DIMACS Center, CoRE Building, Rutgers University

**Organizers:****Daniel Bienstock**, Columbia University, dano at columbia dot edu**Steven Low**, Caltech, slow at caltech dot edu

Title: DC Power Flow in Rectangular Coordinates

The phrase "DC power flow'' refers to the use of an analogy between approximations to the real power components of the power flow equations and a direct current resistive circuit. The approximation can also be interpreted as a linearization of the real power components expressed in terms of phasor voltage magnitude and phase, linearized about a "flat start,'' where the voltage phasors each have magnitude one and angle zero. The accuracy of DC power flow for estimating real power flow is surprisingly good in many cases, although it has large errors when angle differences across lines are large, which is the most critical case for evaluating satisfaction of line flow limits.

Recently, there has been significant progress in analyzing optimal power flow expressed in terms of phasor voltage real and imaginary parts. In this paper, we explore linearization of both the real and reactive power equations expressed in terms of real and imaginary parts of the voltage phasor. We focus on linearization about a flat start, which in rectangular coordinates has the voltage phasors each with real part one and imaginary part zero.

The resulting approximation has relatively good performance for real power. Because of the analog with linearization in terms of polar voltage representation, we call this approximation "DC power flow in rectangular coordinates.''

Title: Robust and Chance-constrained Optimal Power Flow for Integration of Renewables

When uncontrollable resources fluctuate, Optimum Power Flow (OPF), routinely use d by the electric power industry to re-dispatch hourly controllable generation ( coal, gas and hydro plants) over control areas of transmission networks, can res ult in grid instability, and, potentially, cascading outages. This risk arises b ecause OPF dispatch is computed without awareness of major uncertainty, in parti cular fluctuations in renewable output. As a result, grid operation under OPF w ith renewable variability can lead to frequent conditions where power line flow ratings are significantly exceeded. Such a condition, which is borne by simulati ons of real grids, would likely resulting in automatic line tripping to protect lines from thermal stress, a risky and undesirable outcome which compromises sta bility. Smart grid goals include a commitment to large penetration of highly flu ctuating renewables, thus calling to reconsider current practices, in particular the use of standard OPF. Our Chance Constrained (CC) OPF corrects the problem a nd mitigates dangerous renewable fluctuations with minimal changes in the curren t operational procedure. Assuming availability of a reliable wind forecast param eterizing the distribution function of the uncertain generation, our CC-OPF sati sfies all the constraints with high probability while simultaneously minimizing the cost of economic re-dispatch. CC-OPF allows efficient implementation, e.g. s olving a typical instance over the 2746-bus Polish network in 20 seconds on a st andard laptop.

Title: Power Market Participation of Distributed Flexible Loads that Require Energy and Provide Regulation Reserve Capacity

We investigate the shortcomings of uniform price quantity bids and offers in current multi-period power markets, notably day-ahead-markets, in eliciting socially optimal demand response and equally importantly obtaining regulation service reserve offers from flexible loads. More specifically, we show that under the current day-ahead-market uniform price quantity bidding rules, individual flexible loads at the distribution network as well as storage units connected at the transmission system have the perverse incentive to self-schedule based on their estimate of market clearing price trajectories rather than reveal their true utility. We proceed to show that the hierarchical game of iterative individual self-scheduling followed by ISO market clearing price recalculation, converges under reasonable conditions to the socially optimal equilibrium. Moreover, we propose revised bidding rules that, if adopted, achieve the social optimum schedule of distributed flexible loads under mild competitive conditions. A three bus transmission system with nine distribution feeders representing commercial and residential consumers is employed to provide illustrative numerical results.

Title: Large-Scale Optimal Power Flow (SuperOPF) with Stability Constraints: Methodology and Implementation

The stochastic contingency-based security constrained AC Optimal Power Flow formulation behind the SuperOPF makes it very applicable to a variety of problems arising in power system planning and operations under deregulation. The ultimate goal of this development is to develop a commercial-grade SuperOPF in the context of co-optimization framework that correctly accounts for contingencies, ancillary services, static and dynamic constraints in determining both dispatch, price and operating reserve.

This paper is focused on the following: (i) enhancing SuperOPF (into SuperOPF-VS (voltage stability)) in its capability to deal with a large set of contingencies subject to voltage stability constraints, (ii) adjusting (or redispatching) both real and reactive control variables so that SuperOPF-contingency can perform the application functions needed in a ISO/RTO-scale Energy Management System (EMS), (iii) enhancing SuperOPF in its capability to deal with different objective functions needed in power system operation and planning.

