There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality.

Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

I and i

(1990). ENERGY FUNCTION ANALYSIS FOR POWER SYSTEM STABILITY. Electric Machines & Power Systems: Vol. 18, No. 2, pp. 209-210.

Energy function analysis for power system stability

The smart grid concept continues to evolve and various methods have been developed to enhance the energy efficiency of the electricity infrastructure. Demand Response (DR) is considered as the most cost-effective and reliable solution for the smoothing of the demand curve, when the system is under stress. DR refers to a procedure that is applied to motivate changes in the customers' power consumption habits, in response to incentives regarding the electricity prices. In this paper, we provide a comprehensive review of various DR schemes and programs, based on the motivations offered to the consumers to participate in the program. We classify the proposed DR schemes according to their control mechanism, to the motivations offered to reduce the power consumption and to the DR decision variable. We also present various optimization models for the optimal control of the DR strategies that have been proposed so far. These models are also categorized, based on the target of the optimization procedure. The key aspects that should be considered in the optimization problem are the system's constraints and the computational complexity of the applied optimization algorithm.

A Survey on Demand Response Programs in Smart Grids: Pricing Methods and Optimization Algorithms

https://nacto.org/wp-content/uploads/2016/02/2016_Fishman_Bikeshare-A-Review-of-Recent-Literature.pdf

Bikeshare: A Review of Recent Literature

Optimization models have been widely used in the power industry to aid the decision-making process of scheduling and dispatching electric power generation resources, a process known as unit commitment (UC). Since UC’s birth, there have been two major waves of revolution on UC research and real life practice. The first wave has made mixed integer programming stand out from the early solution and modeling approaches for deterministic UC, such as priority list, dynamic programming, and Lagrangian relaxation. With the high penetration of renewable energy, increasing deregulation of the electricity industry, and growing demands on system reliability, the next wave is focused on transitioning from traditional deterministic approaches to stochastic optimization for unit commitment. Since the literature has grown rapidly in the past several years, this paper is to review the works that have contributed to the modeling and computational aspects of stochastic optimization (SO) based UC. Relevant lines of future research are also discussed to help transform research advances into real-world applications.

Stochastic Optimization for Unit Commitment—A Review

This paper considers the efficient operation of shared mobility systems via the combination of intelligent routing decisions for staff-based vehicle redistribution and real-time price incentives for customers. The approach is applied to London's Barclays Cycle Hire scheme, which the authors have simulated based on historical data. Using model-based predictive control principles, dynamically varying rewards are computed and offered to customers carrying out journeys, based on the current and predicted state of the system. The aim is to encourage them to park bicycles at nearby underused stations, thereby reducing the expected cost of redistributing them using dedicated staff. In parallel, routing directions for redistribution staff are periodically recomputed using a model-based heuristic. It is shown that it is possible to trade off reward payouts to customers against the cost of hiring staff to redistribute bicycles, in order to minimize operating costs for a given desired service level.

Dynamic Vehicle Redistribution and Online Price Incentives in Shared Mobility Systems

This paper introduces the concept of affine reserve policies for accommodating large, fluctuating renewable in feeds in power systems. The approach uses robust optimization with recourse to determine operating rules for power system entities such as generators and storage units. These rules, or policies, establish several hours in advance how these entities are to respond to errors in the prediction of loads and renewable infeeds once their values are discovered. Affine policies consist of a nominal power schedule plus a series of planned linear modifications that depend on the prediction errors that will become known at future times. We describe how to choose optimal affine policies that respect the power network constraints, namely matching supply and demand, respecting transmission line ratings, and the local operating limits of power system entities, for all realizations of the prediction errors. Crucially, these policies are time-coupled, exploiting the spatial and temporal correlation of these prediction errors. Affine policies are compared with existing reserve operation under standard modeling assumptions, and operating cost reductions are reported for a multi-day benchmark study featuring a poorly-predicted wind infeed. Efficient prices for such “policy-based reserves” are derived, and we propose new reserve products that could be traded on electricity markets.

Policy-Based Reserves for Power Systems

https://www.utdallas.edu/~tyler.summers/papers/SummersWarringtonMorariLygeros2015.pdf

Stochastic optimal power flow based on conditional value at risk and distributional robustness

This paper presents a computationally-efficient approach for solving stochastic, multiperiod optimal power flow problems. The objective is to determine power schedules for controllable devices in a power network, such as generators, storage, and curtailable loads, which minimize expected short-term operating costs under various device and network constraints. These schedules include planned power output adjustments, or reserve policies, which track errors in the forecast of power requirements as they are revealed, and which may be time-coupled. Such an approach has previously been shown to be an attractive means of accommodating uncertainty arising from highly variable renewable energy sources. Given a probabilistic forecast describing the spatio-temporal variations and dependencies of forecast errors, we formulate a family of stochastic network and device constraints based on convex relaxations of chance constraints, and show that these allow economic efficiency and system security to be traded off with varying levels of conservativeness. The results are illustrated using a simple case study, in which conventional generators plan schedules around an uncertain but time-correlated wind power injection.

Stochastic optimal power flow based on convex approximations of chance constraints

We present a rolling decision-making process for electrical power systems, in which unit commitment, dispatch and reserve policies are co-optimized in order to minimize expected short-run operating costs in the presence of uncertainty arising from demand and renewable infeeds. The uncertainty is assumed to be bounded, with estimated first and second moment statistics available. The rolling “look-ahead” process employs a planning horizon of several hours, with re-optimization taking place each time the first step has been implemented. We present an expected-cost formulation incorporating multi-stage recourse on continuous decision variables—plans for adjusting the dispatch in the light of future information to be discovered at each stage of the optimization horizon. The generic formulation allows the flexibility of devices such as energy storage units to be exploited in the reserve mechanism. We demonstrate using closed-loop numerical tests that significant reductions in the cost of accommodating uncertainty are attainable relative to a time-decoupled reserve mechanism. In contrast to previous results, we show that a time-coupled cost function is not required for this benefit to be observed. In addition, we show that relaxing binary unit commitment decisions after the first step of the horizon brings significant computational speed-ups, and in some cases also reduce closed-loop system operation costs.

Joseph Warrington

Papers

Dynamic Vehicle Redistribution and Online Price Incentives in Shared Mobility Systems

Policy-Based Reserves for Power Systems

Stochastic optimal power flow based on conditional value at risk and distributional robustness

Stochastic optimal power flow based on convex approximations of chance constraints

Rolling Unit Commitment and Dispatch With Multi-Stage Recourse Policies for Heterogeneous Devices