Harsha Nagarajan

Harsha Nagarajan, Mowen Lu, Site Wang et al.

In this work, we develop an adaptive, multivariate partitioning algorithm for solving mixed-integer nonlinear programs (MINLP) with multi-linear terms to global optimality. This iterative algorithm primarily exploits the advantages of piecewise polyhedral relaxation approaches via disjunctive formulations to solve MINLPs to global optimality in contrast to the conventional spatial branch-and-bound approaches. In order to maintain relatively small-scale mixed-integer linear programs at every iteration of the algorithm, we adaptively partition the variable domains appearing in the multi-linear terms. We also provide proofs on convergence guarantees of the proposed algorithm to a global solution. Further, we discuss a few algorithmic enhancements based on the sequential bound-tightening procedure as a presolve step, where we observe the importance of solving piecewise relaxations compared to basic convex relaxations to speed-up the convergence of the algorithm to global optimality. We demonstrate the effectiveness of our disjunctive formulations and the algorithm on well-known benchmark problems (including Pooling and Blending instances) from MINLPLib and compare with state-of-the-art global optimization solvers. With this novel approach, we solve several large-scale instances which are, in some cases, intractable by the global optimization solver. We also shrink the best known optimality gap for one of the hard, generalized pooling problem instance.

Mowen Lu, Harsha Nagarajan, Emre Yamangil et al.

In recent years, there have been increasing concerns about how geomagnetic disturbances (GMDs) impact electrical power systems. Geomagnetically-induced currents (GICs) can saturate transformers, induce hot spot heating and increase reactive power losses. These effects can potentially cause catastrophic damage to transformers and severely impact the ability of a power system to deliver power. To address this problem, we develop a model of GIC impacts to power systems that includes 1) GIC thermal capacity of transformers as a function of normal Alternating Current (AC) and 2) reactive power losses as a function of GIC. We use this model to derive an optimization problem that protects power systems from GIC impacts through line switching, generator redispatch, and load shedding. We employ state-of-the-art convex relaxations of AC power flow equations to lower bound the objective. We demonstrate the approach on a modified RTS96 system and the UIUC 150-bus system and show that line switching is an effective means to mitigate GIC impacts. We also provide a sensitivity analysis of optimal switching decisions with respect to GMD direction.

Mowen Lu, Harsha Nagarajan, Russell Bent et al.

Since the alternating current optimal power flow (ACOPF) problem was introduced in 1962, developing efficient solution algorithms for the problem has been an active field of research. In recent years, there has been increasing interest in convex relaxations-based solution approaches that are often tight in practice. Based on these approaches, we develop tight piecewise convex relaxations with convex-hull representations, an adaptive, multivariate partitioning algorithm with bound tightening that progressively improves these relaxations and, given sufficient time, converges to the globally optimal solution. We illustrate the strengths of our algorithm using benchmark ACOPF test cases from the literature. Computational results show that our novel algorithm reduces the best-known optimality gaps for some hard ACOPF cases.

Harsha Nagarajan, Russell Bent, Pascal Van Hentenryck et al.

As illustrated in recent years (Superstorm Sandy, the Northeast Ice Storm of 1998, etc.), extreme weather events pose an enormous threat to the electric power transmission systems and the associated socio-economic systems that depend on reliable delivery of electric power. Besides inevitable malfunction of power grid components, deliberate malicious attacks can cause high risks to the service. These threats motivate the need for approaches and methods that improve the resilience of power systems. In this paper, we develop a model and tractable methods for optimizing the upgrade of transmission systems through a combination of hardening existing components, adding redundant lines, switches, generators, and FACTS and phase-shifting devices. While many of these controllable components are included in traditional design (expansion planning) problems, we uniquely assess their benefits from a resiliency point of view. More importantly, perhaps, we evaluate the suitability of using state-of-the-art AC power flow relaxations versus the common DC approximation in resilience improvement studies. The resiliency model and algorithms are tested on a modified version of the RTS-96 (single area) system.

Kaarthik Sundar, Harsha Nagarajan, Miles Lubin et al.

