Daniel Bienstock, Columbia University
Title: Solving large-scale multi-time period ACOPF problems (slides)
Abstract: This is joint work with Matias Villagra, PhD student, Columbia Univeristy. The ACOPF, or Alternating Current Power Flow problem, is a nonlinear and nonconvex optimization problem that arises in the operation of power grids and has recently acquired increased relevance as power grids evolve around the world. In this talk we first present linear relaxations for ACOPF that prove simultaneously fastest and tightest over all other relaxations, especially on large, multi-time period cases. As an additional attribute, the relaxations are especially fast when warm started, a feature that is important since in real-world applications there is always a prior solution. Second, we present heuristics for computing feasible solutions using Knitro as a subroutine. These heuristics prove effective on the same cases (e.g., large multi time-period instances) as certified by our relaxations.
Filippo Pecci, Princeton University
Title: Regularized Benders Decomposition for High Performance Capacity Expansion Models (slides)
Abstract: We consider electricity capacity expansion models, which optimize investment and retirement decisions by minimizing both investment and operation costs. In order to provide credible support for planning and policy decisions, these models need to include detailed operations and time-coupling constraints, and allow modeling of discrete planning decisions. Such requirements result in large-scale mixed integer optimization problems that are intractable with off-the-shelf solvers. Hence, practical solution approaches often rely on carefully designed abstraction techniques to find the best compromise between reduced temporal and spatial resolutions and model accuracy. Benders decomposition methods offer scalable approaches to leverage distributed computing resources and enable models with both high resolution and computational performance. Unfortunately, such algorithms are known to suffer from instabilities, resulting in oscillations between extreme planning decisions that slows convergence. In this study, we implement and evaluate several level-set regularization schemes to avoid the selection of extreme planning decisions. Using a large capacity expansion model of the Continental United States with over 70 million variables as a case study, we find that a regularization scheme that selects planning decisions in the interior of the feasible set shows superior performance compared to previously published methods, enabling high-resolution, mixed-integer planning problems with unprecedented computational performance.
Antonio Frangioni, Dipartimento di Informatica, Università di Pisa
Title: The Segmented Pay-as-Clear Approach for (Energy) Markets (slides)
Abstract: Motivated by the recent crisis of the European electricity markets, we propose the concept of Segmented Pay-as-Clear (SPaC) market, introducing a new family of market clearing problems that achieve a degree of decoupling between groups of participants. This requires a relatively straightforward modification of the standard PaC model and retains its crucial features by providing both long- and short-term sound price signals. The approach is based on dynamically partitioning demand across the segmented markets, where the partitioning is endogenous, i.e., controlled by the model variables, and is chosen to minimise the total system cost. The thusly modified model leads to solving Bilevel Programming problems, or more generally Mathematical Programs with Complementarity Constraints; these have a higher computational complexity than those corresponding to the standard PaC, but in the same ballpark as the models routinely used in real-world Day Ahead Markets (DAMs) to represent "nonstandard" requirements, e.g., the unique buying price in the Italian DAM. Thus, SPaC models should still be solvable in a time compatible with market operation with appropriate algorithmic tools. Like all market models, SPaC is not immune to \emph{strategic bidding} techniques, but some theoretical results indicate that, under the right conditions, the effect of these could be limited. An initial experimental analysis of the proposed models, carried out through Agent Based simulations, seems to indicate a good potential for significant system cost reductions and an effective decoupling of the two markets.
Bissan Ghaddar, Ivey Business School
Title: Polynomial Optimization Applied to Power Network Operations (slides)
Abstract: Several challenging optimization problems in power networks involve operational decisions, non-linear models of the underlying physics described by the network as well as uncertainty in the system parameters. However, these networks exhibit a nice structure. This talk provides an overview of approaches that combine recent advances in robust optimization and conic relaxations of polynomial optimization problems along with exploiting the sparse structure of the underlying problem. These approaches are demonstrated in applications arising in power networks.
