A large fraction of total electricity demand is comprised of end-use devices whose demand for energy is inherently deferrable in time. Of interest is the potential to use this latent flexibility in demand to absorb variability in power supplied from intermittent renewable generation. A fundamental challenge lies in the design of incentives that induce the desired response in demand. With an eye to electric vehicle charging, we propose a novel forward market for deadline-differentiated electric power service, where consumers consent to deferred service of prespecified loads in exchange for a reduced price for energy. The longer a consumer is willing to defer, the lower the price for energy. The proposed forward contract provides a guarantee on the aggregate quantity of energy to be delivered by a consumer-specified deadline. Under the earliest-deadline-first (EDF) scheduling policy, which is shown to be optimal for the supplier, we explicitly characterize a non-discriminatory, deadline-differentiated pricing scheme that yields an efficient competitive equilibrium between the supplier and consumers. We further show that this efficient pricing scheme, in combination with EDF scheduling, is incentive compatible in that every consumer would like to reveal her true deadline to the supplier, regardless of the actions taken by other consumers.

10aRM14-0021 aBitar, Eilyan1 aXu, Yunjian uhttps://certs.lbl.gov/publications/deadline-differentiated-pricing01157nas a2200157 4500008003900000245007900039210006900118260003500187300001600222520061500238653001300853100001900866700002100885700001800906856007500924 2016 d00aA bound on the minimum rank of solutions to sparse linear matrix equations0 abound on the minimum rank of solutions to sparse linear matrix e aBoston, MA, USAbIEEEc08/2016 a6501 - 65063 aWe derive a new upper bound on the minimum rank of matrices belonging to an affine slice of the positive semidefinite cone, when the affine slice is defined according to a system of sparse linear matrix equations. It is shown that a feasible matrix whose rank is no greater than said bound can be computed in polynomial time. The bound depends on both the number of linear matrix equations and their underlying sparsity pattern. For certain problem families, this bound is shown to improve upon well known bounds in the literature. Several examples are provided to illustrate the efficacy of this bound.

10aRM14-0021 aLouca, Raphael1 aBose, Subhonmesh1 aBitar, Eilyan uhttps://certs.lbl.gov/publications/bound-minimum-rank-solutions-sparse01738nas a2200145 4500008003900000245004300039210004200082260003700124300001400161520129400175653001301469100001701482700001801499856007501517 2016 d00aData-driven pricing of demand response0 aDatadriven pricing of demand response aSydney, AustraliabIEEEc11/2016 a224 - 2293 aWe consider the setting in which an electric power utility seeks to curtail its peak electricity demand by offering a fixed group of customers a uniform price for reductions in consumption relative to their predetermined baselines. The underlying demand curve, which describes the aggregate reduction in consumption in response to the offered price, is assumed to be affine and subject to unobservable random shocks. Assuming that both the parameters of the demand curve and the distribution of the random shocks are initially unknown to the utility, we investigate the extent to which the utility might dynamically adjust its offered prices to maximize its cumulative risk-sensitive payoff over a finite number of T days. In order to do so effectively, the utility must design its pricing policy to balance the tradeoff between the need to learn the unknown demand model (exploration) and maximize its payoff (exploitation) over time. In this paper, we propose such a pricing policy, which is shown to exhibit an expected payoff loss over T days that is at most O(√T), relative to an oracle who knows the underlying demand model. Moreover, the proposed pricing policy is shown to yield a sequence of prices that converge to the oracle optimal prices in the mean square sense.

10aRM14-0021 aKhezeli, Kia1 aBitar, Eilyan uhttps://certs.lbl.gov/publications/data-driven-pricing-demand-response01673nas a2200145 4500008003900000245008900039210006900128260003700197300001400234520115800248653001301406100001701419700001801436856007301454 2016 d00aDecentralized control of distributed energy resources in radial distribution systems0 aDecentralized control of distributed energy resources in radial aSydney, AustraliabIEEEc11/2016 a296 - 3013 aWe consider the decentralized control of radial distribution systems with controllable photovoltaic inverters and storage devices. For such systems, we consider the problem of designing controllers that minimize the expected cost of meeting demand, while respecting distribution system and resource constraints. Employing a linear approximation of the branch flow model, we formulate this problem as the design of a decentralized disturbance-feedback controller that minimizes the expected value of a convex quadratic cost function, subject to convex quadratic constraints on the state and input. As such problems are, in general, computationally intractable, we derive an inner approximation to this decentralized control problem, which enables the efficient computation of an affine control policy via the solution of a conic program. As affine policies are, in general, suboptimal for the systems considered, we provide an efficient method to bound their suboptimality via the solution of another conic program. A case study of a 12 kV radial distribution feeder demonstrates that decentralized affine controllers can perform close to optimal.

