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Generation Expansion Planning (GEP) can inform regulation, electricity market design, and regional system planning by identifying adaptive investment strategies. Relevant uncertainties include hourly variability in load and renewable generation and decadal-scale uncertainty in technology, markets, and regulation. A multi-stage and multi-scale stochastic GEP model that represents these uncertainties at sufficient resolution becomes intractable. We present an approach for representing this multi-scale uncertainty, and compare it to existing methods, applied to a two-stage stochastic GEP model with a cumulative carbon emission target. For long-term uncertainty, we compare partitioning methods, which reduce the number of decision variables but retain all scenarios, to representative scenario methods, which retain only a subset of the original scenarios. For short-term uncertainty, we compare methods that select representative weeks based on distance metrics in the parameter space to methods that use the covariance of outcomes across feasible decisions to select weeks. We find that scenario reduction methods struggle to find the appropriate investment levels for variable renewable generation and consequently produce more costly plans than scenario partitioning methods. While simple approximating methods perform well with larger models, covariance-based approximations have the best performance overall. [ABSTRACT FROM AUTHOR]

Gas-fired generators, with their ability to quickly ramp up and down their electricity production, play an important role in managing renewable energy variability. However, these changes in electricity production translate into variability in the consumption of natural gas, and propagation of uncertainty from the electric grid to the natural gas system. To ensure that both systems are operating safely, there is an increasing need for coordination and uncertainty management among the electricity and gas networks. A challenging aspect of this coordination is the consideration of natural gas dynamics, which play an important role in intra-day operation, but give rise to a set of non-linear and non-convex equations that are hard to optimize even in the deterministic case. Ideally, the problem is formulated as a stochastic problem but many conventional methods for stochastic optimization are numerically intractable because they either incorporate a large number of scenarios directly or require the underlying problem to be convex. To address these challenges, we propose using a Stochastic Hybrid Approximation algorithm to more efficiently solve these problems and investigate the efficacy of several different variants of this algorithm. Our case study demonstrates that the proposed technique is able to quickly obtain high quality solutions and outperforms existing benchmarks such as Generalized Benders Decomposition. We demonstrate that coordinated uncertainty management that accounts for the gas system can significantly reduce both electric and gas system load shed in stressed conditions. [ABSTRACT FROM AUTHOR]

Addressing non-convexity plays a fundamental role in solving the optimal electricity-gas flow models. In this paper, an improved spatial branch-and-bound algorithm is proposed to solve the non-convex problem, which is formulated as a mixed-integer bilinear programming, for its exact solution. The core of the algorithm is to divide the non-convex model into convex and small sub-models by branching on specific continuous variables, so that the non-convex problem can be equivalent to a rooted tree for exploration. The exactness of the algorithm is guaranteed by the same criterion as the classical branch-and-bound algorithm. To alleviate the computational burden, a novel two-stage spatial branching strategy is developed to improve the effectiveness and efficiency of the branching operations. The performance of the proposed algorithm is verified on two integrated electricity-gas systems with different sizes. Numerical results demonstrate that our method achieves a balance among feasibility, optimality, and efficiency. The comparison with another 6 convexification-based methods, 3 state-of-the-art non-convex optimization solvers, and 2 spatial branch-and-bound algorithms with classical branching rules further shows the superiority of our algorithm. [ABSTRACT FROM AUTHOR]

The nonlinear programming (NLP) problem to solve distribution-level optimal power flow (D-OPF) poses convergence issues and does not scale well for unbalanced distribution systems. The existing scalable D-OPF algorithms either use approximations that are not valid for an unbalanced power distribution system, or apply relaxation techniques to the nonlinear power flow equations that do not guarantee a feasible power flow solution. In this paper, we propose scalable D-OPF algorithms that simultaneously achieve optimal and feasible solutions by solving multiple iterations of approximate, or relaxed, D-OPF subproblems of low complexity. The first algorithm is based on a successive linear approximation of the nonlinear power flow equations around the current operating point, where the D-OPF solution is obtained by solving multiple iterations of a linear programming (LP) problem. The second algorithm is based on the relaxation of the nonlinear power flow equations as conic constraints together with directional constraints, which achieves optimal and feasible solutions over multiple iterations of a second-order cone programming (SOCP) problem. It is demonstrated that the proposed algorithms are able to reach an optimal and feasible solution while significantly reducing the computation time as compared to an equivalent NLP D-OPF model for the same distribution system. [ABSTRACT FROM AUTHOR]