Title: On investment in power systems (slides)
Abstract: In this talk we will discuss the question of investment in power systems wherein the need for flexibility would be a driving force for investments. The investment cost itself consists of a capital investment cost and an operational cost, wherein the latter evaluates the cost of operating the system for a given installed capacity. We argue that the operational cost is best dealt with through the use of the Lagrangian dual. Not only does the latter offer clear computational advantages, but it also naturally incorporates precise modelling of "flexibility". The suggested formalism accounts for accurate models of complex systems. Even though the use of the Lagrangian entails an implicit convexification, we show that if sufficiently many units are to be invested in, the approximation gap vanishes. The laid out computational advantages specific to investment, on top of the well-known decomposition advantages of the Lagrangian, make the suggested formalism, we believe, a serious competitor to balance accuracy and computational performance as investment is concerned.
Marco Capelletti and Giuseppe De Nicolao, Università di Pavia
Title: Probabilistic forecasting of wind power production: a generalized linear model approach (slides)
Abstract: Power grid planning and management must cope with the inherent uncertainty of renewable energy production. For this reason, it is essential to move from point forecasts to probabilistic forecasts that provide the statistical distribution of future production several hours in advance. In the talk, the forecasting of wind power from wind speed forecasts is addressed. The heteroschedasticity and skewness of the wind power distribution conditional on predicted wind speed preclude the direct application of classical regression methods. A promising approach is provided by generalized linear models (GLMs), which provide the required flexibility, while preserving many properties of linear regression models. Specifically, we discuss a two-step beta regression model that includes a preconditioning step followed by maximum likelihood estimation of the GLM. The approach is illustrated through the analysis of real data available in a public repository.
Cristian Bovo, Università di Pavia
Title: The N-1 security constraints modelling in AC-OPF problems
Abstract: In recent years, voltage and reactive power control has become increasingly critical in the operation of the power system due to the trend of operating transmission networks as close as possible to their maximum capacity. In assessing operational security, it is important to also consider the security aspects related to reactive power dispatch. To account for the security associated with reactive power dispatch, in addition to current constraints, it is necessary to adopt the AC PF model instead of the DC PF model. In this formulation, given the strong nonlinearity of the problem, integrating N-1 constraints becomes complex: a set of constraints for each critical N-1 condition should be considered, consisting of all AC PF equations and related security conditions, greatly increasing the problem size to the point where it becomes unsolvable within reasonable computation times, as well as making the resolution process less robust. Therefore, it is necessary to identify a simplified and reasonable formulation of the constraints. Specifically, in the optimization model, N-1 security constraints have been modeled by calculating a reasonable estimate of voltage and/or current variation following a line or generator outage. To perform this analysis, it is necessary to calculate the voltage/current variations in the post-contingency state as accurately as possible. A specific problem is the nonlinearity of the reactive sub-problem: while changes in actual power flows following an outage can typically be calculated using first-order coefficients (via the Inverse Matrix Modification Lemma - IMML), the reactive sub-problem is highly nonlinear, and therefore the same methodology for determining voltage variations would not show the same accuracy. To mitigate this problem, second-order derivatives have been introduced to estimate the network voltage effects due to line or generation contingency tripping.
Title: Integrated planning of multi-energy systems (slides)
Abstract: The multi-energy systems optimal planning (OptiMES) main challenge is the equilibrium between model accuracy and computational feasibility. This is due to the need of large-scale, nationwide, generation and transmission ex-pansion model that incorporates aspects such as optimal units and storage scheduling, space constraints, and renewable energy sources availability. Furthermore, the deep decarbonization required by future energy systems necessitates interconnected energy infrastructures—such as electric, heat, gas, and long-term energy storage—and accounts for all possible interactions over multi-year periods, resulting in increased model complexity. OptiMES is modeled as a Mixed integer linear problem and aims to include all the key features of state-of-the-art models obtainable from a linear formulation. The objective function is the minimization of the overall system cost, and it in-cludes investment, maintenance, network expansion, transmission hourly prices, energy regulations, and fuel prices. The model has been implemented in Python language using Gurobi solver with an hourly time granularity. It can compute a yearly instance in few minutes, depending on the number of nodes of the network. The model flexibility enabled to run various sensitivi-ty analyses to tackle uncertainty factors, considering fuel availability, natural resources, different technologies, regulations, load profiles and their evolu-tion in time. This model has been incorporated by OPTIT in a Decision Sup-port System, consisting in a Web application that is being developed and will be proposed on the market for TSO, DSO, and National/Regional energy planners, in order to design optimal decarbonization pathways.