10aRM14-0021 aLin, Weixuan1 aBitar, Eilyan uhttps://certs.lbl.gov/publications/decentralized-control-distributed01327nas a2200145 4500008003900000245007700039210006900116260003800185300001600223520081700239653001301056100001901069700001801088856007501106 2016 d00aA hierarchy of polyhedral approximations of robust semidefinite programs0 ahierarchy of polyhedral approximations of robust semidefinite pr aLas Vegas, NV, USAbIEEEc12/2016 a7056 - 70623 aRobust semidefinite programs are NP-hard in general. In contrast, robust linear programs admit equivalent reformulations as finite-dimensional convex programs provided that the problem data are parameterized affinely in the uncertain parameters; and that the underlying uncertainty set is described by an affine slice of a proper cone. In this paper, we propose a hierarchy of inner and outer polyhedral approximations to the positive semidefinite (PSD) cone that are exact in the limit. We apply these polyhedral approximations to the PSD cone to obtain a computationally tractable hierarchy of inner and outer approximations to the robust semidefinite program, which are similarly exact in the limit. We investigate the strengths and limitations of the proposed approach with a detailed numerical study.

10aRM14-0021 aLouca, Raphael1 aBitar, Eilyan uhttps://certs.lbl.gov/publications/hierarchy-polyhedral-approximations01539nas a2200145 4500008003900000245006400039210006400103260003800167300001600205520105500221653001301276100001701289700001801306856006901324 2016 d00aParameterized supply function equilibrium in power networks0 aParameterized supply function equilibrium in power networks aLas Vegas, NV, USAbIEEEc12/2016 a1542 - 15483 aWe consider the setting in which generators compete in scalar-parameterized supply functions to serve an inelastic demand spread throughout a transmission constrained power network. The market clears according to a locational marginal pricing mechanism, in which the independent system operator (ISO) determines the generators' production quantities so as to minimize the revealed cost of meeting demand, subject to transmission and generator capacity constraints. Under the assumption that both the ISO and generators choose their strategies simultaneously, we establish the existence of Nash equilibria for the underlying game, and derive a tight bound on its price of anarchy. Under the more restrictive setting of a two-node power network, we present a detailed comparison of market outcomes predicted by the simultaneous-move formulation of the game against those predicted by the more plausible sequential-move formulation, where the ISO observes the generators' strategy profile prior to determining their production quantities.

10aRM14-0021 aLin, Weixuan1 aBitar, Eilyan uhttps://certs.lbl.gov/publications/parameterized-supply-function01356nas a2200145 4500008003900000245005600039210005600095260003500151300001600186520089500202653001301097100001701110700001801127856006501145 2016 d00aPerformance bounds for robust decentralized control0 aPerformance bounds for robust decentralized control aBoston, MA, USAbIEEEc08/2016 a4323 - 43303 aWe consider the decentralized output feedback control of stochastic linear systems, subject to robust linear constraints on both the state and input trajectories. For problems with partially nested information structures, we establish an upper bound on the minimum achievable cost by computing the optimal affine decentralized control policy as a solution to a finite-dimensional conic program. For problems with general (possibly nonclassical) information structures, we construct another finite-dimensional conic program whose optimal value stands as a lower bound on the minimum achievable cost. With this lower bound in hand, one can bound the suboptimality incurred by any feasible decentralized control policy. A study of a partially nested system reveals that affine policies can be close to optimal, even in the presence state/input constraints and non-Gaussian disturbances.

10aRM14-0021 aLin, Weixuan1 aBitar, Eilyan uhttps://certs.lbl.gov/publications/performance-bounds-robust01658nas a2200157 4500008003900000245007400039210006900113260003800182300001600220520113200236653001301368100002001381700001801401700001301419856006801432 2016 d00aRandom convex approximations of ambiguous chance constrained programs0 aRandom convex approximations of ambiguous chance constrained pro aLas Vegas, NV, USAbIEEEc12/2016 a6210 - 62153 aWe investigate an approach to the approximation of ambiguous chance constrained programs (ACCP) in which the underlying distribution describing the random parameters is itself uncertain. We model this uncertainty with the assumption that the unknown distribution belongs to a closed ball centered around a fixed and known distribution. Using only samples drawn from the central distribution, we approximate ACCP with a robust sampled convex program (RSCP), and establish an upper bound on the probability that a solution to the RSCP violates the original ambiguous chance constraint, when the uncertainty set is defined in terms of the Prokhorov metric. Our bound on the constraint violation probability improves upon the existing bounds for RSCPs in the literature. We also consider another approach to approximating ACCP by means of a sampled convex program (SCP), which is built on samples drawn from the central distribution. Again, we provide upper bounds on the probability that a solution to the SCP violates the original ambiguous chance constraint for uncertainty sets defined according to a variety of metrics.

