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Preprint.[show/hide abstract]We prove that nonconvex quadratically constrained quadratic programs can be solved in polynomial time when their underlying graph is acyclic, provided the constraints satisfy a certain technical condition. When this condition is not satisfied, we propose a heuristic to obtain a feasible point. We demonstrate this approach on optimal power flow problems over radial networks.

Proceedings of IEEE IGCC, 2014.[show/hide abstract]This paper surveys the opportunities and challenges in an emerging area of research that has the potential to significantly ease the incorporation of renewable energy into the grid as well as electric power peakload shaving: data center demand response. Data center demand response sits at the intersection of two growing fields: energy efficient data centers and demand response in the smart grid. As such, the literature related to data center demand response is sprinkled across multiple areas and worked on by diverse groups. Our goal in this survey is to demonstrate the potential of the field while also summarizing the progress that has been made and the challenges that remain.

Proceedings of IEEE CDC, 2014.[show/hide abstract]We study the role of a market maker (or system operator) in a transmission constrained electricity market. We model the market as a oneshot networked Cournot competition where generators supply quantity bids and load serving entities provide downward sloping inverse demand functions. This mimics the operation of a spot market in a deregulated market structure. In this paper, we focus on possible mechanisms employed by the market maker to balance demand and supply. In particular, we consider three candidate objective functions that the market maker optimizes  social welfare, residual social welfare, and consumer surplus. We characterize the existence of Generalized Nash Equilibrium (GNE) in this setting and demonstrate that market outcomes at equilibrium can be very different under the candidate objective functions.

Proceedings of ACM Sigmetrics, 2014.[show/hide abstract]Demand response is a crucial tool for the incorporation of renewable energy into the grid. In this paper, we focus on a particularly promising industry for demand response: data centers. We use simulations to show that, not only are data centers large loads, but they can provide as much (or possibly more) flexibility as large scale storage, if given the proper incentives. However, due to the market power most data centers maintain, it is difficult to design programs that are efficient for data center demand response. To that end, we propose that predictionbased pricing is an appealing market design, and show that it outperforms more traditional supplyfunction bidding mechanisms in situations where market power is an issue. However, predictionbased pricing may be inefficient when predictions are not accurate, and so we provide analytic, worstcase bounds on the impact of prediction accuracy on the efficiency of predictionbased pricing. These bounds hold even when network constraints are considered, and highlight that predictionbased pricing is surprisingly robust to prediction error.

IEEE Transactions on Power Systems, 2013. 28:25542572.[show/hide abstract]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) that consists of two relaxation steps. The first step eliminates the voltage and current angles and the second step approximates the resulting problem by a conic program that can be solved efficiently. For radial networks,we prove that both relaxation steps are always exact, provided there are no upper bounds on loads. For mesh networks, the conic relaxation is always exact but the angle relaxation may not be exact, and we provide a simple way to determine if a relaxed solution is globally optimal. We propose convexification of mesh networks using phase shifters so that OPF for the convexified network can always be solved efficiently for an optimal solution. We prove that convexification requires phase shifters only outside a spanning tree of the network and their placement depends only on network topology, not on power flows, generation, loads, or operating constraints. Part I introduces our branch flow model, explains the two relaxation steps, and proves the conditions for exact relaxation. Part II describes convexification of mesh networks, and presents simulation results.

IEEE Transactions on Power Systems, 2012. 27(1):92107.[show/hide abstract]The optimal power flow (OPF) problem is nonconvex and generally hard to solve. In this paper, we propose a semidefinite programming (SDP) optimization, which is the dual of an equivalent form of the OPF problem. A global optimum solution to the OPF problem can be retrieved from a solution of this convex dual problem whenever the duality gap is zero. A necessary and sufficient condition is provided in this paper to guarantee the existence of no duality gap for the OPF problem. This condition is satisfied by the standard IEEE benchmark systems with 14, 30, 57, 118 and 300 buses as well as several randomly generated systems. Since this condition is hard to study, a sufficient zerodualitygap condition is also derived. This sufficient condition holds for IEEE systems after small resistance (10^{âˆ’5} per unit) is added to every transformer that originally assumes zero resistance. We investigate this sufficient condition and justify that it holds widely in practice. The main underlying reason for the successful convexification of the OPF problem can be traced back to the modeling of transformers and transmission lines as well as the nonnegativity of physical quantities such as resistance and inductance.

Proceedings of IEEE CDC, 2011.[show/hide abstract]We propose a simple model that integrates twoperiod electricity markets, uncertainty in renewable generation, and realtime dynamic demand response. A loadserving entity decides its dayahead procurement to optimize expected social welfare a day before energy delivery. At delivery time when renewable generation is realized, it sets prices to manage demand and purchase additional power on the realtime market, if necessary, to balance supply and demand. We derive the optimal dayahead decision, propose realtime demand response algorithm, and study the effect of volume and variability of renewable generation on these optimal decisions and on social welfare.
Preprints

Preprint.[show/hide abstract]Power systems are undergoing unprecedented transformations with the incorporation of larger amounts of renewable energy sources, distributed generation and demand response. All these changes, while potentially making power grids more responsive, efficient and resilient, also pose significant implementation challenges. In particular, operating the new power grid will require new tools and algorithms capable of predicting if the current state of the system is operationally safe. In this paper we study and generalize the socalled energy function as a tool to design algorithms to test if a highvoltage power transmission system is within the allowed operational limits. In the past the energy function technique was utilized primarily to access the power system transient stability. In this manuscript, we take a new look at energy functions and focus on an aspect that has previously received little attention: Convexity. We characterize the domain of voltage magnitudes and phases within which the energy function is convex. We show that the domain of the energy function convexity is sufficiently large to include most operationally relevant and practically interesting cases. We show how the energy function convexity can be used to analyze power flow equations, e.g. to certify solution uniqueness or nonexistence within the domain of convexity. This and other useful features of the generalized energy function are described and illustrated on IEEE 14 and 118 bus models.

Preprint.[show/hide abstract]We explore two different frequency control strategies to ensure stability of power networks and achieve economic dispatch between generators and controllable loads. We first show the global asymptotic stability of a completely decentralized frequency integral control. Then we design a distributed averagingbased integral (DAI) control which operates by local frequency sensing and neighborhood communication. Equilibrium analysis shows that DAI recovers nominal frequency with minimum total generation cost and user disutility for load control after a change in generation or load. Local asymptotic stability of DAI is established with a Lyapunov method. Simulations demonstrate improvement in both transient and steadystate performance achieved by the proposed control strategies, compared to droop control.

IEEE Transactions on Power Systems, Preprint.[show/hide abstract]The feeder reconfiguration problem chooses the on/off status of the switches in a distribution network in order to minimize a certain cost such as power loss. It is a mixed integer nonlinear program and hence hard to solve. In this paper we propose a heuristic algorithm that is based on the recently developed convex relaxation of the AC optimal power flow problem. The algorithm is computationally efficient and scales linearly with the number of redundant lines. It requires neither parameter tuning nor initialization for different networks. It successfully computes an optimal configuration on all four networks we have tested. Moreover we have proved that the algorithm solves the feeder reconfiguration problem optimally under certain conditions for the case where only a single redundant line needs to be opened. We also propose a more computationally efficient algorithm and show that it incurs a loss in optimality of less than 3\% on the four test networks.

