19 research outputs found
Network Synchronization and Control Based on Inverse Optimality : A Study of Inverter-Based Power Generation
This thesis dwells upon the synthesis of system-theoretical tools to understand and control the behavior of nonlinear networked systems. This work is at the crossroads of three topics: synchronization in coupled high-order oscillators, inverse optimal control and the application of inverter-based power systems. The control and stability of power systems leverages the theoretical results obtained for synchronization in coupled high-order oscillators and inverse optimal control.First, we study the dynamics of coupled high-order nonlinear oscillators. These are characterized by their rotational invariance, meaning that their dynamics remain unchanged following a static shift of their angles. We provide sufficient conditions for local frequency synchronization based on both direct, indirect Lyapunov methods and center manifold theory. Second, we study inverse optimal control problems, embedded in networked settings. In this framework, we depart from a given stabilizing control law, with an associated control Lyapunov function and reverse engineer the cost functional to guarantee the optimality of the controller. In this way, inverse optimal control generates a whole family of optimal controllers corresponding to different cost functions. This provides analytically explicit and numerically feasible solutions in closed-form. This approach circumvents the complexity of solving partial differential equations descending from dynamic programming and Bellman's principle of optimality. We show this to be the case also in the presence of disturbances in the dynamics and the cost. In networks, the controller obtained from inverse optimal control has a topological structure (e.g., it is distributed) and thus feasible for implementation. The tuning is analogous to that of linear quadratic regulators.Third, motivated by the pressing changes witnessed by the electrical grid toward renewable energy generation, we consider power system stability and control as the main application of this thesis. In particular, we apply our theoretical findings to study a network of power electronic inverters. We first propose a controller we term the matching controller, a control strategy that, based on DC voltage measurements, endows the inverters with an oscillatory behavior at a common desired frequency. In closed-loop with the matching control, inverters can be considered as nonlinear oscillators. Our study of the dynamics of nonlinear oscillator network provides feasible physical conditions that ask for damping on DC- and AC-side of each converter, that are sufficient for system-wide frequency synchronization.Furthermore, we showcase the usefulness of inverse optimal control for inverter-based generation at two different settings to synthesize robust angle controllers with respect to common disturbances in the grid and provable stability guarantees. All the controllers proposed in this thesis, provide the electrical grid with important services, namely power support whenever needed, as well as power sharing among all inverters
Distributed learning for optimal allocation of synchronous and converter-based generation
Motivated by the penetration of converter-based generation into the
electrical grid, we revisit the classical log-linear learning algorithm for
optimal allocation {of synchronous machines and converters} for mixed power
generation. The objective is to assign to each generator unit a type (either
synchronous machine or DC/AC converter in closed-loop with droop control),
while minimizing the steady state angle deviation relative to an optimum
induced by unknown optimal configuration of synchronous and DC/AC
converter-based generation. Additionally, we study the robustness of the
learning algorithm against a uniform drop in the line susceptances and with
respect to a well-defined feasibility region describing admissible power
deviations. We show guaranteed probabilistic convergence to maximizers of the
perturbed potential function with feasible power flows and demonstrate our
theoretical findings via simulative examples of power network with six
generation units.Comment: 7 pages, 3 figure
On cost design in applications of optimal control
A new approach to feedback control design based on optimal control is
proposed. Instead of expensive computations of the value function for different
penalties on the states and inputs, we use a control Lyapunov function that
amounts to be a value function of an optimal control problem with suitable cost
design and then study combinations of input and state penalty that are
compatible with this value function. This drastically simplifies the role of
the Hamilton-Jacobi-Bellman equation, since it is no longer a partial
differential equation to be solved, but an algebraic relationship between
different terms of the cost. The paper illustrates this idea in different
examples, including control and optimal control of
coupled oscillators.Comment: 6 pages, 4 figure
Fully decentralized conditions for local convergence of DC/AC converter network based on matching control