A four-stage, multi-level, adaptive homotopy-enhanced Interior Point OPF solver will be presented in this paper. This solver is composed of four stages for robustness and efficiency, This four-stage Solver has been evaluated on practical power system models with dimensions ranging from 6,000-bus to 13,500-bus. The results will be presented in this paper.

Title: The best-deterministic method for the stochastic unit commitment problem

Assuming a large penetration of wind energy, with an accurate forecast of wind energy variations except for stochastic timing errors, we investigate numerical situations where a two-stage stochastic unit commitment problem over a few scenarios becomes hopelessly hard to solve. We present an approach for solving approximately such programs by separately optimizing second-stage decisions given a fixed first-stage solution, obtained by solving a parametric deterministic approximation. Methods are presented to set the parameters of the deterministic approximation: (i) by direct search, (ii) by solving an auxiliary stochastic program, (iii) by iterative search. As parameters, we consider quantile levels of wind distributions. Situations where this choice is fully justified are identified.

Title: A Price-Based Approach to Control of Networked Distributed Energy Resources

We introduce a framework for controlling the energy provided or absorbed by distributed energy resources (DERs) in power distribution networks. In this framework, there is a set of agents referred to as aggregators that interact with the wholesale electricity market, and through some market-clearing mechanism, are requested (and will be compensated for) to provide (or absorb) certain amount of active (or reactive) power over some period of time. In order to fulfill the request, each aggregator interacts with a set of DERs and offers them some price per unit of active (or reactive) power they provide (or absorb); the objective is for the aggregator to design a pricing strategy for incentivizing DERs to change its active (or reactive) power consumption (or production) so as they collectively provide the amount that the aggregator has been asked for. In order to make a decision, each DER uses the price information provided by the aggregator and some estimate of the average active (or reactive) power that neighboring DERs can provide computed through some exchange of information among them; this exchange is described by a connected undirected graph. The focus is on the DER strategic decision-making process, which we cast as a game. In this context, we provide sufficient conditions on the aggregator's pricing strategy under which this game has a unique Nash equilibrium. Then, we propose a distributed iterative algorithm that adheres to the graph that describes the exchange of information between DERs that allows them to seek for this Nash equilibrium. We illustrate our results through several numerical simulations.

(Joint work with Bahman Gharesifard and Tamer Basar)

Speaker's Bio

Alejandro Domininguez-Garcia is an Assistant Professor in the Electrical and Computer Engineering Department at the University of Illinois, Urbana, where he is affiliated with the Power and Energy Systems area. His research interests lie at the interface of system reliability theory and control, with special emphasis on applications to electric power systems and power electronics.

Dr. Alejandro Domininguez-Garcia received the Ph.D. degree in Electrical Engineering and Computer Science from the Massachusetts Institute of Technology, Cambridge, MA, in 2007 and the degree of Electrical Engineer from the University of Oviedo (Spain) in 2001.

After finishing his Ph.D., he spent some time as a post-doctoral research associate at the Laboratory for Electromagnetic and Electronic Systems of the Massachusetts Institute of Technology. Alejandro Domininguez-Garcia received the NSF CAREER Award in 2010, and the Young Engineer Award from the IEEE Power and Energy Society in 2012. He is an editor of the IEEE Transactions on Power Systems and the IEEE Power Engineering Letters. He is also a Grainger Associate since August 2011.

Title: Toward Making the Most Out of Available Electric Energy Technologies at Value

In this talk we introduce a definition of technology value for the changing electric energy systems. We propose that the value of new technologies has contextual, temporal and spatial dimensions. In particular, an "optimal grid" represents a non-unique combination of different technologies and algorithms for their utilization, each of which bringing a break-even value to the system as a whole when compared to other candidate optimal technologies.

These somewhat abstract notions of technology value are illustrated by simulating and comparing the effects of candidate technologies such as: 1) investment in transmission lines vs. investment in FACTS in systems with intermittent resources and responsive demand; 2) AC OPF vs. DC OPF as an on-line resource management tool; and, 3) various candidate technologies for stable load following. We show that any combination of technologies which is capable of providing electricity services according to the well-defined terms and conditions is a possible mix of technologies for ensuring reliable and efficient services. We propose that meeting terms of pre-agreed service conditions should be considered as an acceptable service approach, without requiring today's unconditional (N-1) reliability reserves or following rigid operational "nomograms". Simple examples of optimal energy systems with new technology mixes are provided.