IEEE Transactions on Automatic Control, Preprint.[show/hide abstract]To schedule a large number of EVs with the presence of practical nonconvex charging constraints, a distributed and randomized algorithm is proposed in this paper. The algorithm assumes the availability of a coordinator which can communicate with all EVs. In each iteration of the algorithm, the coordinator receives tentative charging profiles from the EVs and computes a broadcast control signal. After receiving this broadcast control signal, each EV generates a probability distribution over its admissible charging profiles, and samples from the distribution to update its tentative charging profile. We prove that the algorithm converges almost surely to a charging profile in finite iterations. The final charging profile (that the algorithm converges to) is random, i.e., it depends on the realization. We characterize the final charging profileÑa charging profile can be a realization of the final charging profile if and only if it is a Nash equilibrium of the game in which each EV seeks to minimize the inner product of its own charging profile and the aggregate electricity demand. Furthermore, we provide a uniform suboptimality upper bound, that scales O(1/n) in the number n of EVs, for all realizations of the final charging profile.

IEEE Transactions on Power Systems, Preprint.[show/hide abstract]Market power assessment is a prime concern when designing a deregulated electricity market. In this paper, we propose a new functional market power measure, termed \emphtransmission constrained network flow TCNF, that takes into account an AC model of the network. The measure unifies three large classes of longterm transmission constrained market power indices in the literature: residual supply based, network flow based, and minimal generation based. Furthermore it is built upon the recent advances in semidefinite relaxations of AC power flow equations to model the underlying power network. Previously, market power measure that took into account the network did so via DC approximations of power flow models. Our results highlight that using the more accurate AC model can yield fundamentally different conclusions both about whether market power exists and about which generators can exploit market power.

Preprint.[show/hide abstract]We formulate an optimal load control (OLC) problem in power networks where the objective is to minimize the aggregate cost of tracking an operating point subject to power balance over the network. We prove that the swing dynamics and the branch power flows, coupled with frequencybased load control, serve as a distributed primaldual algorithm to solve OLC. Even though the system has multiple equilibrium points, we prove that it nonetheless converges to an optimal point. This result implies that the local frequency deviations at each bus convey exactly the right information about the global power imbalance for the loads to make individual decisions that turn out to be globally optimal. It allows a completely decentralized solution without explicit communication among the buses. Simulations show that the proposed OLC mechanism can resynchronize bus frequencies with significantly improved transient performance.

Preprint.[show/hide abstract]We prove that nonconvex quadratically constrained quadratic programs can be solved in polynomial time when their underlying graph is acyclic, provided the constraints satisfy a certain technical condition. When this condition is not satisfied, we propose a heuristic to obtain a feasible point. We demonstrate this approach on optimal power flow problems over radial networks.
2016
2015

2015.

Proceedings of Hawaii International Conference on System Sciences, 2015.[show/hide abstract]We consider joint control of a switchable capacitor and a DSTATCOM for voltage regulation in a distribution circuit with intermittent load. The control problem is formulated as a twotimescale optimal power flow problem with chance constraints, which minimizes power loss while limiting the probability of voltage violations due to fast changes in load. The control problem forms the basis of an optimization problem which determines the sizes of the control devices by minimizing sum of the expected power loss cost and the capital cost. We develop computationally efficient heuristics to solve the optimal sizing problem and implement realtime control. Numerical experiments on a circuit with highperformance computing (HPC) load show that the proposed sizing and control schemes significantly improve the reliability of voltage regulation on the expense of only a moderate increase in cost.

IEEE Transactions on Automatic Control, 2015. 7287.[show/hide abstract]The optimal power flow (OPF) problem determines a network operating point that minimizes a certain objective such as generation cost or power loss. It is nonconvex. We prove that a global optimum of OPF can be obtained by solving a secondorder cone program, under a mild condition after shrinking the OPF feasible set slightly, for radial power networks. The condition can be checked a priori, and holds for the IEEE 13, 34, 37, 123bus networks and two realworld networks.

2015.
2014

Proceedings of American Control Conference, 2014.[show/hide abstract]Automatic generation control (AGC) regulates mechanical power generation in response to load changes through local measurements. Its main objective is to maintain system frequency and keep energy balanced within each control area so as to maintain the scheduled net interchanges between control areas. The scheduled interchanges as well as some other factors of AGC are determined at a slower time scale by considering a centralized economic dispatch (ED) problem among different generators. However, how to make AGC more economically efficient is less studied. In this paper, we study the connections between AGC and ED by reverse engineering AGC from an optimization view, and then we propose a distributed approach to slightly modify the conventional AGC to improve its economic efficiency by incorporating ED into the AGC automatically and dynamically.

Proceedings of IEEE CDC, 2014.[show/hide abstract]We augment existing generatorside primary frequency control with loadside control that are local, ubiquitous, and continuous. The mechanisms on both the generator and the load sides are decentralized in that their control decisions are functions of locally measurable frequency deviations. These local algorithms interact over the network through nonlinear power flows. We design the local frequency feedback control so that any equilibrium point of the closedloop system is the solution to an optimization problem that minimizes the total generation cost and user disutility subject to power balance across entire network. With Lyapunov method we derive a sufficient condition for any equilibrium point of the closedloop system to be asymptotically stable. A simulation demonstrates improvement in both the transient and steadystate performance over the traditional control only on generators, even when the total control capacity remains the same.

Proceedings of IEEE CDC, 2014.[show/hide abstract]The optimal power flow (OPF) problem is fundamental in power system operations and planning. Largescale renewable penetration calls for realtime feedback control, and hence the need for fast and distributed solutions for OPF. This is difficult because OPF is nonconvex and KirchhoffÕs laws are global. In this paper we propose a solution for radial networks. It exploits recent results that suggest solving for a globally optimal solution of OPF over a radial network through the secondorder cone program (SOCP) relaxation. Our distributed algorithm is based on alternating direction method of multiplier (ADMM), but unlike standard ADMM algorithms that often require iteratively solving optimization subproblems in each ADMM iteration, our decomposition allows us to derive closed form solutions for these subproblems, greatly speeding up each ADMM iteration. We present simulations on a realworld 2,065bus distribution network to illustrate the scalability and optimality of the proposed algorithm.

IEEE Transactions on Power Systems, 2014. 29:28922904.[show/hide abstract]The optimal power flow (OPF) problem determines power generations/demands that minimize a certain objective such as generation cost or power loss. It is nonconvex and NPhard in general. In this paper, we study the OPF problem in direct current (DC) networks. A secondorder cone programming (SOCP) relaxation is considered for solving the OPF problem. We prove that the SOCP relaxation is exact if either 1) voltage upper bounds do not bind; or 2) voltage upper bounds are uniform and power injection lower bounds are negative. Based on 1), a modified OPF problem is proposed, whose corresponding SOCP is guaranteed to be exact. We also prove that SOCP has at most one optimal solution if it is exact. Finally, we discuss how to improve numerical stability and how to include line constraints.

Power Systems Computation Conference, 2014.[show/hide abstract]Distribution networks are usually multiphase and radial. To facilitate power flow computation and optimization, two semidefinite programming (SDP) relaxations of the optimal power flow problem and a linear approximation of the power flow are proposed. We prove that the first SDP relaxation is exact if and only if the second one is exact. Case studies show that the second SDP relaxation is numerically exact and that the linear approximation obtains voltages within 0.0016 per unit of their true values for the IEEE 13, 34, 37, 123bus networks and a realworld 2065bus network.

