, one testbed

Everything below runs against the same islanded AC microgrid model, so the methods can actually be compared rather than just described.

• Thesis core

  Koopman–EDMDc model predictive control

  A mode-aware lifting pipeline for droop-controlled grid-forming inverters: sine and cosine embeddings of the power angle in grid-tied mode, angle observables dropped when islanded. Feeding a velocity-form MPC with a Hildreth QP gives offset-free tracking under input and rate constraints. MATLAB · Simulink · EDMDc · constrained QP

  Plant

  Islanded four-bus AC microgrid, droop-controlled grid-forming inverters

  Identification

  Extended DMD with control, mode-dependent observable dictionary

  Controller

  Velocity-form MPC, Hildreth QP, offset-free via integrator augmentation

  Constraints

  Secondary set-point magnitude and ramp rate

  The dictionary has to change with the operating mode. Grid-tied, the power angle carries real information and the sine/cosine pair earns its place. Islanded, there is no stiff reference to hold an angle against — keeping those observables leaves the regression badly conditioned rather than merely redundant, so they get dropped.

  The part that took the longest was not the lift. It was discovering that the identified model and the Simulink plant disagreed about actuation authority: the plant moves frequency by roughly 12.45 mHz per kW of secondary set-point, the identified model predicts about 12.97. A four percent gain error is small enough to look like noise and large enough to leave a standing offset, which is what the velocity form plus an integrating disturbance state is there to absorb.

  The plot at the top of this page is a reduced version of the same argument, small enough to run in a browser.

• Baseline

  Multi-agent RL for secondary control

  A MADDPG implementation of distributed secondary control, replacing a centralised DDPG critic with centralised training and decentralised execution. The interesting case isn't nominal performance — it's what happens when the communication link between agents drops. Python · PyTorch · Gymnasium · pandapower

  Algorithm

  MADDPG — per-inverter actor, critic sees the joint observation during training only

  Environment

  Gymnasium wrapper over the same islanded testbed

  Reward

  Frequency and voltage regulation with an effort penalty

  Role

  Learning-based baseline for the Koopman MPC comparison

  A single centralised critic is the easy version of this problem and not the one a real microgrid has. Centralised training with decentralised execution keeps the deployment story honest: at run time each inverter acts on what it can actually measure locally.

  The comparison worth making is the degraded one. Nominal performance between a tuned MPC and a converged policy is close enough to be an argument about tuning effort. Drop the communication link between agents and the two approaches fail in genuinely different ways — the MPC degrades toward its constraint set, the learned policy degrades toward whatever it saw in training.

• Comparison track

  Physics-informed neural control

  Two strands: a PINC-style distributed secondary voltage controller for a low-voltage inverter microgrid, and a physics-informed predictor driving MPC on a DC–DC buck converter feeding a constant power load. PINN · PINC · averaged CCM model

  Strand one

  PINC distributed secondary voltage control, low-voltage inverter microgrid

  Strand two

  PINN predictor inside an MPC loop, buck converter with constant power load

  Physics

  Averaged continuous-conduction-mode model as the residual term

  Training

  DAgger-style aggregation against a supervising controller

  A constant power load is the interesting part of the buck problem: its incremental resistance is negative, so the small-signal system is destabilising exactly where a naive linear predictor is most confident. Putting the averaged model in as a physics residual is what stops the network from learning a comfortable, wrong linearisation.

  Two findings from the MPC loop worth recording. Asking the controller to track two outputs at once collapsed the bus rather than trading the two off — the single-output formulation is the one that holds. And the achievable step excursion is set by the LC filter, not by the controller, which is a useful thing to establish before blaming a predictor for tracking error it was never able to remove.

• Side build

  CircuitLab, a browser circuit simulator

  Modified nodal analysis with companion models, running entirely client-side, with a phosphor-style oscilloscope readout. Verified against buck, boost, RC, RLC, and half-wave rectifier circuits. React · MNA · numerical integration

  Formulation

  Modified nodal analysis, companion models for reactive elements

  Nonlinearity

  Newton iteration per time step for diodes and switches

  Runtime

  Entirely client-side — no solver on a server anywhere

  Verified against

  Buck, boost, RC, RLC, half-wave rectifier

  Companion models turn each capacitor and inductor into a conductance in parallel with a current source that carries the previous step's state, so every time step reduces to the same linear solve. The switching elements are where it stops being tidy: a hard diode model makes the Newton loop chatter near the conduction boundary.

  The verification set is deliberately boring. Circuits with closed-form or well-known answers are the only ones that tell you the integrator is right, and a rectifier is the cheapest test that exercises the nonlinear path at all.

Notes from the middle of the work

• 23 Jul 2026 What a grid-forming inverter is actually doing
• 20 Jul 2026 The affine lift that fixed my Koopman model

Everything →