Next, a value-based approach to providing incentives for meeting well-defined terms of electricity services is proposed. Such an approach begins to open doors to technology choices capable of aligning closely with the system-level objectives. Notably, it can be shown that this proposed value-based approach to incentivizing technology deployment and utilization guarantees near-certain cost recovery. The implications on deployment of IT, responsive demand, solar and wind power are discussed.

Finally, a general problem posing, modeling and decision-making framework for supporting value-based decision-making and electricity services is proposed. Typical performance using current cost-based operations and planning approach is revisited and compared to the proposed value-based operations and planning approach.

Title: Various Techniques for Nonlinear Energy-Related Optimizations

The operation of next generation electric grids will likely rely on solving large-scale, dynamic optimization problems involving hundreds of thousands of devices jointly optimizing millions of variables. This is due in part to the presence of distributed generators, batteries, deferrable loads and curtailable loads. These problems are not only large scale but also non-convex, where the non-convexity imposed by nonlinear physical laws can introduce inferior local solutions. To address this issue, we study various convexification techniques (SDP relaxation, sum-of-squares and regularization) for optimal power flow (OPF) and its variants (say stability-constrained OPF). We prove that the non-convexity can be eliminated in several scenarios due to the physics of transmission lines and transformers. In particular, we show that certain energy-related optimizations are easy to solve in two cases: (i) for acyclic networks and (ii) for mesh networks where the angle difference across each line is not too large at optimality.

Title: Branch Flow Model: Relaxations, Convexification, Equivalence

We propose a branch flow model for the analysis and optimization of mesh as well as radial networks. The model leads to a new approach to solving optimal power flow (OPF) problems that consists of two relaxation steps. The first step eliminates the voltage and current angles and the second step approximates the resulting problem by an second-order conic program (SOCP) that can be solved efficiently. For radial networks, we prove that both relaxation steps are always exact, provided some mild conditions are satisfied. For mesh networks, the conic relaxation is always exact and we characterize when the angle relaxation may fail. We propose a simple method to convexify a mesh network using phase shifters so that both relaxation steps are always exact and OPF for the convexified network can always be solved efficiently for a globally optimal solution. We prove that convexification requires phase shifters only outside a spanning tree of the network graph and their placement depends only on network topology, not on power flows, generation, loads, or operating constraints. Finally, we prove that our branch flow model is equivalent to the traditional bus injection model and its associated semidefinite relaxations.

(Joint work with Masoud Farivar, Lingwen Gan, Lina Li, Subhonmesh Bose, Lijun Chen, Ufuk Topcu, Mani Chandy, Caltech)

Title: Solving the Economic Power Dispatch and Related Problems More Efficiently (and Reliably)

To optimize the benefits electric power service to society, we need to solve the optimization problem that includes AC power flows, unit commitment, transmission switching and investment. This problem is non-convex in binary variables and nonconvex in continuous complex variables. The basic model can be used to formulate and solve the optimal plan for generation and transmission, the optimal maintenance schedule, the unit commitment, the real-time market look ahead and corrective switching to recover from a contingency. In addition, the time intervals need to be shortened and stochastic elements need to be included. This quickly makes the problem large and complex. This paper will examine the current state of the art, approximations, high-valued research and the next generation of models. We will also present results solving the ACOPF using different formulations and the linear approximation to the current-voltage formulation.

Title: Intelligence by Design in an Entropic Power Grid

In this talk, the term Entropic Grid is used to describe a power grid with increased levels of uncertainty and dynamics. These new features will require the reconsideration of well-established paradigms in the way of planning and operating the grid and its associated markets. The leverage of this knowledge will facilitate the design of new architectures to organize power and energy systems.

A key element for evaluating new designs is to understand the impact that salient features of an entropic grid---uncertainty, dynamics, constraints---can have on the electricity markets. Using a multi-settlement dynamic electricity market, the impact of volatility and dynamic costs are investigated. The following conclusions are obtained:

- Not surprisingly, responsive generation is needed to cope with the volatility of renewable sources, but responsiveness (ramping generation) can be costly.
- It is found that ramping costs, when included in the dynamic electricity market model, in average terms are not reflected in electricity prices in a spot market.

These results reveal several difficulties with today's markets that are based on spot pricing. Alternative market architectures will be proposed, along with several future research questions.

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Document last modified on February 12, 2013.