Heuristics for Lagrangian Relaxation Formulations for the Unit Commitment Problem

The expansion of distributed energy resources (DER), demand response (DR), and virtual bidding in many power systems and energy markets are creating new challenges for unit commitment (UC) and economic dispatch (ED) techniques. Instead of a small number of traditionally large generators, the power system resource mix is moving to one with a high percentage of a large number of small units. These can increase the number of similar or identical units, leading to chattering (switching back and forth among committed units between iterations). This research investigates alternative and scalable ways of increasing the high penetration of these resources. First, the mathematical formulations for UC and ED models are reviewed. Then a new heuristic is proposed that takes advantage of the incremental nature of Lagrangian relaxation (LR). The heuristic linearizes and distributes the network transmission losses to appropriately penalize line flow and mitigate losses. Second, a mixed integer programming (MIP) is used as a benchmark for the proposed LR formulation. The impact of similar and identical units on the solution quality and simulation run time of UC and ED was investigated using the proposed formulation. Third, a system flexibility study is done using DR and a load demand pattern with a high penetration of renewables, creating a high daily ramp rate requirement. This work investigates the impact of available DR on spikes in locational marginal pricing (LMP). Fourth, two studies are done on improving LR computational efficiency. The first proposes a heuristic that focuses on trade-offs between solution quality and simulation run time. The heuristic iterates over lambda and energy marginal price while the convergence issue is handled using Augmented LR (ALR). The second study proposes a heuristic that penalizes transmission lines with binding line limits. The proposed method can reduce power flow in the transmission lines of interest, and considerably reduce the simulation time in optimization problems with a high number of transmission constraints. Finally, the effect of a large number of similar and identical units on simulation run time is considered. The proposed formulation scales linearly with the increase in system size

Large-scale unit commitment under uncertainty: an updated literature survey

The Unit Commitment problem in energy management aims at finding the optimal production schedule of a set of generation units, while meeting various system-wide constraints. It has always been a large-scale, non-convex, difficult problem, especially in view of the fact that, due to operational requirements, it has to be solved in an unreasonably small time for its size. Recently, growing renewable energy shares have strongly increased the level of uncertainty in the system, making the (ideal) Unit Commitment model a large-scale, non-convex and uncertain (stochastic, robust, chance-constrained) program. We provide a survey of the literature on methods for the Uncertain Unit Commitment problem, in all its variants. We start with a review of the main contributions on solution methods for the deterministic versions of the problem, focussing on those based on mathematical programming techniques that are more relevant for the uncertain versions of the problem. We then present and categorize the approaches to the latter, while providing entry points to the relevant literature on optimization under uncertainty. This is an updated version of the paper "Large-scale Unit Commitment under uncertainty: a literature survey" that appeared in 4OR 13(2), 115--171 (2015); this version has over 170 more citations, most of which appeared in the last three years, proving how fast the literature on uncertain Unit Commitment evolves, and therefore the interest in this subject

Essays on the ACOPF Problem: Formulations, Approximations, and Applications in the Electricity Markets

The alternating current optimal power flow (ACOPF) problem, also referred to as the optimal power flow (OPF) problem, is at the core of competitive wholesale electricity markets and vertically integrated utility operations. ACOPF simultaneously co-optimizes real and reactive power. First formulated over half a century ago in 1962 by Carpentier, the ACOPF is the most representative mathematical programming-based formulation of steady-state operations in AC networks. However the ACOPF is not solved in practice due to the nonconvex structure of the problem, which is known to be NP-hard. Instead, least-cost unit commitment and generation dispatch in the day-ahead, intra-day, and real-time markets is determined with numerous simplifications of the ACOPF constraint set. This work presents a series of essays on the ACOPF problem, which include formulations, approximations, and applications in the electricity markets. The main themes center around ACOPF modeling fundamentals, followed by local and global solution methods for a variety of applications in the electricity markets. Original contributions of these essays include an alternative formulation of the ACOPF, a successive linear programming algorithm to solving the ACOPF for the real-time energy market, an outer approximation method to solving integrated ACOPF-unit commitment as a mixed-integer linear program for the day-ahead market, and applications of convex relaxations to the ACOPF and its approximations for the purpose of globally optimal storage integration. These contributions are concluded with a discussion of potential future directions for work

A fast penalty-based Gauss-Seidel method for solving large-scale stochastic network constrained unit commitment problems