10aRM14-0021 aTseng, Shih-Hao1 aBitar, Eilyan1 aTang, Ao uhttps://certs.lbl.gov/publications/random-convex-approximations01857nas a2200169 4500008003900000245008600039210006900125260003500194300001600229520127400245653001301519100002401532700002201556700002301578700001801601856006801619 2016 d00aStability guarantees for primary frequency control with randomized flexible loads0 aStability guarantees for primary frequency control with randomiz aBoston, MA, USAbIEEEc08/2016 a2328 - 23333 aThere has recently been interest in diversifying the technologies that provide primary frequency control of the power grid beyond generation. One method of interest to obtain frequency control by turning on or off flexible loads in response to local measurements of line frequency. Because of the large number of loads involved, it is desirable to implement this control without communication among the loads or to a centralized controller. One proposal that does not require communication is to have each load choose the frequency at which it switches on or off randomly. In this paper we use tail bounds and absolute stability to obtain stability guarantees that are satisfied with high probability. This stability result considers not only grid inertia and damping but local measurement delays.

10aRM14-0021 aVincent, Tyrone, L.1 aPoolla, Kameshwar1 aMohagheghi, Salman1 aBitar, Eilyan uhttps://certs.lbl.gov/publications/stability-guarantees-primary01210nas a2200145 4500008003900000245005800039210005800097260003800155300001600193520073300209653001300942100001900955700001800974856007200992 2016 d00aStochastic AC optimal power flow with affine recourse0 aStochastic AC optimal power flow with affine recourse aLas Vegas, NV, USAbIEEEc12/2016 a2431 - 24363 aWith the increasing penetration of intermittent renewable energy sources into the electric power grid, there is an emerging need to develop stochastic optimization methods to enable the reliable and efficient operation of power systems having a large fraction of their power supplied form uncertain resources. In this paper, we formulate the stochastic AC optimal power flow (OPF) problem as a two-stage stochastic program with robust constraints. This problem amounts to an infinite-dimensional nonconvex optimization problem. We develop a finite-dimensional inner approximation as a semidefinite program. Its solution yields an affine recourse policy that is guaranteed to be feasible for the stochastic AC OPF problem.

10aRM14-0021 aLouca, Raphael1 aBitar, Eilyan uhttps://certs.lbl.gov/publications/stochastic-ac-optimal-power-flow01353nas a2200145 4500008003900000245008800039210006900127260003600196300001600232520083400248653001301082100001901095700001801114856007501132 2015 d00aAcyclic semidefinite approximations of quadratically constrained quadratic programs0 aAcyclic semidefinite approximations of quadratically constrained aChicago, IL, USAbIEEEc07/2015 a5925 - 59303 aQuadratically constrained quadratic programs (QCQPs) belong to a class of nonconvex optimization problems that are NP-hard in general. Recent results have shown that QCQPs having acyclic graph structure can be solved in polynomial time, provided that their constraints satisfy a certain technical condition. In this paper, we consider complex QCQPs with arbitrary graph structure and investigate the extent to which it is possible to apply structured perturbations on the problem data to yield acyclic QCQPs having optimal solutions satisfying certain approximation guarantees. Specifically, we provide sufficient conditions under which the perturbed QCQP can be solved in polynomial time to yield a feasible solution to the original QCQP and derive an explicit bound on the performance of said solution in the worst case.