IEEE Transactions on Power Systems, 2014.[show/hide abstract]Distribution networks are usually multiphase and radial. To facilitate power flow computation and optimization, two semidefinite programming (SDP) relaxations of the optimal power flow problem and a linear approximation of the power flow are proposed. We prove that the first SDP relaxation is exact if and only if the second one is exact. Case studies show that the second SDP relaxation is numerically exact and that the linear approximation obtains voltages within 0.0016 per unit of their true values for the IEEE 13, 34, 37, 123bus networks and a realworld 2065bus network.

Proceedings of IEEE CDC, 2014.[show/hide abstract]Deferrable load control is an essential tool for handling the uncertainties associated with increasing penetration of renewable load control. Model predictive control has emerged as an effective approach for managing deferrable loads, and has received considerable attention. In particular, previous work has derived tight bounds on the averagecase performance of model predictive deferrable load control. However, to this point, distributional analysis of model predictive deferrable load control has been elusive. In this paper, we adapt the Martingale bounded difference approach in order to prove strong concentration results on the distribution of the load variance that results from model predictive deferrable load control. These concentration results highlight, among other things, the impact of shortrange and longrange dependencies in the prediction errors.

Proceedings of IEEE IGCC, 2014.[show/hide abstract]This paper surveys the opportunities and challenges in an emerging area of research that has the potential to significantly ease the incorporation of renewable energy into the grid as well as electric power peakload shaving: data center demand response. Data center demand response sits at the intersection of two growing fields: energy efficient data centers and demand response in the smart grid. As such, the literature related to data center demand response is sprinkled across multiple areas and worked on by diverse groups. Our goal in this survey is to demonstrate the potential of the field while also summarizing the progress that has been made and the challenges that remain.

Proceedings of IEEE CDC, 2014.[show/hide abstract]We study the role of a market maker (or system operator) in a transmission constrained electricity market. We model the market as a oneshot networked Cournot competition where generators supply quantity bids and load serving entities provide downward sloping inverse demand functions. This mimics the operation of a spot market in a deregulated market structure. In this paper, we focus on possible mechanisms employed by the market maker to balance demand and supply. In particular, we consider three candidate objective functions that the market maker optimizes  social welfare, residual social welfare, and consumer surplus. We characterize the existence of Generalized Nash Equilibrium (GNE) in this setting and demonstrate that market outcomes at equilibrium can be very different under the candidate objective functions.

Optimal power procurement for data centers in dayahead and realtime electricity marketsProceedings of Infocom Workshop on Smart Data Pricing, 2014.[show/hide abstract]With the growing trends in the amount of power consumed by data centers, finding ways to cut electricity bills has become an important and challenging problem. In this paper, we seek to understand the cost reductions that data centers may achieve by exploiting the diversity in the price of electricity in the dayahead and realtime electricity markets. Based on a stochastic optimization framework, we propose to jointly select the data centers' service rates and their demand bids to the dayahead and realtime electricity markets. In our analysis, we take into account service levelagreements, risk management constraints, and also the statistical characteristics of the workload and the electricity prices. Using empirical electricity price and Internet workload data, our numerical studies show that directly participating in the dayahead and realtime electricity markets can significantly help data centers to reduce their energy expenditure.

Proceedings of ACM Sigmetrics, 2014.[show/hide abstract]Demand response is a crucial tool for the incorporation of renewable energy into the grid. In this paper, we focus on a particularly promising industry for demand response: data centers. We use simulations to show that, not only are data centers large loads, but they can provide as much (or possibly more) flexibility as large scale storage, if given the proper incentives. However, due to the market power most data centers maintain, it is difficult to design programs that are efficient for data center demand response. To that end, we propose that predictionbased pricing is an appealing market design, and show that it outperforms more traditional supplyfunction bidding mechanisms in situations where market power is an issue. However, predictionbased pricing may be inefficient when predictions are not accurate, and so we provide analytic, worstcase bounds on the impact of prediction accuracy on the efficiency of predictionbased pricing. These bounds hold even when network constraints are considered, and highlight that predictionbased pricing is surprisingly robust to prediction error.

Proceedings of ACM Sigmetrics, 2014.[show/hide abstract]The increasing penetration of intermittent, unpredictable renewable energy sources, such as wind energy, pose significant challenges for the utility companies trying to incorporate renewable energy in their portfolio. In this talk, we discuss inventory management issues that arise in the presence of intermittent renewable resources. We model the problem as a three stage newsvendor problem with uncertain supply and model the estimates of wind using a martingale model of forecast evolution. We describe the optimal procurement strategy and use it to study the impact of proposed market changes and of increased renewable penetration. A key insight from our results is to show a separation between the impact of the structure of electricity markets and the impact of increased penetration. In particular, the effect of market structure on the optimal procurement policy is independent of the level of wind penetration. Additionally, we study two proposed changes to the market structure: the addition and the placement of an intermediate market. Importantly, we show that addition of an intermediate market does not necessarily reduce the total amount of energy procured by the utility company.

2014.

2014.

IEEE Transactions on Automatic Control, 2014. 59:11771189.[show/hide abstract]We present a systematic method to design ubiquitous continuous fastacting distributed load control for primary frequency regulation in power networks, by formulating an optimal load control (OLC) problem where the objective is to minimize the aggregate cost of tracking an operating point subject to power balance over the network. We prove that the swing dynamics and the branch power flows, coupled with frequencybased load control, serve as a distributed primaldual algorithm to solve OLC. We establish the global asymptotic stability of a multimachine network under such type of loadside primary frequency control. These results imply that the local frequency deviations at each bus convey exactly the right information about the global power imbalance for the loads to make individual decisions that turn out to be globally optimal. Simulations confirm that the proposed algorithm can rebalance power and resynchronize bus frequencies after a disturbance with significantly improved transient performance.
2013

Proceedings of IEEE CDC, 2013.[show/hide abstract]The feeder reconfiguration problem chooses the on/off status of the switches in a distribution network in order to minimize a certain cost such as power loss. It is a mixed integer nonlinear program and hence hard to solve. A popular heuristic search consists of repeated application of branch exchange, where some loads are transferred from one feeder to another feeder while maintaining the radial structure of the network, until no load transfer can further reduce the cost. Optimizing each branch exchange step is itself a mixed integer nonlinear program. In this paper we propose an efficient algorithm for optimizing a branch exchange step. It uses an AC power flow model and is based on the recently developed convex relaxation of optimal power flow. We provide a bound on the gap between the optimal cost and that of our solution. We prove that our algorithm is optimal when the voltage magnitudes are the same at all buses. We illustrate the effectiveness of our algorithm through the simulation of realworld distribution feeders.

The need for new measures to assess market power in deregulated electricity marketsIEEE Smart Grid Newsletter, 2013.

Proceedings of IEEE CDC, 2013.[show/hide abstract]The feeder reconfiguration problem chooses the on/off status of the switches in a distribution network in order to minimize a certain cost such as power loss. It is a mixed integer nonlinear program and hence hard to solve. A popular heuristic search consists of repeated application of branch exchange, where some loads are transferred from one feeder to another feeder while maintaining the radial structure of the network, until no load transfer can further reduce the cost. Optimizing each branch exchange step is itself a mixed integer nonlinear program. In this paper we propose an efficient algorithm for optimizing a branch exchange step. It uses an AC power flow model and is based on the recently developed convex relaxation of optimal power flow. We provide a bound on the gap between the optimal cost and that of our solution. We prove that our algorithm is optimal when the voltage magnitudes are the same at all buses. We illustrate the effectiveness of our algorithm through the simulation of realworld distribution feeders.

