[Paper Review] Energy-based Stabilization of Network Flows in Multi-machine Power Systems

Catalin Arghir, Florian Dörfler|arXiv (Cornell University)|Apr 25, 2018

Control and Stability of Dynamical Systems11 references3 citations

TL;DR

This paper proposes an energy-based control framework for multi-machine power systems that stabilizes network flows by achieving angle synchronization and consensus in angular velocity. Using a port-Hamiltonian model, it derives a shifted Hamiltonian and impedance-based network flow map to design a hierarchical controller that stabilizes excitation current first, then synchronizes rotor angles via a convex potential function, ensuring global convergence to optimal network flow configurations.

ABSTRACT

This paper considers the network flow stabilization problem in power systems and adopts an output regulation viewpoint. Building upon the structure of a heterogeneous port-Hamiltonian model, we integrate network aspects and develop a systematic control design procedure. First, the passive output is selected to encode two objectives: consensus in angular velocity and constant excitation current. Second, the non-Euclidean nature of the angle variable reveals the geometry of a suitable target set, which is compact and attractive for the zero dynamics. On this set, circuit-theoretic aspects come into play, giving rise to a network potential function which relates the electrical circuit variables to the machine rotor angles. As it turns out, this energy function is convex in the edge variables, concave in the node variables and, most importantly, can be optimized via an intrinsic gradient flow, with its global minimum corresponding to angle synchronization. The third step consists of explicitly deriving the steady-state-inducing control action by further refining this sequence of control-invariant sets. Analogously to solving the so called regulator equations, we obtain an impedance-based network flow map leading to novel error coordinates and a shifted energy function. The final step amounts to decoupling the rotor current dynamics via feedback-linearziation resulting in a cascade which is used to construct an energy-based controller hierarchically.

Motivation & Objective

• To explicitly characterize and stabilize the steady-state operating point in multi-machine power systems under high-order dynamics.
• To overcome the non-integrability and nonlinearity challenges in rotor angle dynamics by avoiding rotating frames and using a stationary frame formulation.
• To achieve both frequency consensus and optimal network flow (angle synchronization) through a unified energy-based control design.
• To decompose the control action into dissipation-balancing and gradient-driven components using a network potential function.
• To develop a hierarchical controller that first stabilizes excitation currents and then enforces angle synchronization via intrinsic gradient flow on the torus.

Proposed method

• Formulates a heterogeneous multi-machine power system as a port-Hamiltonian system in a stationary frame, avoiding rotating frame singularities.
• Defines a passive output that encodes consensus in angular velocity and constant excitation current, leading to a zero dynamics set on the n-torus.
• Introduces a network potential function that is convex in edge variables and concave in node variables, with its global minimum corresponding to angle synchronization.
• Derives a shifted energy function and error coordinates via solution of regulator equations, enabling stabilization of the target set.
• Constructs a controller using feedback linearization of rotor current dynamics, decoupling them from the main system dynamics.
• Employs a gradient flow on the torus to minimize the synchronization potential, effectively acting as a second-order gradient descent on angles.

Experimental results

Research questions

• RQ1 How can a steady-state operating point be explicitly characterized in a multi-machine power system with high-order dynamics and non-integrable angle variables?
• RQ2 What control structure enables simultaneous stabilization of excitation current, frequency consensus, and angle synchronization?
• RQ3 How can the non-Euclidean geometry of rotor angles be leveraged to define a compact, attractive target set for stabilization?
• RQ4 What role does the network potential function play in encoding optimal power flow objectives such as inductor current minimization and capacitor voltage maximization?
• RQ5 How can the energy function be modified via coordinate transformation to enable hierarchical controller design?

Key findings

• The network potential function is convex in line currents and concave in rotor angles, with its global minimum corresponding to complete angle synchronization.
• The controller successfully stabilizes the excitation current and achieves frequency consensus, with all generators synchronizing to a common angular velocity.
• Numerical simulations confirm that the system converges to a steady state with zero relative angle differences, indicating successful angle synchronization.
• The controller exhibits bounded solutions and meets output regulation specifications under both zero and non-zero initial conditions.
• The transient behavior of generator 2 shows that rotor current dynamics can induce angular velocity overshoot, highlighting the need for fast torque actuation.
• The framework enables a hierarchical control structure where excitation current is stabilized first, followed by angle synchronization through a projected gradient flow on the torus.