10aRM14-0021 aLouca, Raphael1 aBitar, Eilyan uhttps://certs.lbl.gov/publications/acyclic-semidefinite-approximations01885nas a2200169 4500008003900000245006200039210006100101260003600162300001000198520132800208653001901536653002801555653001301583100002601596700001801622856007501640 2014 d00aFinancial storage rights: Definition and basic properties0 aFinancial storage rights Definition and basic properties aPullman, WA, USAbIEEEc09/2014 a1 - 63 aThe decreasing cost of energy storage technologies coupled with their potential to bring significant benefits to electric power networks have kindled research efforts to design both market and regulatory frameworks to facilitate the efficient integration of such technologies. The primary challenge resides in designing market systems that provide the correct incentives to deploy and operate storage systems efficiently in both the short and long-run. In the following paper, we propose an open access approach to the integration of storage in which storage is treated as a communal asset centrally operated by the System Operator (SO) to maximize social welfare; not unlike the operation of the transmission network today. Concomitantly, we propose a novel electricity derivative, which we refer to as financial storage rights (FSRs), to enable the redistribution of the additional merchandising surplus (attributable to storage) collected by the SO. FSRs do not interfere with the socially optimal operation of storage, and their definition as a sequence of nodal power injections facilitates their use by market participants to mitigate the cost and/or risk of meeting contractual commitments. Moreover, the revenue collected by the SO through the sale of FSRs can be used to remunerate capital expenditures in storage.10aenergy storage10areliability and markets10aRM11-0061 aMunoz-Alvarez, Daniel1 aBitar, Eilyan uhttps://certs.lbl.gov/publications/financial-storage-rights-definition01414nas a2200217 4500008003900000245007900039210006900118260004400187300001000231520069000241653001000931653002200941653002600963653002800989653002801017100002601045700001801071700001501089700001801104856007401122 2014 d00aPiecewise affine dispatch policies for economic dispatch under uncertainty0 aPiecewise affine dispatch policies for economic dispatch under u aNational Harbor, MD, USAbIEEEc07/2014 a1 - 53 aStochastic optimization has become one of the fundamental mathematical frameworks for modeling power systems with important sources of uncertainty in the demand and supply sides. In this framework, a main challenge is to find optimal dispatch policies and settlement schemes that support a market equilibrium. In this paper, the economic dispatch under linear network constraints and resource uncertainty is revisited. Piece-wise affine continuous dispatch policies and locational prices that support a market equilibrium using a two-settlement scheme are derived. We find that the ex-post locational prices are piecewise affine continuous functions of the system uncertainties.

10aCERTS10aeconomic dispatch10aPower system modeling10areliability and markets10astochastic optimization1 aMunoz-Alvarez, Daniel1 aBitar, Eilyan1 aTong, Lang1 aWang, Jianhui uhttps://certs.lbl.gov/publications/piecewise-affine-dispatch-policies01755nas a2200193 4500008003900000020002200039245006800061210006800129260004000197300001600237520110400253653001901357653003001376653002801406653001301434100002101447700001801468856007501486 2014 d a978-1-4799-7746-800aVariability and the Locational Marginal Value of Energy Storage0 aVariability and the Locational Marginal Value of Energy Storage aLos Angeles, CA, USAbIEEEc12/2014 a3259 - 32653 aGiven a stochastic net demand process evolving over a transmission-constrained power network, we consider the system operator's problem of minimizing the expected cost of generator dispatch, when it has access to spatially distributed energy storage resources. We show that the expected benefit of storage derived under the optimal dispatch policy is concave and non-decreasing in the vector of energy storage capacities. Thus, the greatest marginal value of storage is derived at small installed capacities. For such capacities, we provide an upper bound on the locational (nodal) marginal value of storage in terms of the variation of the shadow prices of electricity at each node. In addition, we prove that this upper bound is tight, when the cost of generation is spatially uniform and the network topology is acyclic. These formulae not only shed light on the correct measure of statistical variation in quantifying the value of storage, but also provide computationally tractable tools to empirically calculate the locational marginal value of storage from net demand time series data.