As renewable wind energy penetration rates continue to increase, one of the major challenges facing grid operators is the question of how to control transmission grids in a reliable and a cost-efficient manner. The stochastic nature of wind forces an alteration of traditional methods for solving day-ahead and look-ahead unit commitment and dispatch. In particular, uncontrollable wind generation increases the risk of random component failures. To address these questions, we present an N-1 Security and Chance-Constrained Unit Commitment (SCCUC) that includes the modeling of generation reserves that respond to wind fluctuations and tertiary reserves to account for single component outages. The basic formulation is reformulated as a mixed-integer second-order cone problem to limit the probability of failure. We develop three different algorithms to solve the problem to optimality and present a detailed case study on the IEEE RTS-96 single area system. The case study assesses the economic impacts due to contingencies and various degrees of wind power penetration into the system and also corroborates the effectiveness of the algorithms.

Kaarthik Sundar, Carleton Coffrin, Harsha Nagarajan et al.

This paper considers a probabilistic generalization of the $N$-$k$ failure-identification problem in power transmission networks, where the probability of failure of each component in the network is known a priori and the goal of the problem is to find a set of $k$ components that maximizes disruption to the system loads weighted by the probability of simultaneous failure of the $k$ components. The resulting problem is formulated as a bilevel mixed-integer nonlinear program. Convex relaxations, linear approximations, and heuristics are developed to obtain feasible solutions that are close to the optimum. A general cutting-plane algorithm is proposed to solve the convex relaxation and linear approximations of the $N$-$k$ problem. Extensive numerical results corroborate the effectiveness of the proposed algorithms on small-, medium-, and large-scale test instances, the test instances include the IEEE 14-bus system, the IEEE single-area and three-area RTS96 systems, the IEEE 118-bus system, the WECC 240-bus test system, the 1354-bus PEGASE system, and the 2383-bus Polish winter-peak test system.

Kaarthik Sundar, Harsha Nagarajan, Sidhant Misra et al.

This article develops a strengthened convex quadratic convex (QC) relaxation of the AC Optimal Power Flow (AC-OPF) problem and presents an optimization-based bound-tightening (OBBT) algorithm to compute tight, feasible bounds on the voltage magnitude variables for each bus and the phase angle difference variables for each branch in the network. Theoretical properties of the strengthened QC relaxation that show its dominance over the other variants of the QC relaxation studied in the literature are also derived. The effectiveness of the strengthened QC relaxation is corroborated via extensive numerical results on benchmark AC-OPF test networks. In particular, the results demonstrate that the proposed relaxation consistently provides the tightest variable bounds and optimality gaps with negligible impacts on runtime performance.

Deepjyoti Deka, Harsha Nagarajan, Scott Backhaus

The transient response of power grids to external disturbances influences their stable operation. This paper studies the effect of topology in linear time-invariant dynamics of different power grids. For a variety of objective functions, a unified framework based on $H_2$ norm is presented to analyze the robustness to ambient fluctuations. Such objectives include loss reduction, weighted consensus of phase angle deviations, oscillations in nodal frequency, and other graphical metrics. The framework is then used to study the problem of optimal topology design for robust control goals of different grids. For radial grids, the problem is shown as equivalent to the hard "optimum communication spanning tree" problem in graph theory and a combinatorial topology construction is presented with bounded approximation gap. Extended to loopy (meshed) grids, a greedy topology design algorithm is discussed. The performance of the topology design algorithms under multiple control objectives are presented on both loopy and radial test grids. Overall, this paper analyzes topology design algorithms on a broad class of control problems in power grid by exploring their combinatorial and graphical properties.

Sai Krishna Kanth Hari, Kaarthik Sundar, Harsha Nagarajan et al.

