Ashish Cherukuri

Distributed Generator Coordination for Initialization and Anytime Optimization in Economic Dispatch

This paper considers the economic dispatch problem for a group of generator units communicating over an arbitrary weight-balanced digraph. The objective of the individual units is to collectively generate power to satisfy a certain load while minimizing the total generation cost, which corresponds to the sum of individual arbitrary convex functions. We propose a class of distributed Laplacian-gradient dynamics that are guaranteed to asymptotically find the solution to the economic dispatch problem with and without generator constraints. The proposed coordination algorithms are anytime, meaning that its trajectories are feasible solutions at any time before convergence, and they become better solutions as time elapses. In addition, we design the provably correct determine feasible allocation strategy that handles generator initialization and the addition and deletion of units via a message passing routine over a spanning tree of the network. Our technical approach combines notions and tools from algebraic graph theory, distributed algorithms, nonsmooth analysis, set-valued dynamical systems, and penalty functions. Simulations illustrate our results.

Asymptotic convergence of constrained primal–dual dynamics

This paper studies the asymptotic convergence properties of the primal-dual dynamics designed for solving constrained concave optimization problems using classical notions from stability analysis. We motivate the need for this study by providing an example that rules out the possibility of employing the invariance principle for hybrid automata to study asymptotic convergence. We understand the solutions of the primal-dual dynamics in the Caratheodory sense and characterize their existence, uniqueness, and continuity with respect to the initial condition. We use the invariance principle for discontinuous Caratheodory systems to establish that the primal-dual optimizers are globally asymptotically stable under the primal-dual dynamics and that each solution of the dynamics converges to an optimizer.

Initialization-free distributed coordination for economic dispatch under varying loads and generator commitment

This paper considers the economic dispatch problem for a network of power generating units communicating over a strongly connected, weight-balanced digraph. The collective aim is to meet a power demand while respecting individual generator constraints and minimizing the total generation cost. In power networks, this problem is also referred to as tertiary control. We design a distributed coordination algorithm consisting of two interconnected dynamical systems. One block uses dynamic average consensus to estimate the evolving mismatch in load satisfaction given the generation levels of the units. The other block adjusts the generation levels based on the optimization objective and the estimate of the load mismatch. Our convergence analysis shows that the resulting strategy provably converges to the solution of the dispatch problem starting from any initial power allocation, and therefore does not require any specific procedure for initialization. We also characterize the algorithm robustness properties against the addition and deletion of units (capturing scenarios with intermittent power generation) and its ability to track time-varying loads. Our technical approach employs a novel refinement of the LaSalle Invariance Principle for differential inclusions, that we also establish and is of independent interest. Several simulations illustrate our results.

Saddle-Point Dynamics: Conditions for Asymptotic Stability of Saddle Points

This paper considers continuously differentiable functions of two vector variables that have (possibly a continuum of) min-max saddle points. We study the asymptotic convergence properties of the associated saddle-point dynamics (gradient-descent in the first variable and gradient-ascent in the second one). We identify a suite of complementary conditions under which the set of saddle points is asymptotically stable under the saddle-point dynamics. Our first set of results is based on the convexity-concavity of the function defining the saddle-point dynamics to establish the convergence guarantees. For functions that do not enjoy this feature, our second set of results relies on properties of the linearization of the dynamics, the function along the proximal normals to the saddle set, and the linearity of the function in one variable. We also provide global versions of the asymptotic convergence results. Various examples illustrate our discussion.

The Role of Convexity in Saddle-Point Dynamics: Lyapunov Function and Robustness

This paper studies the projected saddle-point dynamics associated to a convex–concave function, which we term saddle function. The dynamics consists of gradient descent of the saddle function in variables corresponding to convexity and (projected) gradient ascent in variables corresponding to concavity. We examine the role that the local and/or global nature of the convexity–concavity properties of the saddle function plays in guaranteeing convergence and robustness of the dynamics. Under the assumption that the saddle function is twice continuously differentiable, we provide a novel characterization of the omega-limit set of the trajectories of this dynamics in terms of the diagonal blocks of the Hessian. Using this characterization, we establish global asymptotic convergence of the dynamics under local strong convexity–concavity of the saddle function. When strong convexity–concavity holds globally, we establish three results. First, we identify a Lyapunov function (that decreases strictly along the trajectory) for the projected saddle-point dynamics when the saddle function corresponds to the Lagrangian of a general constrained convex optimization problem. Second, for the particular case when the saddle function is the Lagrangian of an equality-constrained optimization problem, we show input-to-state stability (ISS) of the saddle-point dynamics by providing an ISS Lyapunov function. Third, we use the latter result to design an opportunistic state-triggered implementation of the dynamics. Various examples illustrate our results.