Proceedings of IEEE CDC, 2013.[show/hide abstract]We consider a class of local volt/var control schemes where the control decision on the reactive power at a bus depends only on the local bus voltage. These local algorithms form a feedback dynamical system and collectively determine the bus voltages of a power network. We show that the dynamical system has a unique equilibrium by interpreting the dynamics as a distributed algorithm for solving a certain convex optimization problem whose unique optimal point is the system equilibrium. Moreover, the objective function serves as a Lyapunov function implying global asymptotic stability of the equilibrium. The optimization based model does not only provide a way to characterize the equilibrium, but also suggests a principled way to engineer the control. We apply the results to study the parameter setting for the inverterbased volt/var control in the proposed IEEE 1547.8 standard.

Proceedings of IEEE CDC, 2013.[show/hide abstract]The optimal power flow (OPF) problem seeks to control power generation/demand to optimize certain objectives such as minimizing the generation cost or power loss. It is becoming increasingly important for tree distribution networks due to the emerging distributed generation and controllable loads. The OPF problem is nonconvex. We prove that after modifying the OPF problem, its global optimum can be recovered via a secondorder cone programming (SOCP) relaxation for tree networks, under a condition that can be checked in advance. Empirical studies justify that the modification is

Proceedings of IEEE CDC, 2013.[show/hide abstract]The optimal power flow (OPF) problem determines power generations/demands that minimize a certain objective such as generation cost or power loss. It is nonconvex and NPhard in general. In this paper, we study the OPF problem in direct current (DC) networks. A secondorder cone programming (SOCP) relaxation is considered for solving the OPF problem. We prove that the SOCP relaxation is exact if either 1) voltage upper bounds do not bind; or 2) voltage upper bounds are uniform and power injection lower bounds are negative. Based on 1), a modified OPF problem is proposed, whose corresponding SOCP is guaranteed to be exact. We also prove that SOCP has at most one optimal solution if it is exact. Finally, we discuss how to improve numerical stability and how to include line constraints.

Proceedings of IEEE CDC, 2013.[show/hide abstract]A large fraction of the total electric load is comprised of enduse devices whose demand is inherently deferrable in time. While this latent flexibility in demand can be leveraged to absorb variability in supply from renewable generation, the challenge lies in designing incentives to induce the desired response in demand. In the following, we study a novel forward market, where consumers consent to deferred service of prespecified loads in exchange for a reduced perunit price for energy. The longer a customer is willing to defer, the larger the reduction in price. The proposed deadline differentiated forward contract provides a guarantee on the aggregate quantity to be delivered by a consumerspecified deadline. Under the earliestdeadlinefirst (EDF) scheduling policy, which is shown to be optimal for the supplier, we explicitly characterize differentiated prices yielding an efficient competitive equilibrium between supply and demand. We also show that such prices are incentive compatible (IC) in that every consumer would like to reveal her true deadline type to the supplier, provided that the other consumers are truthtelling.

Energy Policy, 2013. 62:830843.[show/hide abstract]The price of electricity supplied from home rooftop photo voltaic (PV) solar cells has fallen below the retail price of grid electricity in some areas. A number of residential households have an economic incentive to install rooftop PV systems and reduce their purchases of electricity from the grid. A significant portion of the costs incurred by utility companies are fixed costs which must be recovered even as consumption falls. Electricity rates must increase in order for utility companies to recover fixed costs from shrinking sales bases. Increasing rates will, in turn, result in even more economic incentives for customers to adopt rooftop PV. In this paper, we model this feedback between PV adoption and electricity rates and study its impact on future PV penetration and netmetering costs. We find that the most important parameter that determines whether this feedback has an effect is the fraction of customers who adopt PV in any year based solely on the money saved by doing so in that year, independent of the uncertainties of future years. These uncertainties include possible changes in rate structures such as the introduction of connection charges, the possibility of PV prices dropping significantly in the future, possible changes in tax incentives, and confidence in the reliability and maintainability of PV.

Data center demand response: avoiding the coincident peak via workload shifting and local generationSIGMETRICS Performance Evaluation Review, 2013. 70:770791.[show/hide abstract]Demand response is a crucial aspect of the future smart grid. It has the potential to provide significant peak demand reduction and to ease the incorporation of renewable energy into the grid. Data centersÕ participation in demand response is becoming increasingly important given the high and increasing energy consumption and the flexibility in demand management in data centers compared to conventional industrial facilities. In this extended abstract we briefly describe recent work in [1] on two demand response schemes to reduce a data centerÕs peak loads and energy expenditure: workload shifting and the use of local power generations. In [1], we conduct a detailed characterization study of coincident peak data over two decades from Fort Collins Utilities, Colorado and then develop two algorithms for data centers by combining workload scheduling and local power generation to avoid the coincident peak and reduce the energy expenditure. The first algorithm optimizes the expected cost and the second one provides a good worstcase guarantee for any coincident peak pattern. We evaluate these algorithms via numerical simulations based on real world traces from production systems. The results show that using workload shifting in combination with local generation can provide significant cost savings (up to 40\% in the Fort Collins UtilitiesÕ case) compared to either alone.

Proceedings of Allerton, 2013.[show/hide abstract]Driven by the national policy to expand renewable generation, as well as the advances in renewable technologies that reduce the cost of smallscale renewable generation units, distributed generation at end users will comprise a significant fraction of electricity generation in the future. We study the problem faced by a social planner who seeks to minimize the longterm discounted costs (associated with both the procurement and the usage of conventional and distributed generation assets), subject to meeting an inelastic demand for electricity. Under mild conditions on the problem parameters, we fully characterize the optimal investment policy for the social planner. We also analyze the impact of problem parameters (e.g., asset lifespans) on the optimal investment policy through numerical examples.

SIGMETRICS Performance Evaluation Review, 2013. 41:7779.[show/hide abstract]Realtime demand response is potential to handle the uncertainties of renewable generation. It is expected that a large number of deferrable loads, including electric vehicles and smart appliances, will participate in demand response in the future. In this paper, we propose a decentralized algorithm that reduces the tracking error between demand and generation, by shifting the power consumption of deferrable loads to match the generation in realtime. At each time step within the control window, the algorithm minimizes the expected tracking error to go with updated predictions on demand and generation. It is proved that the root mean square tracking error vanishes as control window expands, even in the presence of prediction errors.

Proceedings of ACM eEnergy, 2013.[show/hide abstract]Realtime demand response is essential for handling the uncertainties of renewable generation. Traditionally, demand response has been focused on large industrial and commercial loads; however, it is expected that a large number of small residential loads such as air conditioners, dish washers, and electric vehicles will also participate in the coming years. The electricity consumption of these smaller loads, which we call deferrable loads, can be shifted over time, and thus be used (in aggregate) to compensate for the random fluctuations in renewable generation. In this paper, we propose a realtime distributed deferrable load control algorithm to reduce the variance of aggregate load (load minus renewable generation) by shifting the power consumption of deferrable loads to periods with high renewable generation. At every time step, the algorithm minimizes the expected variance to go with updated predictions. We prove that suboptimality of the algorithm vanishes as time horizon expands. Further, we evaluate the algorithm via tracebased simulations.

