Advances in the theory of positive polynomials, semidefinite programming, and sum of squares decomposition have provided a very promising approach to the analysis and design of systems with polynomial vector fields. Recently, an algebraic reformulation technique has been proposed for the analysis of non-polynomial vector fields, by recasting them into rational vector fields. Sum of squares decomposition techniques can then be applied to analyze the stability of the recasted system and to infer the properties of the original, non-polynomial systems. We demonstrate the application of these technique to the transient stability analysis of power systems by providing proof of concept numerical examples that compute the local stability Lyapunov function and estimate the region of attraction of the stable operating point. This is a joint work with Federico Milano (University of Castilla - La Mancha).
The anticipated increase in the number of distributed energy resources will put additional strain on electrical distribution circuits, e.g. electrical vehicles (EV) will strain capacity, photovoltaics (PV) will challenge voltage regulation equipment, and additional switching enabling real-time circuit reconfiguration will test control logic. This and the following three talks will address different aspects of these issues. Here, we focus on a control scheme for EV charging. Many control schemes have been proposed to control EV charging. We develop control algorithms based on randomized EV charging start times and simple one-way broadcast communication allowing for a time delay between communication events. Using arguments from queuing theory and statistical analysis, we seek to maximize the utilization of excess distribution circuit capacity while keeping the probability of a circuit overload negligible.
Within the electric power literature the transmission expansion planning problem (TNEP) refers to the problem of how to upgrade an electric power network to meet future demands. As this problem is a complex, non-linear, and non-convex optimization problem, researchers have traditionally focused on approximate models of power flows. Existing approaches are often tightly coupled to the approximation choice. Until recently, these approximations have produced results that are straight-forward to adapt to the more complex (real) problem. However, the power grid is evolving towards a state where the adaptations are no longer easy (e.g. large amounts of limited control, renewable generation) that necessitates new optimization techniques. In this paper, we propose a local search variation of the powerful Limited Discrepancy Search (LDLS) that encapsulates the complexity of power flows in a black box that may be queried for information about the quality of a proposed expansion. This a llows the development of a new optimization algorithm that is independent of the underlying power model.
Sophisticated (often centralized) embedded optimization systems are widely by system operators and others to carry out generator dispatch, plan hydro system schedules, clear markets, and many other tasks arising in the operation of an energy grid. These systems minimize objectives such as minimum total generation cost, subject to many constraints on transmission, start-up and power change limits, as well as failure or outage contingency planning. We will argue that similar techniques can be applied far more widely, all the way down to the single appliance, with real-time (distributed) embedded optimization serving as a unifying scheme for the operation of a smart grid.
Topological graph models for many networks imply a relationship
between structure and performance. In order to identify the extent to which
results from topological graph models are useful for modeling vulnerability
in power systems, we measure the susceptibility of power networks to random
failures and directed attacks using three measures of vulnerability:
characteristic path lengths, connectivity loss and blackout sizes. The first
two are purely topological measures, following the procedure described in
Albert, et al (2004). The blackout size calculation results from a
simplified model of cascading failure in power networks. Tests with randomly
selected sections of the Eastern US power grid indicate that in topological
dynamics power grids are similar to random graphs, which is to be expected
given the observed exponential degree distribution. However the connectivity
loss model and the cascading failure model indicate that power grids behave
more like scale free networks, in that they are acutely more vulnerable to
directed attacks than random failures. These results suggest caution in
drawing conclusions about grid vulnerability from simple topological
metrics.
Joint work with P. Hines and E. C. Sanchez.
Ref: R. Albert, I. Albert, and G. Nakarado, “Structural vulnerability of the
north american power grid,” Physical Review E, vol. 69, no. 2, Feb 2004.
We discuss the modelling and synchronization problem for network-reduced and structure-preserving power system models. First, we focus on the network-reduced power system model with non-trivial transfer conductances - the classic swing equations. We exploit the relationship between the network-reduced power system model and a first-order model of coupled oscillators. Extending methods from transient stability, synchronization theory and consensus protocols, we establish sufficient conditions for synchronization of non-uniform Kuramoto oscillators. These conditions reduce to and improve upon previously-available tests for the well-known Kuramoto model. Combining our singular perturbation and Kuramoto analyses, we derive concise and purely algebraic conditions that establish synchronization and transient stability in a network-reduced power system. Second, we analyze the network-reduction process relating the network-reduced and the more detailed structure-preserving power system model. The network reduction process, termed Kron reduction, is characterized by iterative Schur complementation of the admittance matrix. A detailed algebraic and graph-theoretic analysis of the Kron reduction process allows us to extend the synchronization conditions obtained for the network-reduced model to the structure-preserving model. In the end, we are able state one spectral and one resistance-based condition that relate synchronization in a power network to the underlying network state, parameters, and topology. This is a joint work with Florian Dorfler.