Marco Rossi, Ricerca sul Sistema Energetico - RSE S.p.A.
Title: Coordinated planning of transmission and distribution networks based on optimization (slides)
Abstract: Network operators are currently facing the challenges of renewable generation increase and demand electrification, which lead to the necessity of adapting the electricity system to future scenarios. Thanks to the development of ancillary services markets, flexibility from both transmission and distribution resources can be considered a competitive alternative to the conventional grid reinforcement, and current regulation promotes flexibility exploitation for planning and operation purposes. The presentation will analyse the optimization tools aimed at evaluating the cost-effectiveness of storage and demand-side-management solutions with respect to reinforced lines and transformers, particularly focusing on the difficult task behind the coordination of the necessarily-separated planning procedures of transmission and distribution networks.
Maria Teresa Vespucci, Department of Management, Information and Production Engineering, University of Bergamo
Title: Analyzing the market power in day-ahead electricity market and provision of flexibility services under different TSO-DSOs coordination schemes (slides)
Abstract: The ambitious decarbonisation targets set by the European Commission and the need to make Europe as independent as possible from gas and fossil fuels are driving energy systems towards an ever increasing penetration of renewable energy sources, characterised by a typically intermittent generation pattern. The growth of Distributed Energy Resources (DER), most frequently connected to the distribution networks, leads on the one hand to an increased demand for flexibility services from system operators, and on the other hand provides an additional opportunity for DER owners to provide flexibility services to both distribution and transmission networks. Energy markets are oligopolies with a limited number of participating entities: for example, in a hypothetical market for local distribution services, the radial topology of the distribution system greatly reduces the number of entities that can be active in solving congestion in a given branch of the system. The ability to exercise market power (i.e. the ability to adopt bidding strategies to maximise profits), also known as “gaming”, includes the ability to arbitrage between the day-ahead market and the services market, which could be seen as two successive stages of a single process. Therefore, the mathematical representation of market gaming needs to include both day-ahead and services markets. In this two-stage market (day-ahead followed by a services market), different architectures can be considered. The services market can be a single session open to both transmission and distribution connected entities, and in this case the market constraints should include transit limits for the whole system. Another possibility is that the services market is, in turn, split into a transmission services market and several distribution (local) services markets.
Oligopolistic competition between flexibility resource aggregators in the two markets is formulated as a "game" in which each subject brings its own optimisation problem: bidders optimise their own revenues, market operators optimise social welfare. We develop a procedure to determine the Nash equilibrium between aggregators (a solution from which no aggregator has an interest in deviating unilaterally). A bilevel model is used to determine the profit-maximising bid prices for each aggregator, i.e. the day-ahead market bid prices, the prices of upward and downward regulation bids, and the prices of load curtailment bids.
Different alternatives are considered for the coordination scheme between the transmission and distribution services markets: (a) a single services market for both transmission and distribution, in which resources connected to both transmission and distribution operate; (b) separate services markets for transmission and distribution, in which (b1) either only resources connected to the respective network operate in each market, or (b2) resources in distribution that are not used in the respective networks can provide transmission services. A CIGRE benchmark network (transmission and distribution) has been used for preliminary numerical tests.
Authors: Maria Teresa Vespucci (1), Giovanni Micheli (1), Gianluigi Migliavacca (2), Dario Siface (2)
1 Department of Management, Information and Production Engineering, University of Bergamo, Italy
2 Ricerca sul Sistema Energetico - RSE SpA, Milano, Italy
Andrea Pitto, Ricerca sul Sistema Energetico - RSE S.p.A.