Nonconvexity induced by the nonlinear AC power flow equations challenges solution algorithms for AC optimal power flow (OPF) problems. While significant research efforts have focused on reliably computing high-quality OPF solutions, it is not always clear that there exists a feasible path to reach the desired operating point. Transitioning between operating points while avoiding constraint violations can be challenging since the feasible space of the OPF problem is nonconvex and potentially disconnected. To address this problem, we propose an algorithm that computes a provably feasible path from an initial operating point to a desired operating point. Given an initial feasible point, the algorithm solves a sequence of convex quadratically constrained optimization problems over conservative convex inner approximations of the OPF feasible space. In each iteration, we obtain a new, improved operating point and a feasible transition from the operating point in the previous iteration. In addition to computing a feasible path to a known desired operating point, this algorithm can also be used to improve the operating point locally. Extensive numerical studies on a variety of test cases demonstrate the algorithm and the ability to arrive at a high-quality solution in few iterations. [ABSTRACT FROM AUTHOR]

One major difficulty in transmission expansion planning is selecting the representative scenarios to use to evaluate candidate transmission networks. The variability in demand and renewable generation makes the inclusion of several scenarios critical when calculating reliability and cost, but including too many scenarios in an optimization is computationally intractable. To reduce the number of representative operating conditions needed to obtain an accurate approximation of the system, we propose a method rooted in multivariate statistics that exploits the latent correlative structure between different scenarios and network configurations. The proposed algorithm includes an objective and rigorous way to select a subset of scenarios that provide as much information about the system as possible, and a method to accurately approximate the system cost from that scenario subset. The result is a set of scenarios and weights that are easily incorporated into traditional transmission expansion planning formulations. We apply this to a 312-bus WECC model with 8,736 distinct operating conditions. The transmission plans found with the proposed method are more reliable and have a lower total cost than those from other scenario reduction techniques, as well as a smaller error between the expected system performance from the optimization objective and the actual system performance. [ABSTRACT FROM AUTHOR]

This paper focuses on the interchange and generation scheduling problem of multi-area power systems, where decentralized decision procedure is preferred and the uncertainties of nodal injections should be considered. The problem is formulated as a multi-area robust security constrained unit commitment model, with a novel uncertainty set specified by the variance of system net load. We prove that such an uncertainty set can provide a more precise probabilistic guarantee than conventional budgeted sets. To solve the problem in decentralized manner, an upper-level robust convex optimization model is proposed to optimize the tie-line power flow and derive a generation interval for each area. Then through the coordinated uncertainty requirement decided by the generation interval, each regional system operator can solve a decoupled robust unit commitment problem independently. Besides, a modified outer approximation algorithm is developed to obtain higher-quality solution of the bilinear programming involved in the process. Simulation results on a two-area system demonstrate the effectiveness of the uncertainty set, and show the economic efficiencies of unit commitment solutions acquired from the hierarchical method. [ABSTRACT FROM AUTHOR]

Bi-level optimization constitutes the most popular mathematical methodology for modeling the deregulated electricity market. However, state-of-the-art models neglect the physical non-convex operating characteristics of market participants, due to their inherent inability to capture binary decision variables in their representation of the market clearing process, rendering them problematic in modeling markets with complex bidding and unit commitment (UC) clearing mechanisms. This paper addresses this fundamental limitation by proposing a novel modeling approach enabling incorporation of these non-convexities into bi-level optimization market models, which is based on the relaxation and primal-dual reformulation of the original, non-convex lower level problem and the penalization of the associated duality gap. Case studies demonstrate the ability of the proposed approach to closely approximate the market clearing solution of the actual UC clearing algorithm and devise more profitable bidding decisions for strategic producers than the state-of-the-art bi-level optimization approach, and reveal the potential of strategic behavior in terms of misreporting non-convex operating characteristics. [ABSTRACT FROM AUTHOR]