In recent years, microgrids, i.e., disconnected distribution systems, have received increasing interest from power system utilities to support the economic and resiliency posture of their systems. The economics of long distance transmission lines prevent many remote communities from connecting to bulk transmission systems and these communities rely on off-grid microgrid technology. Furthermore, communities that are connected to the bulk transmission system are investigating microgrid technologies that will support their ability to disconnect and operate independently during extreme events. In each of these cases, it is important to develop methodologies that support the capability to design and operate microgrids in the absence of transmission over long periods of time. Unfortunately, such planning problems tend to be computationally difficult to solve and those that are straightforward to solve often lack the modeling fidelity that inspires confidence in the results. To address these issues, we first develop a high fidelity model for design and operations of a microgrid that include component efficiencies, component operating limits, battery modeling, unit commitment, capacity expansion, and power flow physics; the resulting model is a mixed-integer quadratically-constrained quadratic program (MIQCQP). We then develop an iterative algorithm, referred to as the Model Predictive Control (MPC) algorithm, that allows us to solve the resulting MIQCQP. We show, through extensive computational experiments, that the MPC-based method can scale to problems that have a very long planning horizon and provide high quality solutions that lie within 5\% of optimal.

Harsha Nagarajan, Mowen Lu, Emre Yamangil et al.

In this work, we propose a two-stage approach to strengthen piecewise McCormick relaxations for mixed-integer nonlinear programs (MINLP) with multi-linear terms. In the first stage, we exploit Constraint Programing (CP) techniques to contract the variable bounds. In the second stage we partition the variables domains using a dynamic multivariate partitioning scheme. Instead of equally partitioning the domains of variables appearing in multi-linear terms, we construct sparser partitions yet tighter relax- ations by iteratively partitioning the variable domains in regions of interest. This approach decouples the number of partitions from the size of the variable domains, leads to a significant reduction in computation time, and limits the number of binary variables that are introduced by the partitioning. We demonstrate the performance of our algorithm on well-known benchmark problems from MINLPLIB and discuss the computational benefits of CP-based bound tightening procedures.

David M. Fobes, Harsha Nagarajan, Manuel Garcia et al.

The distribution system restoration (DSR) problem has received considerable attention over the last decade or more. Solutions to the DSR problem identify the best set or sequence of actions to perform on a distribution circuit to restore service after a disruption. The problem is challenging from a computational perspective, with engineering constraints specific to distribution systems, such as radial operations, that are difficult to effectively model. In this paper, we revisit the model for how specific loads are shed, energized and restored--and develop a formulation that more accurately models the requirements of load shedding, load energizing and restoration in distribution systems.

Md Umar Hashmi, Harsha Nagarajan, Dirk Van Hertem1

The increasing interconnection of power systems through AC and DC links enables energy storage units to access multiple electricity markets yet most existing arbitrage models remain limited to singlemarket participation This gap restricts understanding of the economic value and operational constraints associated with crossborder storage operation To address this an optimal multiregion energy storage arbitrage model is developed for a gridscale battery located at one end of an interconnector linking two distinct dayahead markets The formulation incorporates battery capacity and ramping limits converter and interconnector losses and marketspecific buying and selling prices Using disjunctive linearization of nonlinear terms this work exactly reformulates the multiregion energy arbitrage optimization as a mixedinteger linear programming problem The proposed formulation ensures that the battery either charges or discharges from all participating energy markets simultaneously at any given time Case studies using eight years of BelgianUK price data demonstrate that multiregion participation can increase arbitrage revenue by more than 40% compared to local energy arbitrage operation only while also highlighting the negative impact of interconnector congestion on achievable gains The results indicate that crossborder market access substantially enhances storage profitability while considering the cycle of battery and that the proposed formulation provides a computationally efficient framework for evaluating and operating storage assets in interconnected power systems Finally a pseudoefficiency term is introduced to improve battery utilization by discarding less profitable charging and discharging battery cycles

Duong Thuy Anh Nguyen, Harsha Nagarajan, Robert Ferrando et al.

This paper presents an optimal network topology control framework using cutting-plane methods for efficient network partitioning with controllable edges. The objective is to enable real-time reconfiguration of interconnected sub-networks while ensuring radial connectivity, resource feasibility, and structured leader allocation, which are essential for distributed control, stability, and coordination. The problem is formulated as a mixed-integer program that integrates graph-theoretic constraints, resource flow, and network structural properties to enforce an operational hierarchy. To address the combinatorial complexity of cycle elimination and leader assignment, we propose an iterative cutting-plane framework that ensures convergence to an optimal and feasible network topology. Theoretical guarantees on optimality preservation, feasibility, and convergence are established, ensuring systematic elimination of infeasible configurations while maintaining distributed controllability. Simulations on a modified Iowa 240-bus power distribution grid demonstrate the framework's effectiveness in network reconfiguration under resource constraints. The approach achieves median and best-case speedups of 57.5x and over 64x in a 46-switch configuration, highlighting its applicability to other networked control systems.