Proceedings of IEEE Power & Energy Society General Meeting, 2013. ''Best Paper on System Operations and Market Economics'' award recipient.[show/hide abstract]A competitive deregulated electricity market with increasingly active market players is foreseen to be the future of the electricity industry. In such settings, market power assessment is a primary concern. In this paper, we propose a novel functional approach for measuring long term market power that unifies a variety of popular market power indices. Specifically, the new measure, termed transmission constrained network flow (TCNF), unifies three large classes of market power measures: residual supply based, network flow based, and minimal generation based. Further, TCNF provides valuable information about market power not captured by prior indices. We derive its analytic properties and test its efficacy on IEEE test systems.

Energy Policy, 2013. 62:830843.[show/hide abstract]The price of electricity supplied from home rooftop photovoltaic (PV) solar cells has fallen below the retail price of grid electricity in some areas. A number of residential households have an economic incentive to install rooftop PV systems and reduce their purchases of electricity from the grid. A significant portion of the costs incurred by utility companies are fixed costs which must be recovered even as consumption falls. Electricity rates must increase in order for utility companies to recover fixed costs from shrinking sales bases. Increasing rates will, in turn, result in even more economic incentives for customers to adopt rooftop PV. In this paper, we model this feedback between PV adoption and electricity rates and study its impact on future PV penetration and netmetering costs. We find that the most important parameter that determines whether this feedback has an effect is the fraction of customers who adopt PV in any year based solely on the money saved by doing so in that year, independent of the uncertainties of future years. These uncertainties include possible changes in rate structures such as the introduction of connection charges, the possibility of PV prices dropping significantly in the future, possible changes in tax incentives, and confidence in the reliability and maintainability of PV.

IEEE Transactions on Power Systems, 2013. 28:940951.[show/hide abstract]We propose decentralized algorithms for optimally scheduling electric vehicle (EV) charging. The algorithms exploit the elasticity and controllability of electric vehicle loads in order to fill the valleys in electric demand profiles. We first formulate a global optimization problem, whose objective is to impose a generalized notion of valleyfilling, and study the properties of optimal charging profiles. We then give two decentralized algorithms, one synchronous (i.e., information update takes place in each iteration) and one asynchronous (i.e., EVs may use outdated information with bounded delay in some of the iterations) to solve the problem. In each iteration of the proposed algorithms, EVs choose their own charging profiles according to the price profile broadcast by the utility, and the utility updates the price profile to guide their behavior. Both algorithms are guaranteed to converge to optimal charging profiles (that are as 'flat' as they can possibly be) irrespective of the specifications (e.g., maximum charging rate and deadline) of electric vehicles. Furthermore, both algorithms only require each EV solving its local problem, hence their implementation requires low computation capability. We also extend the algorithms to accommodate the objective of tracking a given demand profile and to realtime implementations.

IEEE Transactions on Power Systems, 2013. 28:25542572.[show/hide abstract]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) that consists of two relaxation steps. The first step eliminates the voltage and current angles and the second step approximates the resulting problem by a conic program that can be solved efficiently. For radial networks,we prove that both relaxation steps are always exact, provided there are no upper bounds on loads. For mesh networks, the conic relaxation is always exact but the angle relaxation may not be exact, and we provide a simple way to determine if a relaxed solution is globally optimal. We propose convexification of mesh networks using phase shifters so that OPF for the convexified network can always be solved efficiently for an optimal solution. We prove that convexification requires phase shifters only outside a spanning tree of the network and their placement depends only on network topology, not on power flows, generation, loads, or operating constraints. Part I introduces our branch flow model, explains the two relaxation steps, and proves the conditions for exact relaxation. Part II describes convexification of mesh networks, and presents simulation results.

IEEE Transactions on Power Systems, 2013. 28:35763587.[show/hide abstract]We propose a decentralized optimal load control scheme that provides contingency reserve in the presence of sudden generation drop. The scheme takes advantage of flexibility of frequency responsive loads and neighborhood area communication to solve an optimal load control problem that balances load and generation while minimizing enduse disutility of participating in load control. Local frequency measurements enable individual loads to estimate the total mismatch between load and generation. Neighborhood area communication helps mitigate effects of inconsistencies in the local estimates due to frequency measurement noise. Case studies show that the proposed scheme can balance load with generation and restore the frequency within seconds of time after a generation drop, even when the loads use a highly simplified power system model in their algorithms. We also investigate tradeoffs between the amount of communication and the performance of the proposed scheme through simulation based experiments.
2012

Integrating distributed energy resource pricing and controlProceedings of CIGRE USNC Grid of the Future Symposium, 2012.[show/hide abstract]As the market adoption of distributed energy resources (DER) reaches regional scale it will create significant challenges in the management of the distribution system related to existing protection and control systems. This is likely to lead to issues for power quality and reliability because of three issues. In this paper, we describe a framework for the development of a class of pricing mechanisms that both induce deep customer participation and enable efficient management of their enduse devices to provide both distribution and transmission side support. The basic challenge resides in reliably extracting the desired response from customers on short timescales. Thus, new pricing mechanisms are needed to create effective closed loop systems that are tightly coupled with distribution control systems to ensure reliability and power quality.

Proceedings of ACM Sigmetrics, 2012. Sigmetrics held jointly with IFIP Performance. An extension of this work is used in HP's Netzero Data Center Architecture, which was named a 2013 Computerworld Honors Laureate. It was one of the ten most downloaded papers of ACM SIGMETRICS in the Summer, Fall, and Winter of 2013..[show/hide abstract]The demand for data center computing increased significantly in recent years resulting in huge energy consumption. Data centers typically comprise three main subsystems: IT equipment provides services to customers; power infrastructure supports the IT and cooling equipment; and the cooling infrastructure removes the generated heat. This work presents a novel approach to model the energy flows in a data center and optimize its holistic operation. Traditionally, supplyside constraints such as energy or cooling availability were largely treated independently from IT workload management. This work reduces cost and environmental impact using a holistic approach that integrates energy supply (e.g., renewable supply, dynamic pricing) and cooling supply (e.g., chiller, outside air cooling) with IT workload planning to improve the overall attainability of data center operations. Specifically, we predict renewable energy as well as IT demand and design an IT workload management plan that schedules IT workload and allocates IT resources within a data center according to time varying power supply and cooling efficiency. We have implemented and evaluated our approach using traces from real data centers and production systems. The results demonstrate that our approach can reduce the recurring power costs and the use of nonrenewable energy by as much as 60 percent compared to existing, nonintegrated techniques, while still meeting operational goals and SLAs.

Proceedings of IEEE CDC, 2012.[show/hide abstract]The optimal power flow problem is nonconvex, and a convex relaxation has been proposed to solve it. We prove that the relaxation is exact, if there are no upper bounds on the voltage, and any one of some conditions holds. One of these conditions requires that there is no reverse real power flow, and that the resistance to reactance ratio is nondecreasing as transmission lines spread out from the substation to the branch buses. This condition is likely to hold if there are no distributed generators. Besides, avoiding reverse real power flow can be used as rule of thumb for placing distributed generators.