David Chassin will discuss how retail electricity markets can be used to optimize the cost of power delivery at the distribution level when capacity is limited. He will examine the issues related to how markets themselves can be used as optimizers and how doing so affect the real-time price of electricity. Data from the Olympic Peninsula Demonstration Testbed project will be examined and the results from that project considered in the context of the ARRA projects that currently under way.
Biography: David P. Chassin is staff scientist with the Energy Science & Technology Directorate at Pacific Northwest National Laboratory where he has worked since 1992. He was Vice-President of Development for Image Systems Technology from 1987 to 1992, where he pioneered a hybrid raster/vector computer aided design (CAD) technology called CAD OverlayT. He led the development of building energy simulation and diagnostic systems, including Softdesk Energy and DOE's Whole Building Diagnostician. His recent research focuses on modeling of the Smart Grid with particular attention to load behavior, demand response, and how markets affect system behavior at the distribution level. He is currently the principal investigation and project manager for the development of DOE GridLAB-D Smart Grid simulation environment. He contributes to the Western Electricity Coordinating Council's (WECC) Load Modeling Task Force, the North American Electricity Reliability Council (NERC) Load Foreca
sting Work Group, a member of the WECC's Market Integration Committee, and he is the current chair of the OASIS Blue Steering Committee.
Here we develop an approach to predict power grid weak points, and specifically to efficiently identify the most probable failure modes in load distribution for a given power network. This approach is applied to two examples: Guam's power system and also the IEEE RTS-96 system, both modeled within the static Direct Current power flow model. Our algorithm is a power network adaption of the worst configuration heuristics, originally developed to study low probability events in physics and failures in error-correction. One finding is that, if the normal operational mode of the grid is sufficiently healthy, the failure modes, also called instantons, are sufficiently sparse, i.e. the failures are caused by load fluctuations at only a few buses. The technique is useful for discovering weak links which are saturated at the instantons. It can also identify overutilized and underutilized generators, thus providing predictive capability for improving the reliability of any power networ k. This is a joint work with F. Pan and M. Stepanov.
Despite all the changes happening in the power system, we need to maintain high reliability as an essential part of a smarter transmission grid Cascading failure is a substantial risk to electric power system reliability, especially for the large blackouts that have the largest impact. I will discuss probabilistic models that seem to capture essential features of cascading failure at a high level. These branching process models are complementary to more detailed models and offer opportunities to quantify and monitor cascading blackout risk from real or simulated data. I will show how branching process models are consistent with some industry data and show how blackout extent can be estimated.
We propose an optimization approach to design cost-effective electrical power transmission networks. That is, we aim to select both the network structure and the line conductances (line sizes) so as to optimize the trade-off between network efficiency (low power dissipation within the transmission network) and the cost to build the network. We begin with a convex optimization method based on the paper "Minimizing Effective Resistance of a Graph" [Ghosh, Boyd & Saberi]. We show that this (DC) resistive network method can be adapted to the context of AC power flow. However, that does not address the combinatorial aspect of selecting network structure. We approach this problem as selecting a subgraph within an over-complete network, posed as minimizing the (convex) network power dissipation plus a non-convex cost on line conductances that encourages sparse networks where many line conductances are set to zero. We develop a heuristic approach to solve this non-convex optimizat ion problem using: (1) a continuation method to interpolate from the smooth, convex problem to the (non-smooth, non-convex) combinatorial problem, (2) the majorization-minimization algorithm to perform the necessary intermediate smooth but non-convex optimization steps. Ultimately, this involves solving a sequence of convex optimization problems in which we iteratively reweight a linear cost on line conductances to fit the actual non-convex cost. Several examples are presented which suggest that the overall method is a good heuristic for network design. We also consider how to obtain sparse networks that are still robust against failures of lines and/or generators. Joint work with M. Chertkov.
As energy networks become increasingly interconnected and interdependent, individual plants and different energy infrastructures can no longer operate independently without incurring significant efficiency losses when compared to a network-wide, optimized policy. We propose a new optimization model that is robust, extensible, object oriented, and most importantly, amenable to modern optimization methods. This formulation has the advantage that convexity is preserved; that is, when the conversion efficiency maps, power losses, and generator efficiencies are compatible with convex optimization, so will be the problem of operating the system at maximum efficiency. The resulting model can then be solved globally and efficiently.