Doctor of PhilosophyDepartment of Electrical and Computer EngineeringHongyu WuThe exponential growth of variable renewable energy (VRE) such as wind and solar generation brings grand challenges to the operational planning of power systems. The instantaneous penetration of VRE reaches over 50% in certain balancing areas in the United States. The VRE generation is characterized by a large amount of uncertainties and variabilities. Consequently, power system operators, planners and researchers have made substantial efforts to manage VRE uncertainties in the power system scheduling, such as Network-Constrained Unit Commitment (NCUC). In order to account for the impact of VRE uncertainties, there are several noteworthy NCUC approaches in the literature, each with distinctive objectives, theories, computational requirements and economic outcomes. A common approach presented in the literature is the use of stochastic programming, namely Stochastic NCUC (S-NCUC), in which the expected system operating cost is minimized across a number of scenarios, each representing a possible realization of uncertainties. S-NCUC is typically a large-scale, non-convex, and mixed-integer programming (MIP) problem. It is modeled as a two-stage stochastic problem where the first-stage unit commitment decisions are the same for all the scenarios. Generally, S-NCUC solutions can be categorized into two main approaches. First, the most straightforward approach is to use a commercially available off-the-shelf solver to solve an extensive form (EF) of S-NCUC. However, for any large-scale system with a reasonable number of scenarios, the resulting EF of SNCUC may become computationally intractable. To overcome this issue, the second approach is based on stage-wise or scenario-wise decomposition methods, which solve each individual scenario separately, usually in parallel, and a final solution is generated by coordinating all individual scenario solutions. Progressive Hedging Algorithm (PHA) is one of the main decomposition methods for solving the stochastic MIP. However, PHA is originally devised for a continuous convex program and is not provably convergent for the non-convex S-NCUC problem. The solution to the dual problem is generally primal infeasible and the once relaxed system-wide constraints may not be satisfied. An additional effort is required to restore the primal feasibility from a Lagrangian dual solution. Therefore, it is desirable to directly obtain a primally feasible solution from the Lagrangian dual iterations. This gives rise to exact augmented Lagrangian, a class of exact penalty methods whose objective is to solve a constrained optimization (primal) problem through an unconstrained optimization problem that has the same local (global) solutions as the primal problem. Nevertheless, the following two critical research questions remain unresolved: 1) How can we devise an effective penalty function such that an exact solution can be obtained with a zero-duality gap? 2) If an exact solution is attained, how can we find a robust yet tight lower bound that is capable of measuring the quality of the exact solution accurately? This dissertation addresses the aforementioned first question by applying a novel Penalty- Based Gauss-Seidel (PBGS) algorithm with an exact augmented Lagrangian representation to solve S-NCUC within a scenario-based decomposition framework. To improve the computational efficiency of PBGS, an accelerating technique that skips solving scenarios meeting certain conditions has been proposed. The proposed algorithm is named “Fast PBGS.” A proof of the Fast PBGS method is given, along with the proof of convergence of PBGS. Numerical validation of these algorithms on the IEEE 118-bus and Electric Reliability Council of Texas (ERCOT)-like large-scale systems has been carried out. Fast PBGS saves computational time by an average 35% for ERCOT-like Large System and 50% for IEEE 118-bus System with 50 scenarios compared with PBGS. Numerical results demonstrate the high quality of the PBGS solution and the efficacy of the proposed algorithms. Additionally, comparing the proposed algorithms with other prevailing S-NCUC methods such as PHA and extensive-form-based MIP solutions has been completed. The comparison of Fast PBGS shows the results are closer to EF (average difference 0.92%) than the PHA solution with EF (average difference 2.18%). When it came to the computational time, the Fast PBGS outperformed both EF and PHA. An average Fast PBGS took 48% less time than EF to obtain a solution. Compared with PHA, Fast PBGS was 142% faster. The second question is addressed by applying the combined Frank Wolfe with PHA algorithm (FW-PHA). Our research shows that FW-PHA obtains superior lower bounds, i.e., up to 6% better than the PHA does on the IEEE 118-bus system. We further improve the computational efficiency of FW-PHA with a warm start technique that initializes the algorithm with a Fast PBGS solution. An out-of-sample analysis including a large number of samples is conducted to demonstrate the efficacy of the Fast PBGS