Chapter in Control and Optimization Theory for Electric Smart Grids, 2012.[show/hide abstract]We consider a set of users served by a single loadserving entity (LSE). The LSE procures capacity a day ahead. When random renewable energy is realized at delivery time, it manages user load through realtime demand response and purchases balancing power on the spot market to meet the aggregate demand. Hence optimal supply procurement by the LSE and the consumption decisions by the users must be coordinated over two timescales, a day ahead and in real time, in the presence of supply uncertainty. Moreover, they must be computed jointly by the LSE and the users since the necessary information is distributed among them. In this paper we present a simple yet versatile user model and formulate the problem as a dynamic program that maximizes expected social welfare. When random renewable generation is absent, optimal demand response reduces to joint scheduling of the procurement and consumption decisions. In this case, we show that optimal prices exist that coordinate individual user decisions to maximize social welfare, and present a decentralized algorithm to optimally schedule a day in advance the LSE's procurement and the users' consumptions. The case with uncertain supply is reported in a companion paper.

IEEE Transactions on Power Systems, 2012. 27(1):92107.[show/hide abstract]The optimal power flow (OPF) problem is nonconvex and generally hard to solve. In this paper, we propose a semidefinite programming (SDP) optimization, which is the dual of an equivalent form of the OPF problem. A global optimum solution to the OPF problem can be retrieved from a solution of this convex dual problem whenever the duality gap is zero. A necessary and sufficient condition is provided in this paper to guarantee the existence of no duality gap for the OPF problem. This condition is satisfied by the standard IEEE benchmark systems with 14, 30, 57, 118 and 300 buses as well as several randomly generated systems. Since this condition is hard to study, a sufficient zerodualitygap condition is also derived. This sufficient condition holds for IEEE systems after small resistance (10^{âˆ’5} per unit) is added to every transformer that originally assumes zero resistance. We investigate this sufficient condition and justify that it holds widely in practice. The main underlying reason for the successful convexification of the OPF problem can be traced back to the modeling of transformers and transmission lines as well as the nonnegativity of physical quantities such as resistance and inductance.

Proceedings of American Control Conference, 2012.[show/hide abstract]Increased penetration of renewable energy sources poses new challenges to the power grid. Grid integrated energy storage combined with fastramping conventional generation can help to address challenges associated with power output variability. This paper proposes a risk mitigating optimal power flow (OPF) framework to study the dispatch and placement of energy storage units in a power system with wind generators that are supplemented by fastramping conventional backup generators. This OPF with storage charge/discharge dynamics is solved as a finitehorizon optimal control problem. Chance constraints are used to implement the risk mitigation strategy. The model is applied to case studies based on the IEEE 14 bus benchmark system. First, we study the scheduling of spinning reserves and storage when generation and loads are subject to uncertainties. The framework is then extended to investigate the optimal placement of storage across different network topologies. The results of the case studies quantify the need for storage and reserves as well as suggest a strategy for their scheduling and placement.

Proceedings of American Control Conference, 2012.[show/hide abstract]Maintaining demandsupply balance and regulating frequency are key issues in power system control. Conventional approaches focus on adjusting the generation so that it follows the load. However, relying on solely regulating generation is inefficient, especially for power systems where contingencies like a sudden loss in generation or a sudden change in load frequently occur. We present a frequencybased load control scheme for demandsupply balancing and frequency regulation. We formulate a load control optimization problem which aims to balance the change in load with the change in supply while minimizing the overall enduse disutility. By studying the power system model that characterizes the frequency response to real power imbalance between demand and supply, we design decentralized synchronous and asynchronous algorithms which take advantage of local frequency measurements to solve the load control problem. Case studies show that the proposed load control scheme is capable of relatively quickly balancing the power and restoring the frequency under generationloss like contingencies, even when users only have the knowledge of a simplified system model instead of an accurate one.

Proceedings of IEEE Power & Energy Society General Meeting, 2012.[show/hide abstract]Matching demand with supply and regulating frequency are key issues in power system operations. Flexibility and local frequency measurement capability of loads offer new regulation mechanisms through load control. We present a frequencybased fast load control scheme which aims to match total demand with supply while minimizing the global enduse disutility. Local frequency measurement enables loads to make decentralized decisions on their power from the estimates of total demandsupply mismatch. To resolve the errors in such estimates caused by stochastic frequency measurement errors, loads communicate via a neighborhood area network. Case studies show that the proposed load control can balance demand with supply and restore the frequency at the timescale faster than AGC, even when the loads use a highly simplified system model in their algorithms. Moreover, we discuss the tradeoff between communication and performance, and show with experiments that a moderate amount of communication significantly improves the performance.

Proceedings of IEEE Power & Energy Society General Meeting, 2012.[show/hide abstract]To address the gridside challenges associated with the anticipated high electric vehicle (EV) penetration level, various charging protocols have been proposed in the literature. Most if not all of these protocols assume continuous charging rates and allow intermittent charging. However, due to charging technology limitations, EVs can only be charged at a fixed rate, and the intermittency in charging shortens the battery lifespan. We consider these charging requirements, and formulate EV charging scheduling as a discrete optimization problem. We propose a stochastic distributed algorithm to approximately solve the optimal EV charging scheduling problem in an iterative procedure. In each iteration, the transformer receives charging profiles computed by the EVs in the previous iteration, and broadcasts the corresponding normalized total demand to the EVs; each EV generates a probability distribution over its potential charging profiles accordingly, and samples from the distribution to obtain a new charging profile. We prove that this stochastic algorithm almost surely converges to one of its equilibrium charging profiles, and each of its equilibrium charging profiles has a negligible suboptimality ratio. Case studies corroborate our theoretical results.

Proceedings of IEEE Power & Energy Society General Meeting, 2012.[show/hide abstract]The intent of the study detailed in this paper is to demonstrate the benefits of inverter var control on a fast timescale to mitigate rapid and large voltage fluctuations due to the high penetration of photovoltaic generation and the resulting reverse power flow. Our approach is to formulate the volt/var control as a radial optimal power flow (OPF) problem to minimize line losses and energy consumption, subject to constraints on voltage magnitudes. An efficient solution to the radial OPF problem is presented and used to study the structure of optimal inverter var injection and the net benefits, taking into account the additional cost of inverter losses when operating at nonunity power factor. This paper will illustrate how, depending on the circuit topology and its loading condition, the inverter's optimal reactive power injection is not necessarily monotone with respect to their real power output. The results are demonstrated on a distribution feeder on the Southern California Edison system that has a very light load and a 5 MW photovoltaic (PV) system installed away from the substation.

Proceedings of IEEE SmartGridComm Conference, 2012.[show/hide abstract]In electricity transmission networks, loads can provide ßexible, fast responsive, and decentralized sources for frequency regulation and generationdemand balancing, complementary to generation control. We consider an optimal load control (OLC) problem in a transmission network, when a disturbance in generation occurs on an arbitrary subset of the buses. In OLC, the frequencyinsensitive loads are reduced (or increased) in realtime in a way that balances the generation shortfall (or surplus), resynchronizes the bus frequencies, and minimizes the aggregate disutility of load control. We propose a frequencybased load control mechanism and show that the swing dynamics of the network, together with the proposed mechanism, act as a decentralized primaldual algorithm to solve OLC. Simulation shows that the proposed mechanism can resynchronize the bus frequencies, balance demand with generation and achieve the optimum of OLC within several seconds after a disturbance in generation. Through simulation, we also compare the performance of the proposed mechanism with automatic generation control (AGC), and discuss the effect of their incorporation.
2011

Proceedings of IEEE Power & Energy Society General Meeting, 2011.[show/hide abstract]Demand side management will be a key component of future smart grid that can help reduce peak load and adapt elastic demand to fluctuating generations. In this paper, we consider households that operate different appliances including PHEVs and batteries and propose a demand response approach based on utility maximization. Each appliance provides a certain benefit depending on the pattern or volume of power it consumes. Each household wishes to optimally schedule its power consumption so as to maximize its individual net benefit subject to various consumption and power flow constraints. We show that there exist timevarying prices that can align individual optimality with social optimality, i.e., under such prices, when the households selfishly optimize their own benefits, they automatically also maximize the social welfare. The utility company can thus use dynamic pricing to coordinate demand responses to the benefit of the overall system. We propose a distributed algorithm for the utility company and the customers to jointly compute this optimal prices and demand schedules. Finally, we present simulation results that illustrate several interesting properties of the proposed scheme.