Voltage control plays an important role in the operation of electricity distribution networks, especially with high penetration of distributed energy resources. These resources introduce significant and fast varying uncertainties. In this paper, we focus on reactive power compensation to control voltage in the presence of uncertainties. We adopt a chance constraint approach that accounts for arbitrary correlations between renewable resources at each of the buses. We show how the problem can be solved efficiently using historical samples analogously to the stochastic quasi-gradient methods. We also show that this optimization problem is convex for a wide variety of probabilistic distributions. Compared to conventional per-bus chance constraints, our formulation is more robust to uncertainty and more computationally tractable. We illustrate the results using standard IEEE distribution test feeders. [ABSTRACT FROM AUTHOR]

Load tap changers (LTCs) are commonly used for voltage control in distribution systems. The mechanical relays on LTCs are controlled in discrete steps; hence, distribution optimal power flow (DOPF) formulations require LTCs to be modeled with integer variables. Integer variables are generally ignored or relaxed and rounded off to reduce the computational burden in DOPF models. However, in this paper, we highlight and overcome complex infeasibility issues caused by rounding methods. Moreover, recent advances in convex optimization have improved the computational tractability of the DOPF problem. In this paper, we develop and analyze numerical efficiency of different but exact formulations for incorporating LTCs as integer control assets in a relaxed second-order cone program (SOCP) version of the DOPF model. The resulting formulation becomes a mixed-integer SOCP (MISOCP), which is computationally tractable compared to mixed-integer non-linear program (MINLP). To recover the exact optimal solution in the proposed MISOCP-DOPF model, a McCormick relaxation is employed within a sequential bound-tightening algorithm. We show that solutions obtained from MISOCP-DOPF are always ac feasible. Thus, the MISOCP-DOPF yields optimal and realistic ac solutions, and is validated in large distribution feeders and is compared to MINLP counterpart. [ABSTRACT FROM AUTHOR]

We propose a novel global solution algorithm for the network-constrained unit commitment problem that incorporates a nonlinear alternating current (ac) model of the transmission network, which is a nonconvex mixed-integer nonlinear programming problem. Our algorithm is based on the multi-tree global optimization methodology, which iterates between a mixed-integer lower-bounding problem and a nonlinear upper-bounding problem. We exploit the mathematical structure of the unit commitment problem with ac power flow constraints and leverage second-order cone relaxations, piecewise outer approximations, and optimization-based bounds tightening to provide a globally optimal solution at convergence. Numerical results on four benchmark problems illustrate the effectiveness of our algorithm, both in terms of convergence rate and solution quality. [ABSTRACT FROM AUTHOR]

The AC optimal power flow problem is a non-convex optimization problem that is difficult to solve quickly and reliably for large-scale grids. While various approximations have been proposed, they may lead to physically meaningless solutions. This paper presents a computationally efficient algorithm for constructing accurate linear approximations of line flow constraints. These approximations reduce the complexity of the optimization problem while ensuring that the solution is physically meaningful and has a high quality. The algorithm is based on an in-depth analysis of the feasible set of the line flow constraint. Numerical experiments are performed on ten large-scale systems using three nonlinear programming solvers. Obtained results indicate that the proposed formulation helps improve the solvers’ reliability and reduce the computation time for hard large-scale problems. At the same time, the developed algorithm provides high quality approximations of the line flow constraints. [ABSTRACT FROM AUTHOR]

Renewable generation and energy storage are playing an ever increasing role in power systems. Hence, there is a growing need for integrating these resources into the optimal power flow (OPF) problem. While storage devices are important for mitigating renewable variability, they introduce temporal coupling in the OPF constraints, resulting in a multiperiod OPF formulation. This paper explores a solution method for multiperiod AC OPF that combines a successive quadratic programming approach (AC-QP) with a second-order cone programming (SOCP) relaxation of the OPF problem. The SOCP relaxation's solution is used to initialize the AC-QP OPF algorithm. Additionally, the lower bound on the objective value obtained from the SOCP relaxation provides a measure of solution quality. This combined method is demonstrated on several test cases with up to 4259 nodes and a time horizon of eight time steps. A comparison of initialization schemes indicates that the SOCP-based approach offers improved convergence rate, execution time, and solution quality. [ABSTRACT FROM PUBLISHER]