Distributed, anytime optimization in power-generator networks for economic dispatch

This paper considers the economic dispatch problem for a group of power generating units communicating over an arbitrary strongly connected, weight-balanced digraph. The goal of the group is to collectively meet a specified load while respecting individual generation bounds and minimizing the total generation cost, which corresponds to the sum of individual arbitrary convex functions. We introduce a distributed coordination algorithm, termed Laplacian-set-valued dynamics, and establish its asymptotic convergence to the solutions of the economic dispatch problem. In addition, we show that the algorithm is anytime, meaning that its executions are feasible solutions at all times and the total cost monotonically decreases as time elapses. The technical approach combines notions and tools from algebraic graph theory, nonsmooth analysis, set-valued dynamical systems, and penalty functions. Several simulations illustrate our results.

Asymptotic stability of saddle points under the saddle-point dynamics

This paper considers continuously differentiable functions of two vector variables that have (possibly a continuum of) min-max saddle points. We study the asymptotic convergence properties of the associated saddle-point dynamics (gradient-descent in the first variable and gradient-ascent in the second one). We identify a suite of complementary conditions under which the set of saddle points is asymptotically stable under the saddle-point dynamics. Our first set of results is based on the convexity-concavity of the function defining the saddle-point dynamics to establish the convergence guarantees. For functions that do not enjoy this feature, our second set of results relies on properties of the linearization of the dynamics and the function along the proximal normals to the saddle set. We also provide global versions of the asymptotic convergence results. Various examples illustrate our discussion.

Distributed coordination for economic dispatch with varying load and generator commitment

This paper considers the economic dispatch problem for a group of power generating units. The collective aim is to meet a power demand while respecting individual generator constraints and minimizing the total generation cost. Assuming that the units communicate over a strongly connected, weight-balanced digraph, we propose a distributed coordination algorithm that provably converges to the solution of the dispatch problem starting from any initial power allocation. Additionally, we establish that the proposed strategy is robust against mismatch between load and total generation (and thus able to handle time-varying loads), and against intermittent generation commitment, a plausible scenario due to the integration of renewable energy sources into the grid. Our technical approach uses notions and tools from algebraic graph theory, nonsmooth analysis, set-valued dynamical systems, and dynamic average consensus. Several simulations illustrate our results.

Stable Interconnection of Continuous-Time Price-Bidding Mechanisms with Power Network Dynamics

We study price-based bidding mechanisms in power networks fo r real-time dispatch and frequency regulation. On the market side, we consider the interaction between the independent system operator (ISO) and a group of generators involved in a Bertrand game of competition. The generators seek to maximize their individual profit while the ISO aims to solve the economic dispatch problem and to regulate the frequency. Since the generators are strategic and do not share their cost functions, the ISO engages the generators in a continuous-time price-based bidding process. This results in a coupling between the ISO-generator dynamics and swing dynamics of the network. We analyze its stability, establishing frequency regulation and the convergence to the efficient Nash equilibrium and the optimal generation levels. Simulation illustrate our results.

Distributed dynamic economic dispatch of power generators with storage

This paper considers the dynamic economic dispatch problem for a group of generators with storage that communicate over a weight-balanced strongly connected digraph. The objective of the generators is to collectively meet a certain load profile, specified over a finite time horizon, while minimizing the aggregate cost. At each time slot, each generator decides on the amount of generated power and the amount sent to/drawn from the storage unit. The amount injected into the grid by each generator to satisfy the load is equal to the difference between the generated and stored powers. Additional constraints on the generators include bounds on the amount of generated power, ramp constraints on the difference in generation across successive time slots, and bounds on the amount of power in storage. We synthesize a provably-correct distributed algorithm that solves the resulting finite-horizon optimization problem starting from any initial condition. Our design consists of two interconnected dynamical systems, one estimating the mismatch in the injection and the total load at each time slot, and another using this estimate as a feedback to reduce the mismatch and optimize the total cost of generation, while meeting the constraints.