Proceedings of Allerton, 2011.[show/hide abstract]We consider a set of users served by a single load serving entity (LSE) in the electricity grid. The LSE procures capacity a day ahead. When random renewable energy is realized at delivery time, it actively manages user load through realtime demand response and purchases balancing power on the spot market to meet the aggregate demand. Hence, to maximize the social welfare, decisions must be coordinated over two timescales (a day ahead and in real time), in the presence of supply uncertainty, and computed jointly by the LSE and the users since the necessary information is distributed among them. We formulate the problem as a dynamic program. We propose a distributed heuristic algorithm and prove its optimality when the welfare function is quadratic and the LSEâ€™s decisions are strictly positive. Otherwise, we bound the gap between the welfare achieved by the heuristic algorithm and the maximum in certain cases. Simulation results suggest that the performance gap is small. As we scale up the size of a renewable generation plant, both its mean production and its variance will likely increase. We characterize the impact of the mean and variance of renewable energy on the maximum welfare. This paper is a continuation of [2], focusing on timecorrelated demand.

Proceedings of IEEE CDC, 2011.[show/hide abstract]We propose a simple model that integrates twoperiod electricity markets, uncertainty in renewable generation, and realtime dynamic demand response. A loadserving entity decides its dayahead procurement to optimize expected social welfare a day before energy delivery. At delivery time when renewable generation is realized, it sets prices to manage demand and purchase additional power on the realtime market, if necessary, to balance supply and demand. We derive the optimal dayahead decision, propose realtime demand response algorithm, and study the effect of volume and variability of renewable generation on these optimal decisions and on social welfare.

Proceedings of IEEE CDC, 2011.[show/hide abstract]The growth of wind energy production poses several challenges in its integration in current electric power systems. In this work, we study how a wind power producer can bid optimally in existing electricity markets. We derive optimal contract size and expected profit for a wind producer under arbitrary penalty function and generation costs. A key feature of our analysis is to allow for the wind producer to strategically withhold production once the day ahead contract is signed. Such strategic behavior is detrimental to the smooth functioning of electricity markets. We show that under simple conditions on the offered price and marginal imbalance penalty, a risk neutral profit maximizing wind power producer will produce as much as wind power is available (up to its contract size).

Proceedings of ACM Greenmetrics, 2011. ''Best Student Paper'' award recipient.[show/hide abstract]Given the significant energy consumption of data centers, improving their energy efficiency is an important social problem. However, energy efficiency is necessary but not sufficient for sustainability, which demands reduced usage of energy from fossil fuels. This paper investigates the feasibility of powering internetscale systems using (nearly) entirely renewable energy. We perform a tracebased study to evaluate three issues related to achieving this goal: the impact of geographical load balancing, the role of storage, and the optimal mix of renewables. Our results highlight that geographical load balancing can significantly reduce the required capacity of renewable energy by using the energy more efficiently with

Proceedings of ACM Sigmetrics, 2011.[show/hide abstract]Energy expenditure has become a signifficant fraction of data center operating costs. Recently, `geographical load balancing' has been suggested as an approach for taking advantage of the geographical diversity of Internetscale distributed systems in order to reduce energy expenditures by exploiting the electricity price differences across regions. However, the fact that such designs reduce energy costs does not imply that they reduce energy usage. In fact, such designs often increase energy usage.
This paper explores whether the geographical diversity of Internetscale systems can be used to provide environmental gains in addition to reducing data center costs. Specifically, we explore whether geographical load balancing can encourage usage of 'green' energy from renewable sources and reduce usage of 'brown' energy from fossil fuels. We make two contributions. First, we derive three algorithms, with varying degrees of distributed computation, for achieving optimal geographical load balancing. Second, using these algorithms, we show that if dynamic pricing of electricity is done in proportion to the fraction of the total energy that is brown at each time, then geographical load balancing provides signifficant reductions in brown energy usage. However, the benefits depend strongly on the degree to which systems accept dynamic energy pricing and the form of pricing used. 
Proceedings of IFAC World Congress, 2011.[show/hide abstract]The problem to minimize power losses in an electrical network subject to voltage and power constraints is in general hard to solve. However, it has recently been discovered that semidefinite programming relaxations in many cases enable exact computation of the global optimum. Here we point out a fundamental reason for the successful relaxations, namely that the passive network components give rise to matrices with nonnegative offdiagonal entries. Recent progress on quadratic programming with Metzler matrix structure can therefore be applied.

Proceedings of Allerton, 2011.[show/hide abstract]The optimal power flow (OPF) problem is critical to power system operation but it is generally nonconvex and therefore hard to solve. Recently, a sufficient condition has been found under which OPF has zero duality gap, which means that its solution can be computed efficiently by solving the convex dual problem. In this paper we simplify this sufficient condition through a reformulation of the problem and prove that the condition is always satisfied for a tree network provided we allow oversatisfaction of load. The proof, cast as a complex semidefinite program, makes use of the fact that if the underlying graph of an n x n Hermitian positive semidefinite matrix is a tree, then the matrix has rank at least n  1.

Proceedings of IEEE SmartGridComm Conference, 2011.[show/hide abstract]The renewable energy generation such as solar and wind will constitute an important part of the next generation grid. As the variations of renewable sources may not match the time distribution of load, energy storage is essential for grid stability. Supplemented with energy storage, we investigate the feasibility of integrating solar photovoltaic (PV) panels and wind turbines into the grid. To deal with the fluctuation in both the power generation and demand, we borrow the ideas from stochastic network calculus and build a stochastic model for the power supply reliability with different renewable energy configurations. To illustrate the validity of the model, we conduct a case study for the integration of renewable energy sources into the power system of an island off the coast of Southern California. Performance of the hybrid system under study is assessed by employing the stochastic model, e.g., with a set of system configurations, the longterm expected Fraction of Time that energy NotServed (FTNS) of a given period can be obtained.

Proceedings of IEEE SmartGridComm Conference, 2011.[show/hide abstract]Hybrid energy systems with renewable generation are built in many remote areas where the renewable resources are abundant and the environment is clean. We present a case study of the Catalina Island in California for which a system with photovoltaic (PV) arrays, wind turbines, and battery storage is designed based on empirical weather and load data. To determine the system size, we formulate an optimization problem that minimizes the total construction and operation cost subject to maximum tolerable risk. Simulations using the Hybrid Optimization Model for Electric Renewable (HOMER) is used to determine the feasible set of the optimization problem.