Title: An optimization-based methodology for power system resilience enhancement in planning (slides)
Abstract: Achieving a good level of resilience to extreme events caused by severe weather conditions is a major target for operators in modern power systems. Moreover, regulatory authorities are pushing transmission and distribution operators to prepare resilience plans suitably supported by Cost–Benefit Analyses (CBA). The presentation intends to describe an optimization-based methodology developed in RSE for resilience-informed planning.
The methodology aims at selecting the optimal mix of active and passive measures for resilience enhancement, achieved within a Cost-Benefit Analysis (CBA) framework relying on an Optimization via Simulation (OvS) approach. A scenario-based approach is set up, which starts from a set of representative grid scenarios and a comprehensive set of candidate grid hardening (passive) and operational (active) measures; risk-based CBA indicators are quantified, accounting for probabilistic models of climate changes. The adoption of a heuristic direct search-based method in an OvS approach is justified by its good capability to tackle the complexity of the formulated combinatorial problem: on one side, the simulation module in the OvS approach allows a realistic quantification of the Energy Not Served provoked by multiple contingencies, by simulating the actual response of the system, including the potential cascading outages activated by the initiating contingencies. On the other side, the heuristic solution method allows to achieve satisfactory solutions for practical applications with acceptable computational times. The application of the methodology to the model of a large portion of the Italian Extra-High Voltage transmission system demonstrates the practicability of the approach in planning for resilience.
Gabor Riccardi, Università di Pavia
Title: A Parallel Algorithm for Large-Scale Capacity Expansion Models of the Power Grid (slides)
Abstract: This study focuses on decomposition resolution algorithms for large-scale Capacity Expansion models of the Power Grid, designed to identify future capacity expansion and retirement opportunities of the generators. Because of the inherently stochastic nature of the problem, and the size of the European network, the European Networks of Transmission System Operators for Electricity (ENTSO-E) has employed a relaxed deterministic model due to computational limitations. Our research reframes the problem as a two-stage stochastic program and introduces a new parallel computing method based on the sparse interconnectedness of different timesteps inducing an iteratively tightened relaxation of the problem. We prove the convergence to the exact solution of the method and consider some generalizations based on the hypergraph structure of an LP problem.
Title: On cycle inequalities for the Optimal Transmission Switching
Abstract: In the distribution of power energy, electrical current must satisfy Kirchoff and Ohm's laws. Power distribution can be modeled in two main ways: Direct Current (DC) and Alternate Current (AC) models. In this talk we only consider the DC setting.
According to the DC model, electrical current must satisfy two kinds of constraints: flow conservation at each node, taking account of production and consumption of energy, and voltage angle constraints for each link of the distribution network.
The contemporary satisfaction of both these constraints may cause a case of the well-known Braess paradox, that is it is possible that eliminating some links may reduce the total production costs (usually associated only to thermal production units).
Therefore, it arises the need to find the optimal set of links to activate: this problem is named the Optimal Transmission Switching (OTS) problem.
In [1] it is presented a new formulation for the OTS problem based on the definition of particular cycle inequalities. In that paper a heuristic procedure for separating those inequalities was proposed.
In this talk we present a new pseudopolynomial algorithm for the exact separations of cycle inequalities for the OTS problem and we present some preliminary computational results.
Bibliography:
[1] Burak Kocuk, Hyemin Jeon, Santanu S. Dey, Jeff Linderoth, James Luedtke, and Xu Andy Sun. A cycle-based formulation and valid inequalities for DC power transmission problems with switching. Operations Research, 64(4):922–938, 2016
Ambrogio Maria Bernardelli, University of Pavia
Title: Optimal Power Flow problem: a study on Jabr relaxation (slides)
Abstract: The Optimal Power Flow (OPF) problem, in its most realistic form, is large-scale, non-smooth, non-convex, and nonlinear. It generally has multiple local minima and a corresponding feasibility problem that is strongly NP-hard. In recent years, the study of the OPF problem in its polar form has gained attention thanks to the second-order cone program relaxation given by the Jabr inequality. The corresponding equality, together with additional constraints called loop constraints, is proven to be exact if the graph representing the network is a tree. In this work, we focus on the study of the feasibility and exactness of the Jabr equality with the addition of loop constraints with respect to the structure of the network. In particular, we analyze the case in which the network of the OPF problem contains loops.