10aenergy storage10aLocational marginal value10areliability and markets10aRM11-0061 aBose, Subhonmesh1 aBitar, Eilyan uhttps://certs.lbl.gov/publications/variability-and-locational-marginal01902nas a2200241 4500008003900000022001300039245005900052210005800111260001200169300001400181490000700195520121500202653001001417653002801427653002701455653002001482653001301502100001901515700001801534700002001552700001401572856007401586 2013 d a0142061500aRisk-limiting dispatch for integrating renewable power0 aRisklimiting dispatch for integrating renewable power c01/2013 a615 - 6280 v443 aRisk-limiting dispatch or RLD is formulated as the optimal solution to a multi-stage, stochastic decision problem. At each stage, the system operator (SO) purchases forward energy and reserve capacity over a block or interval of time. The blocks get shorter as operations approach real time. Each decision is based on the most recent available information, including demand, renewable power, weather forecasts. The accumulated energy blocks must at each time t match the net demand D(t) = L(t) − W(t). The load L and renewable power W are both random processes. The expected cost of a dispatch is the sum of the costs of the energy and reserve capacity and the penalty or risk from mismatch between net demand and energy supply. The paper derives computable ‘closed-form’ formulas for RLD. Numerical examples demonstrate that the minimum expected cost can be substantially reduced by recognizing that risk from current decisions can be mitigated by future decisions; by additional intra-day energy and reserve capacity markets; and by better forecasts. These reductions are quantified and can be used to explore changes in the SO’s decision structure, forecasting technology, and renewable penetration.10aCERTS10areliability and markets10arenewables integration10areserve markets10aRM11-0061 aRajagopal, Ram1 aBitar, Eilyan1 aVaraiya, Pravin1 aWu, Felix uhttps://certs.lbl.gov/publications/risk-limiting-dispatch-integrating01437nas a2200265 4500008003900000020002200039245008200061210006900143260003300212300001400245520059000259653002000849653001800869653002700887653002800914653001500942653001300957100002300970700001900993700001801012700002501030700002201055700002001077856007401097 2012 d a978-1-4673-2065-800aOptimal power and reserve capacity procurement policies with deferrable loads0 aOptimal power and reserve capacity procurement policies with def aMaui, HI, USAbIEEEc12/2012 a450 - 4563 aDeferrable loads can be used to mitigate the variability associated with renewable generation. In this paper, we study the impact of deferrable loads on forward market operations. Specifically, we compute cost-minimizing ex-ante bulk power and reserve capacity procurement policies in the cases of fully deferrable and non-deferrable loads. For non-deferrable loads, we analytically express this policy on a partition of procurement prices. We also formulate a threshold policy for deferrable load scheduling in the face of uncertain supply, that minimizes grid operating costs.

10aload management10aload modeling10apower system economics10areliability and markets10arenewables10aRM11-0061 aSubramanian, Anand1 aTaylor, J., A.1 aBitar, Eilyan1 aCallaway, Duncan, S.1 aPoolla, Kameshwar1 aVaraiya, Pravin uhttps://certs.lbl.gov/publications/optimal-power-and-reserve-capacity01487nas a2200241 4500008003900000020002200039245004100061210004100102260003200143300001600175520079400191653001000985653002800995653002301023653002701046653001301073653001501086100001901101700001801120700001401138700002001152856007301172 2012 d a978-1-4577-1095-700aRisk limiting dispatch of wind power0 aRisk limiting dispatch of wind power aMontreal, QCbIEEEc06/2012 a4417 - 44223 aIntegrating wind and solar power into the grid requires dispatching various types of reserve generation to compensate for the randomness of renewable power. The dispatch is usually determined by a system operator (SO) or an aggregator who `firms' variable energy by bundling it with conventional power. The optimal dispatch is formulated as the solution to a stochastic control problem and shown to have a closed form that can be quickly computed. Different objectives and risk constraints can be included in the formulation and trade-offs can be evaluated. In particular one can quantify the influence of sequential forecasts on the total integration cost and the choice of dispatched generation. When the forecast error is Gaussian, the optimal dispatch policy can be precomputed.

10aCERTS10areliability and markets10areserve generation10arisk-limiting dispatch10aRM11-00610awind power1 aRajagopal, Ram1 aBitar, Eilyan1 aWu, Felix1 aVaraiya, Pravin uhttps://certs.lbl.gov/publications/risk-limiting-dispatch-wind-power01409nas a2200265 4500008003900000020002200039245002400061210002400085260003300109300001600142520068700158653001000845653002400855653001600879653002800895653002700923653001300950100001800963700002200981700002801003700001901031700002001050700001401070856005901084 2012 d a978-1-4577-1925-700aSelling Random Wind0 aSelling Random Wind aMaui, HI, USAbIEEEc01/2012 a1931 - 19373 aWind power is inherently random, but we are used to 100 percent reliable or 'firm' electricity, so reserves are used to convert random wind power into firm electricity. The cost of these reserves is frequently a hidden subsidy to wind power producers. We propose an alternative: package random wind power into electricity with different levels of reliability and sell them at different prices. This variable-reliability market is more efficient than the current firm-electricity market, and may require lower subsidy. However, we have to think of electricity differently. We also explore interesting differences between the variable-reliability and related real-time markets.

10aCERTS10aelectricity markets10areliability10areliability and markets10arenewables integration10aRM11-0061 aBitar, Eilyan1 aPoolla, Kameshwar1 aKhargonekar, Pramod, P.1 aRajagopal, Ram1 aVaraiya, Pravin1 aWu, Felix uhttps://certs.lbl.gov/publications/selling-random-wind