Proceedings of IEEE SmartGridComm Conference, 2011.[show/hide abstract]Motivated by the need to cope with rapid and random fluctuations of renewable generation, we present a model that augments the traditional Volt/VAR control through switched controllers on a slow timescale with inverter control on a fast timescale. The optimization problem is generally nonconvex and therefore hard to solve. We propose a simple convex relaxation and prove that it is exact provided oversatisfaction of load is allowed. Hence Volt/VAR control over radial networks is efficiently solvable. Simulations of a realworld distribution circuit illustrates that the proposed inverter control achieves significant improvement over the IEEE 1547 standard in terms of power quality and power savings.

Proceedings of IEEE SmartGridComm Conference, 2011.[show/hide abstract]A distributed control and coordination architecture for integrating inherently variable and uncertain generation is presented. The key idea is to distribute the intelligence into the periphery of the grid. This will allow coordination of generation, storage, and adjustable demand on the distribution side of the system and thus reduce the need to build new transmission facilities to accommodate large amounts of renewable generation.

Proceedings of IEEE CDC, 2011. (Journal version submitted to IEEE Transactions on Power Systems.).[show/hide abstract]Motivated by the powergridside challenges in the integration of electric vehicles, we propose a decentralized protocol for negotiating dayahead charging schedules for electric vehicles. The overall goal is to shift the load due to electric vehicles to fill the overnight electricity demand valley. In each iteration of the proposed protocol, electric vehicles choose their own charging profiles for the following day according to the price profile broadcast by the utility, and the utility updates the price profile to guide their behavior. This protocol is guaranteed to converge, irrespective of the specifications (e.g., maximum charging rate and deadline) of electric vehicles. At convergence, the l2 norm of the aggregated demand is minimized, and the aggregated demand profile is as 'flat' as it can possibly be. The proposed protocol needs no coordination among the electric vehicles, hence requires low communication and computation capability. Simulation results demonstrate convergence to optimal collections of charging profiles within few iterations.

Proceedings of IEEE Power & Energy Society General Meeting, 2011.[show/hide abstract]Plugin hybrid electric vehicles (PHEVs) play an important role in making a greener future. Given a group of PHEVs distributed across a power network equipped with the smart grid technology (e.g. wireless communication devices), the objective of this paper is to study how to schedule the charging of the PHEV batteries. To this end, we assume that each battery must be fully charged by a prespecified time, and that the charging rate can be timevarying at discretetime instants. The scheduling problem for the PHEV charging can be augmented into the optimal power flow (OPF) problem to obtain a joint OPFcharging (dynamic) optimization. A solution to this highly nonconvex problem optimizes the network performance by minimizing the generation and charging costs while satisfying the network, physical and inelasticload constraints. A global optimum to the joint OPFcharging optimization can be found efficiently in polynomial time by solving its convex dual problem whenever the duality gap is zero for the joint OPFcharging problem. It is shown in a recent work that the duality gap is expected to be zero for the classical OPF problem. We build on this result and prove that the duality gap is zero for the joint OPFcharging optimization if it is zero for the classical OPF problem. The results of this work are applied to the IEEE 14 bus system.
2010

Proceedings of IEEE CDC, 2010.[show/hide abstract]The integration of renewable energy generation, such as wind power, into the electric grid is difficult because of the source intermittency and the large distance between generation sites and users. This difficulty can be overcome through a transmission network with largescale storage that not only transports power, but also mitigates against fluctuations in generation and supply. We formulate an optimal power flow problem with storage as a finitehorizon optimal control problem. We prove, for the special case with a single generator and a single load, that the optimal generation schedule will cross the timevarying demand profile at most once, from above. This means that the optimal policy will generate more than demand initially in order to charge up the battery, and then generate less than the demand and use the battery to supplement generation in final stages. This is a consequence of the fact that the marginal storage costtogo decreases in time.

Proceedings of Allerton, 2010. Invited paper.[show/hide abstract]The optimal power flow (OPF) problem is nonconvex and generally hard to solve. In this paper, we propose a semidefinite programming (SDP) optimization, which is the dual of an equivalent form of the OPF problem. A global optimum solution to the OPF problem can be retrieved from a solution of this convex dual problem whenever the duality gap is zero. A necessary and sufficient condition is provided in this paper to guarantee the existence of no duality gap for the OPF problem. This condition is satisfied by the standard IEEE benchmark systems with 14, 30, 57, 118 and 300 buses as well as several randomly generated systems. Since this condition is hard to study, a sufficient zerodualitygap condition is also derived. This sufficient condition holds for IEEE systems after small resistance (10^{âˆ’5} per unit) is added to every transformer that originally assumes zero resistance. We investigate this sufficient condition and justify that it holds widely in practice. The main underlying reason for the successful convexification of the OPF problem can be traced back to the modeling of transformers and transmission lines as well as the nonnegativity of physical quantities such as resistance and inductance.

Proceedings of Allerton, 2010. Invited paper.[show/hide abstract]The integration of renewable energy resources, such as solar and wind power, into the electric grid presents challenges partly due to the intermittency in the power output. These difficulties can be alleviated by effectively utilizing energy storage. We consider, as a case study, the integration of renewable resources into the electric power generation portfolio of an island off the coast of Southern California, Santa Catalina Island, and investigate the feasibility of replacing diesel generation entirely with solar photovoltaics (PV) and wind turbines, supplemented with energy storage. We use a simple storage model alongside a combination of renewables and varying loadshedding characterizations to determine the appropriate area of PV cells, number of wind turbines, and energy storage capacity needed to stay below a certain threshold probability for loadshedding over a prespecified period of time and longterm expected fraction of time at loadshedding.

Proceedings of IEEE SmartGridComm Conference, 2010.[show/hide abstract]In this paper, we consider two abstract market models for designing demand response to match power supply and shape power demand, respectively. We characterize the resulting equilibria in competitive as well as oligopolistic markets, and propose distributed demand response algorithms to achieve the equilibria. The models serve as a starting point to include the appliancelevel details and constraints for designing practical demand response schemes for smart power grids.
To appear

Convexity of structure preserving energy functions in power transmission: novel results and applicationsProceedings of American Control Conference, To appear.[show/hide abstract]It is wellknown in the power systems literature that the behavior of the transmission power system (under certain simplifying assumptions) can be used to study the postfault dynamics of a power system and provide principled estimates on dynamic stability margins. In this paper, we study a special feature of the energy function that has previously received little attention: convexity. We prove that the energy function for structure preserving models of power systems is convex under certain reasonable conditions on phases and voltages. Beyond stability analysis, these convexity results have a number of applications, noticeably, building a provably convergent PF solver, which we discuss in detail in this paper. We also outline potential applications to reformulating Optimum Power Flow (OPF), Model Predictive Control (MPC) and identifying the most probable failure (instanton) as convex optimization problems.
To be submitted

To be submitted.[show/hide abstract]Several convex relaxations of the optimal power flow (OPF) problem have recently been developed using both bus injection models and branch flow models. In this paper we prove relations among three convex relaxations: a semidefinite relaxation that computes a full matrix, a chordal relaxation based on a chordal extension of the network graph, and a secondorder cone relaxation that computes the smallest partial matrix. We prove a bijection between the feasible sets of OPF in the bus injection model and the branch flow model, establishing the equivalence of these two models and their secondorder cone relaxations. Our results imply that, for radial networks, all these relaxations are equivalent and one should always solve the secondorder cone relaxation. For mesh networks the semidefinite relaxation is tighter than the secondorder cone relaxation but requires a heavier computational effort, and the chordal relaxation strikes a good balance. Simulations are used to illustrate these results.