TBA
We investigate the problem of allocating energy from renewable sources to flexible consumers in electricity markets. We assume there is a renewable energy supplier that provides energy according to a time-varying (and possibly unpredictable) supply process. The plant must serve consumers within a specified delay window, and incurs a cost of drawing energy from other (possibly non-renewable) sources if its own supply is not sufficient to meet the deadlines. We formulate two stochastic optimization problems: The first seeks to minimize the time average cost of using the other sources (and hence strives for the most efficient utilization of the renewable source). The second allows the renewable source to dynamically set a price for its service, and seeks to maximize the resulting time average profit. These problems are solved via the Lyapunov optimization technique. Our resulting algorithms do not require knowledge of the statistics of the time-varying supply and demand proce ss es and are robust to arbitrary sample path variations.
Computer systems that control the power grid are large scale, involving many computing devices, switches, routers, and firewalls. Restriction of communication to allowable communication partners executing allowable applications is a part of the first line of defense against cyber-attack. However, configuration of protection devices to enforce allowable communication is delicate and error prone. This talk describes NetAPT (Network Access Policy Tool) and its algorithms, which address this problem. NetAPT accepts as input the configuration files of all firewall devices in a network of interest, discovers from these the network topology, and validates the configurations against an abstract statement of allowable access policy. Use of NetAPT in a recent security assessment of a large energy utility will be described.
Plug-in hybrid electric vehicles (PHEVs) are an environmentally friendly technology that is expected to rapidly penetrate the transportation system. Renewable energy sources such as wind and solar have received considerable attention as clean power options for future generation expansion. However, these sources are intermittent and increase the uncertainty in the ability to generate power. The deployment of PHEVs in a vehicle-to-grid (V2G) system provide a potential mechanism for reducing the variability of renewable energy sources. For example, PHEV supporting infrastructures like battery exchange stations that provide battery service to PHEV customers could be used as storage devices to stabilize the grid when renewable energy production is fluctuating. In this paper, we study how to best site these stations in terms of how they can support both the transportation system and the power grid. To model this problem we develop a two-stage stochastic program to optimally loc ate the stations prior to the realization of battery demands, loads, and generation capacity of renewable power sources. We develop two test cases to study the benefits and the performance of these systems. Joint work with R. Bent, A. Berscheid, D. Izraelevitz.
There is an increasing interest in renewable energy production both from economic security and environmental perspectives. The State of California has set a target of thirty percent penetration from all renewable sources by 2020. Wind energy will play a key role in realizing such aggressive targets. At today's modest (order one percent) penetration levels, wind energy is integrated into the grid by legislative fiat. At deep penetration levels called for, integration of utility-scale wind production into the electricity grid poses serious engineering and market challenges. These are due to the variability, intermittency, and uncontrollability of wind power.
In this talk we investigate ways to use a portfolio of available means to achieve deep penetration of wind generation in the current grid. This portfolio includes co-located storage, fast-acting local production, optimized contracts, and novel market instruments.
We introduce a linear programming formulation that enables us to study sensitivities and conduct parametric studies. We argue that co-located storage has a marginal economic utility of approximately 17 MW-hours-per-day for each MW-hour of storage. Our studies suggest that it will become necessary to waste some produced wind energy (when production is lower than thirty percent of nameplate capacity) to permit reliable servicing of electricity contracts. This is due to the difficulty associated with forecasting produced power at low wind levels. Finally, we suggest the use of risk-limiting contracts to achieve firming of wind-power. In these auditable contracts, the producer receives a short reprieve which enables them to offer power predictably by avoiding ramp times. We conclude by discussing how variability risk should be shared among participants in an electricity network while respecting security constraints.
The anticipated increase in the number of plug-in electric vehicles (EV) will put additional strain on electrical distribution circuits. Many control schemes have been proposed to control EV charging. Here, we develop control algorithms based on randomized EV charging start times and simple one-way broadcast communication allowing for a time delay between communication events. Using arguments from queuing theory and statistical analysis, we seek to maximize the utilization of excess distribution circuit capacity while keeping the probability of a circuit overload negligible. Joint work with S. Backhaus, N. Sinitsyn, M. Chertkov.
I will introduce a simplified model of a redundant power grid that we used to study integration of fluctuating renewable generation. The grid consists of large number of generator and consumer nodes. The net power consumption is determined by the difference between the gross consumption and the level of renewable generation. The gross consumption is drawn from a narrow distribution representing the predictability of aggregated loads, and we consider two different distributions representing wind and solar resources. Each generator is connected to $D$ consumers, and redundancy is built in by connecting $R\leq D$ of these consumers to other generators. The lines are switchable so that at any instance each consumer is connected to a single generator. We explore the capacity of the renewable generation by determining the level of "firm" generation capacity that can be displaced for different levels of redundancy $R$. We also develop message-passing control algorithm for finding switch settings where no generator is overloaded. Joint work with S. Backhaus and M. Chertkov.