We present a parallel implementation of Lagrangian relaxation for solving stochastic unit commitment subject to uncertainty in renewable power supply and generator and transmission line failures. We describe a scenario selection algorithm inspired by importance sampling in order to formulate the stochastic unit commitment problem and validate its performance by comparing it to a stochastic formulation with a very large number of scenarios, that we are able to solve through parallelization. We examine the impact of narrowing the duality gap on the performance of stochastic unit commitment and compare it to the impact of increasing the number of scenarios in the model. We report results on the running time of the model and discuss the applicability of the method in an operational setting. [ABSTRACT FROM PUBLISHER]

In development studies carried out on large power systems, a need may arise for a reduction in which whole regions are seen as the nodes of a simplified graph whose edges replace the inter-regional links. The property required from this kind of backbone representation may be that reduced load flows stay close to their physical counterpart on the full grid. In the dc approximation, such a model can consist of a set of impedances attached to each edge, along with a flow characterizing a particular state of the system. The identification of the model accommodating best a large sample of flows, representative of all of their possible diversity, is a nonconvex problem for which two partially convex reformulations are proposed. Simple ad hoc algorithms are introduced as a way to find approximate solutions satisfying first-order optimality conditions. [ABSTRACT FROM PUBLISHER]

Unit commitment, one of the most critical tasks in electric power system operations, faces new challenges as the supply and demand uncertainty increases dramatically due to the integration of variable generation resources such as wind power and price responsive demand. To meet these challenges, we propose a two-stage adaptive robust unit commitment model for the security constrained unit commitment problem in the presence of nodal net injection uncertainty. Compared to the conventional stochastic programming approach, the proposed model is more practical in that it only requires a deterministic uncertainty set, rather than a hard-to-obtain probability distribution on the uncertain data. The unit commitment solutions of the proposed model are robust against all possible realizations of the modeled uncertainty. We develop a practical solution methodology based on a combination of Benders decomposition type algorithm and the outer approximation technique. We present an extensive numerical study on the real-world large scale power system operated by the ISO New England. Computational results demonstrate the economic and operational advantages of our model over the traditional reserve adjustment approach. [ABSTRACT FROM PUBLISHER]

In this paper, we present a unit commitment problem with uncertain wind power output. The problem is formulated as a chance-constrained two-stage (CCTS) stochastic program. Our model ensures that, with high probability, a large portion of the wind power output at each operating hour will be utilized. The proposed model includes both the two-stage stochastic program and the chance-constrained stochastic program features. These types of problems are challenging and have never been studied together before, even though the algorithms for the two-stage stochastic program and the chance-constrained stochastic program have been recently developed separately. In this paper, a combined sample average approximation (SAA) algorithm is developed to solve the model effectively. The convergence property and the solution validation process of our proposed combined SAA algorithm is discussed and presented in the paper. Finally, computational results indicate that increasing the utilization of wind power output might increase the total power generation cost, and our experiments also verify that the proposed algorithm can solve large-scale power grid optimization problems. [ABSTRACT FROM PUBLISHER]

The environmental-economic dispatch (EED) is a bi-objective optimization problem. The Nash bargaining theory provides one way to determine a compromising solution without a clear carbon tax or carbon cap, or any subjective attitudes on both objectives, which yields a non-convex program. This letter proposes a convex formulation for Nash bargaining based EED and a linear programming (LP) based algorithm. Case studies show its scalability on real-world large-scale power systems. [ABSTRACT FROM PUBLISHER]

This letter proposes a novel method to solve the environmental/economic dispatch (EED) problem with DC power flow constraints. A parameterized KKT condition is introduced to characterize the Pareto front. Then Nash's bargain criterion is employed to determine a compromise solution on the Pareto front. The proposed method is applied to the EED of a regional power grid in China. Numerical studies demonstrate that it is valid and promising. [ABSTRACT FROM PUBLISHER]

A multi-objective (MO) optimization approach is proposed in this letter to develop optimal transmission switching strategies with minimal generating cost and maximal probabilistic reliability. The problem is solved via an evolutionary algorithm together with Monte Carlo simulation to capture the probabilistic nature of system component failures. The numerical results show that the identified quasi Pareto-optimal solutions can provide insights into the trade-off between generating cost and system reliability considering transmission switching. [ABSTRACT FROM AUTHOR]