Rohit Dube, Natarajan Gautam, Amarnath Banerjee et al.

Accurate predictions of electricity demands are necessary for managing operations in a small aggregation load setting like a Microgrid. Due to low aggregation, the electricity demands can be highly stochastic and point estimates would lead to inflated errors. Interval estimation in this scenario, would provide a range of values within which the future values might lie and helps quantify the errors around the point estimates. This paper introduces a residual bootstrap algorithm to generate interval estimates of day-ahead electricity demand. A machine learning algorithm is used to obtain the point estimates of electricity demand and respective residuals on the training set. The obtained residuals are stored in memory and the memory is further partitioned. Days with similar demand patterns are grouped in clusters using an unsupervised learning algorithm and these clusters are used to partition the memory. The point estimates for test day are used to find the closest cluster of similar days and the residuals are bootstrapped from the chosen cluster. This algorithm is evaluated on the real electricity demand data from EULR(End Use Load Research) and is compared to other bootstrapping methods for varying confidence intervals.

Kaarthik Sundar, Harsha Nagarajan, Line Roald et al.

As renewable wind energy penetration rates continue to increase, one of the major challenges facing grid operators is the question of how to control transmission grids in a reliable and a cost-efficient manner. The stochastic nature of wind forces an alteration of traditional methods for solving day-ahead and look-ahead unit commitment and dispatch. In particular, the variability of wind generation increases the risk of unexpected overloads and cascading events. To address these questions, we present an N-1 Security and Chance-Constrained Unit Commitment (SCCUC) that includes models of generation reserves that respond to wind fluctuations and component outages. We formulate the SCCUC as a mixed-integer, second-order cone problem that limits the probability of failure. We develop a modified Benders decomposition algorithm to solve the problem to optimality and present detailed case studies on the IEEE RTS-96 three-area and the IEEE 300 NESTA test systems. The case studies assess the economic impacts of contingencies and various degrees of wind power penetration and demonstrate the effectiveness and scalability of the algorithm.

Carleton Coffrin, Harsha Nagarajan, Russell Bent

The recent emergence of novel computational devices, such as adiabatic quantum computers, CMOS annealers, and optical parametric oscillators, present new opportunities for hybrid-optimization algorithms that are hardware accelerated by these devices. In this work, we propose the idea of an Ising processing unit as a computational abstraction for reasoning about these emerging devices. The challenges involved in using and benchmarking these devices are presented and commercial mixed integer programming solvers are proposed as a valuable tool for the validation of these disparate hardware platforms. The proposed validation methodology is demonstrated on a D-Wave 2X adiabatic quantum computer, one example of an Ising processing unit. The computational results demonstrate that the D-Wave hardware consistently produces high-quality solutions and suggests that as IPU technology matures it could become a valuable co-processor in hybrid-optimization algorithms.

Arthur Barnes, Harsha Nagarajan, Emre Yamangil et al.

In the electrical grid, the distribution system is themost vulnerable to severe weather events. Well-placed and coordinatedupgrades, such as the combination of microgrids, systemhardening and additional line redundancy, can greatly reduce thenumber of electrical outages during extreme events. Indeed, ithas been suggested that resilience is one of the primary benefitsof networked microgrids. We formulate a resilient distributiongrid design problem as a two-stage stochastic program andmake use of decomposition-based heuristic algorithms to scaleto problems of practical size. We demonstrate the feasibilityof a resilient distribution design tool on a model of an actualdistribution network. We vary the study parameters, i.e., thecapital cost of microgrid generation relative to system hardeningand target system resilience metrics, and find regions in thisparametric space corresponding to different distribution systemarchitectures, such as individual microgrids, hardened networks,and a transition region that suggests the benefits of microgridsnetworked via hardened circuit segments.