Optimization Methods Applied to Power Systems Ⅱ

Electrical power systems are complex networks that include a set of electrical components that allow distributing the electricity generated in the conventional and renewable power plants to distribution systems so it can be received by final consumers (businesses and homes). In practice, power system management requires solving different design, operation, and control problems. Bearing in mind that computers are used to solve these complex optimization problems, this book includes some recent contributions to this field that cover a large variety of problems. More specifically, the book includes contributions about topics such as controllers for the frequency response of microgrids, post-contingency overflow analysis, line overloads after line and generation contingences, power quality disturbances, earthing system touch voltages, security-constrained optimal power flow, voltage regulation planning, intermittent generation in power systems, location of partial discharge source in gas-insulated switchgear, electric vehicle charging stations, optimal power flow with photovoltaic generation, hydroelectric plant location selection, cold-thermal-electric integrated energy systems, high-efficiency resonant devices for microwave power generation, security-constrained unit commitment, and economic dispatch problems

Energy storage systems and grid code requirements for large-scale renewables integration in insular grids

This thesis addresses the topic of energy storage systems supporting increased penetration of renewables in insular systems. An overview of energy storage management, forecasting tools and demand side solutions is carried out, comparing the strategic utilization of storage and other competing strategies. Particular emphasis is given to energy storage systems on islands, as a new contribution to earlier studies, addressing their particular requirements, the most appropriate technologies and existing operating projects throughout the world. Several real-world case studies are presented and discussed in detail. Lead-acid battery design parameters are assessed for energy storage applications on insular grids, comparing different battery models. The wind curtailment mitigation effect by means of energy storage resources is also explored. Grid code requirements for large-scale integration of renewables are discussed in an island context, as another new contribution to earlier studies. The current trends on grid code formulation, towards an improved integration of distributed renewable resources in island systems, are addressed. Finally, modeling and control strategies with energy storage systems are addressed. An innovative energy management technique to be used in the day-ahead scheduling of insular systems with Vanadium Redox Flow battery is presented.Esta tese aborda a temática dos sistemas de armazenamento de energia visando o aumento da penetração de energias renováveis em sistemas insulares. Uma visão geral é apresentada acerca da gestão do armazenamento de energia, ferramentas de previsão e soluções do lado da procura de energia, comparando a utilização estratégica do armazenamento e outras estratégias concorrentes. É dada ênfase aos sistemas de armazenamento de energia em ilhas, como uma nova contribuição no estado da arte, abordando as suas necessidades específicas, as tecnologias mais adequadas e os projetos existentes e em funcionamento a nível mundial. Vários casos de estudos reais são apresentados e discutidos em detalhe. Parâmetros de projeto de baterias de chumbo-ácido são avaliados para aplicações de armazenamento de energia em redes insulares, comparando diferentes modelos de baterias. O efeito de redução do potencial de desperdício de energia do vento, recorrendo ao armazenamento de energia, também é perscrutado. As especificidades subjacentes aos códigos de rede para a integração em larga escala de energias renováveis são discutidas em contexto insular, sendo outra nova contribuição no estado da arte. As tendências atuais na elaboração de códigos de rede, no sentido de uma melhor integração da geração distribuída renovável em sistemas insulares, são abordadas. Finalmente, é estudada a modelação e as estratégias de controlo com sistemas de armazenamento de energia. Uma metodologia de gestão de energia inovadora é apresentada para a exploração de curto prazo de sistemas insulares com baterias de fluxo Vanádio Redox

Control of transmission system power flows

Power flow (PF) control can increase the utilization of the transmission system and connect lower cost generation with load. While PF controllers have demonstrated the ability to realize dynamic PF control for more than 25 years, PF control has been sparsely implemented. This research re-examines PF control in light of the recent development of fractionally-rated PF controllers and the incremental power flow (IPF) control concept. IPF control is the transfer of an incremental quantity of power from a specified source bus to specified destination bus along a specified path without influencing power flows on circuits outside of the path. The objectives of the research are to develop power system operation and planning methods compatible with IPF control, test the technical viability of IPF control, develop transmission planning frameworks leveraging PF and IPF control, develop power system operation and planning tools compatible with PF control, and quantify the impacts of PF and IPF control on multi-decade transmission planning. The results suggest that planning and operation of the power system are feasible with PF controllers and may lead to cost savings. The proposed planning frameworks may incent transmission investment and be compatible with the existing transmission planning process. If the results of the planning tool demonstration scale to the national level, the annual savings in electricity expenditures would be $13 billion per year (2010$). The proposed incremental packetized energy concept may facilitate a reduction in the environmental impact of energy consumption and lead to additional cost savings.Ph.D