Paul-Niklas Kandora, Karlsruhe Institute of Technology
Title: Strengthening convex relaxations of the radial OPF problem via copositive optimization
Abstract: Copositive reformulations can be used to derive convex underestimators of optimal value functions of QCQPs, which can be dualized to derive piecewise affine approximations. We apply this machinery to the so-called radial OPF problem. First, we reformulate the problem via indicator functions, which can be cast as optimal value functions. Using the copositivity-based convex underestimators we can derive a family of convex relaxations that are at least as strong as the popular positive semidefinite relaxation. We use a Benders Decomposition method for solving the resulting relaxations which require solving copositive subproblems. These are tackled via a cutting plane algorithm where a copositivity test with respect to a special quadratic cone has to be performed in each iteration. Due to the structure of said quadratic cone, this test can be performed in polynomial time.
Alfredo Vaccaro, University of Sannio
Title: Robust and Scalable Optmization Modeling: the enabler for reliable operation of decarbonized power systems (slides)
Abstract: Optimal power system operation requires intensive numerical analysis to study and improve system security and reliability. To address this issue, optimization models are some of the most important tools, since they represent the mathematical foundation of many power engineering applications.
However, the transition to decarbonized power systems is posing unprecedented challenges to the conventional optimization and computation techniques currently deployed in electrical grids operation tools, which should enable rapid decisions in a data rich, but information limited environment. In this context, the research for highly scalable optimization frameworks, which aim at coordinating small, distributed and flexible energy sources, and for robust decision support systems, which aim at modelling the effects of complex and correlated data uncertainties on reliable power system operation, represent two relevant issues to address. In trying and addressing these challenging issues, this talk will discuss the latest advances and the enabling methodologies for solving optimization problems in the context of realistic decarbonized power systems operation scenario. The results of experimental studies obtained on the Italian power transmission system will be presented and discussed in order to emphasize the potential role of modern optimization techniques in solving challenging operation problems.
Bertrand Cornélusse, University of Liège, Belgium
Title: Considering the feedback of grid users when planning distribution networks (slides)
Abstract: The growing electrification of transportation, heating, and cooling will largely impact electricity distribution networks. To determine how to develop distribution networks, it is paramount to consider jointly the multi-year distribution network development plan and the grid users’ energy infrastructure evolution. To this end, we formulate a bilevel program in a one-leader multi-follower setting with the distribution network development plan as the upper level, while the lower level minimizes grid users’ energy costs. Solving this optimization problem allows for assessing the impact of exogenous factors, such as grid tariffs, on network development plans and grid user investment in distributed energy sources and storage. Some initial results are reported using a small test system.
Valentin Ilea, Politecnico di Milano
Title: Transient Stability Constrained Optimal Power Flow Problems (slides)
Abstract: The steady increase of converter-based renewable energy sources, HVDC lines and battery storage systems is steadily substituting the traditional synchronous generation-based power plants. Soon, this will lead to very low levels of inertia in modern power systems, thus reducing their frequency stability. In this context, the optimal dispatch of the resources in power systems, traditionally achieved through deployment of Optimal Power Flow (OPF) tools, will no longer be able to ignore the severity of frequency transients following an unexpected event. Traditionally approached in a static manner through assuring a minimum level of frequency regulation reserve, the frequency security will have to be modeled in OPF tools as a dynamic phenomenon, meaning that the OPF will become a Transient Stability Constrained – OPF (TSC-OPF) problem. In this presentation recent advancements in the formulation of TSC-OPF problems will be presented and analyzed focusing on frequency stability in low-inertia modern power systems.