Sachin Shivakumar

Noah Rhodes, Sachin Shivakumar, Luke S. Baker et al.

We examine market outcomes in energy transport networks with a focus on gas-fired generators, which are producers in a wholesale electricity market and consumers in the natural gas market. Market administrators monitor bids to determine whether a participant wields market power to manipulate the price of energy, reserves, or financial transmission rights. If economic or physical withholding of generation from the market is detected, mitigation is imposed by replacing excessive bids with reference level bids to prevent artificial supply shortages. We review market monitoring processes in the power grid, and present scenarios in small interpretable test networks to show how gas-fired generators can bid in the gas market to alter outcomes in a power market. We develop a framework based on DC optimal power flow (OPF) and steady-state optimal gas flow (OGF) formulations to represent two interacting markets with structured exchange of price and quantity bids. We formulate optimization-based methods to identify market power in a power grid, as well as to identify market conditions that indicate market power being exerted by a generator using gas market bids.

Tristan Thomas, Sachin Shivakumar, Javad Mohammadpour Velni

Impulse-to-peak response (I2P) analysis for state-space ordinary differential equation (ODE) systems is a well-studied classical problem. However, the techniques employed for I2P optimal control of ODEs have not been extended to partial differential equation (PDE) systems due to the lack of a universal transfer function and state-space representation. Recently, however, partial integral equation (PIE) representation was proposed as the desired state-space representation of a PDE, and Lyapunov stability theory was used to solve various control problems, such as stability and optimal ${H}_\infty$ control. In this work, we utilize this PIE framework, and associated Lyapunov techniques, to formulate the I2P response analysis problem as a solvable convex optimization and obtain provable bounds for the I2P-norm of linear PDEs. Moreover, by establishing strong duality between primal and dual formulations of the optimization problem, we develop a constructive method for I2P optimal state-feedback control of PDEs and demonstrate the effectiveness of the method on various examples.

Alexis M. H. Teter, Wenqing Wang, Sachin Shivakumar et al.

For a controllable linear time-varying (LTV) pair $(\boldsymbol{A}_t,\boldsymbol{B}_t)$ and $\boldsymbol{Q}_{t}$ positive semidefinite, we derive the Markov kernel for the Itô diffusion ${\mathrm{d}}\boldsymbol{x}_{t}=\boldsymbol{A}_{t}\boldsymbol{x}_t {\mathrm{d}} t + \sqrt{2}\boldsymbol{B}_{t}{\mathrm{d}}\boldsymbol{w}_{t}$ with an accompanying killing of probability mass at rate $\frac{1}{2}\boldsymbol{x}^{\top}\boldsymbol{Q}_{t}\boldsymbol{x}$. This Markov kernel is the Green's function for an associated linear reaction-advection-diffusion partial differential equation. Our result generalizes the recently derived kernel for the special case $\left(\boldsymbol{A}_t,\boldsymbol{B}_t\right)=\left(\boldsymbol{0},\boldsymbol{I}\right)$, and depends on the solution of an associated Riccati matrix ODE. A consequence of this result is that the linear quadratic non-Gaussian Schrödinger bridge is exactly solvable. This means that the problem of steering a controlled LTV diffusion from a given non-Gaussian distribution to another over a fixed deadline while minimizing an expected quadratic cost can be solved using dynamic Sinkhorn recursions performed with the derived kernel. Our derivation for the $\left(\boldsymbol{A}_t,\boldsymbol{B}_t,\boldsymbol{Q}_t\right)$-parametrized kernel pursues a new idea that relies on finding a state-time dependent distance-like functional given by the solution of a deterministic optimal control problem. This technique breaks away from existing methods, such as generalizing Hermite polynomials or Weyl calculus, which have seen limited success in the reaction-diffusion context. Our technique uncovers a new connection between Markov kernels, distances, and optimal control. This connection is of interest beyond its immediate application in solving the linear quadratic Schrödinger bridge problem.

Sachin Shivakumar, Georgiy A. Bondar, Gabriel Khan et al.

For a given ground cost, approximating the Monge optimal transport map that pushes forward a given probability measure onto another has become a staple in several modern machine learning algorithms. The fourth-order Ma-Trudinger-Wang (MTW) tensor associated with this ground cost function provides a notion of curvature in optimal transport. The non-negativity of this tensor plays a crucial role for establishing continuity for the Monge optimal transport map. It is, however, generally difficult to analytically verify this condition for any given ground cost. To expand the class of cost functions for which MTW non-negativity can be verified, we propose a provably correct computational approach which provides certificates of non-negativity for the MTW tensor using Sum-of-Squares (SOS) programming. We further show that our SOS technique can also be used to compute an inner approximation of the region where MTW non-negativity holds. We apply our proposed SOS programming method to several practical ground cost functions to approximate the regions of regularity of their corresponding optimal transport maps.