We investigate local convergence of identical DC/AC converters interconnected
via identical resistive and inductive lines towards a synchronous equilibrium
manifold. We exploit the symmetry of the resulting vector field and develop a
Lyapunov-based framework, in which we measure the distance of the solutions of
the nonlinear power system model to the equilibrium manifold by analyzing the
evolution of their tangent vectors. We derive sufficient and fully
decentralized conditions to characterize the equilibria of interest, and
provide an estimate of their region of contraction. We provide ways to satisfy
these conditions and illustrate our results based on numerical simulations of a
two-converter benchmark.Comment: 6 page
Steady state characterization and frequency synchronization of a multi-converter power system on high-order manifolds
We investigate the stability properties of a multi-converter power system
model, defined on high-order manifolds than the circle. For this, we identify
its symmetry (i.e., rotational invariance) generated by a static angle shift
and rotation of AC signals and define a suitable equivalence class for the
quotient space. Based on its Jacobian matrix, we characterize the quotient
stable steady states, primarily determined by their steady state angles and DC
power input. We show that local contraction is achieved on a well-defined
region of the space, based on a differential Lyapunov framework and Finsler
distance measure. We demonstrate our results based on a numerical example
involving two test cases consisting of two and three identical DC/AC converter
system.Comment: 15 pages, 8 figure
Leveraging second-order information for tuning of inverse optimal controllers
We leverage second-order information for tuning of inverse optimal
controllers for a class of discrete-time nonlinear input-affine systems. For
this, we select the input penalty matrix, representing a tuning knob, to yield
the Hessian of the Lyapunov function of the closed-loop dynamics. This draws a
link between second-order methods known for their high speed of convergence and
the tuning of inverse optimal stabilizing controllers to achieve a fast decay
of the closed-loop trajectories towards a steady state. In particular, we
ensure quadratic convergence, a feat that is otherwise not achieved with a
constant input penalty matrix. To balance trade-offs, we suggest a practical
implementation of the Hessian and validate this numerically on a network of
phase-coupled oscillators that represent voltage source controlled power
inverters.Comment: 6 pages, 3 figure
Inverse optimal control for angle stabilization in converters-based generation
In inverse optimal control, an optimal controller is synthesized with respect
to a meaningful, a posteriori, defined cost functional. Our work illustrates
the usefulness of this approach in the control of converter-based power systems
and networked systems in general, and thereby in designing controllers with
topological structure and known optimality properties. In particular, we design
an inverse optimal feedback controller that stabilizes the phase angles of
voltage source-controlled DC/AC converters at an induced steady state with {\em
zero} frequency error. The distributed angular droop controller yields active
power to angle droop behavior at steady state. Moreover, we suggest a practical
implementation of the controller and corroborate our results through
simulations on a three-converter system and a numerical comparison with
standard frequency droop control.Comment: 8 pages, 5 figure
Performance analysis and optimization of power systems with spatially correlated noise
Based on stochastic differential equations (SDEs), we analyse the overall
performance of heterogeneous power systems network, subject to spatially
distributed and correlated noise with random initial conditions. We determine
bounds on the H_2 norm of the heterogeneous system based on a closed-form of
the norm of the homogeneous power system. Then, we formulate possible scenarios
for performance optimization and link these to applications for network design
and control problems in power systems. Our results are corroborated by
numerical simulations from Kundur's four-machine two-area network after
adaption to our setup.Comment: 6 pages, 3 figure
Frequency synchronization of a high-order multi-converter system
We investigate the stability properties of a multi-converter power system model, defined on a high-order manifold. For this, we identify its symmetry (i.e., rotational invariance) generated by a static angle shift and rotation of AC signals. We characterize the steady state set, primarily determined by the steady state angles and DC power input. Based on eigenvalue conditions of its Jacobian matrix, we show asymptotic stability of the multi-converter system in a neighborhood of the synchronous steady state set by applying the center manifold theory. We guarantee the eigenvalue conditions via an explicit approach. Finally, we demonstrate our results based on a numerical example involving a network of identical DC/AC converter systems