Planning and flexible operation of storage systems in power grids: from transmission to distribution networks

The first part of the thesis has been devoted to the transmission planning with high penetration of renewable energy sources. Both stationary and transportable battery energy storage (BES, BEST) systems have been considered in the planning model, so to obtain the optimal set of BES, BEST and transmission lines that minimizes the total cost in a power network. First, a coordinated expansion planning model with fixed transportation cost for BEST devices has been presented; then, the model has been extended to a planning formulation with a distance-dependent transportation cost for the BEST units, and its tractability has been proved through a case study based on a 190-bus test system. The second part of this thesis is then devoted to the analysis of planning and management of renewable energy communities (RECs). Initially, the planning of photovoltaic and BES systems in a REC with an incentive-based remuneration scheme according to the Italian regulatory framework has been analysed, and two planning models, according to a single-stage, or a multi-stage approach, have been proposed in order to provide the optimal set of BES and PV systems allowing to achieve the minimum energy procurement cost in a given REC. Further, the second part of this thesis is devoted to the study of the day-ahead scheduling of resources in renewable energy communities, by considering two types of REC. The first one, which we will refer to as “cooperative community”, allows direct energy transactions between members of the REC; the second type of REC considered, which we shall refer to as “incentive-based”, does not allow direct transactions between members but includes economic revenues for the community shared energy, according to the Italian regulation framework. Moreover, dispatchable renewable energy generation has been considered by including producers equipped with biogas power plants in the community

Practical Optimisation of District Energy Systems: Representation of Technology Characteristics, Demand Uncertainty, and System Robustness

District energy systems are an alternative to conventional national-scale networks to meet demand in urban areas. Decentralising electricity production reduces distribution losses, while centralising thermal energy production capitalises on economies of scale. Furthermore, geographic proximity facilitates the use of waste heat from electricity production to meet thermal demand. However, it is difficult to assess which configuration of technologies will best suit a cluster of users. Mathematical optimisation techniques have been extensively researched as a method to resolve this, as they can simplify the design of investment portfolios and operation schedules for a given set of geolocated demands. However, they are not yet practically applicable. This thesis uses the open-source, mixed integer linear programming framework Calliope to present three methodological enhancements which address model simplification, parameter uncertainty, and conflicting decision-maker objectives. These enhancements enable the practical design of district energy systems through data-centric workflows which can readily represent real system complexities in a tractable optimisation model. This thesis first examines the impact of parameter simplification in a linear model. Piecewise linearisation is applied to nonlinear part-load energy consumption curves and a pre-processing step is developed to optimise breakpoint positioning along a piecewise curve. Second, a three-step method is proposed to handle demand uncertainty in linear models, using historical demand data. These steps are scenario generation, scenario reduction, and scenario optimisation. Using out-of-sample tests, robustness of optimal investments to unmet demand is quantified. Furthermore, system resilience to unexpected events is explored by introducing interruptions to the national electricity grid availability for a district in Bangalore, India. Scenario optimisation is extended to account for these interruptions as well as to mitigate unfavourably high levels of CO$_2$ emissions in system design. Finally, this thesis identifies eight possible decision makers, who each hold a different objective in the design of a district energy system. Optimal technology configuration and out-of-sample test results are compared across all decision-maker objectives. These methodological enhancements demonstrate the capability of optimisation models to be reflective of reality whilst being transparent concerning the impact of simplifications, uncertainty, and conflicting objectives. Calliope is extended in this thesis to be practically applicable for district energy systems. Moreover, its extensibility facilitates the continuation of development, including possible future work into data validation and spatial dimension simplification.This research was supported by the Engineering and Physical Sciences Research Council (grant EP\L016095\1). This work was performed using resources provided by the Cambridge Service for Data Driven Discovery (CSD3) operated by the University of Cambridge Research Computing Service (http://www.csd3.cam.ac.uk/), provided by Dell EMC and Intel using Tier-2 funding from the Engineering and Physical Sciences Research Council (capital grant EP\P020259\1), and DiRAC funding from the Science and Technology Facilities Council (www.dirac.ac.uk)