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A tractable ellipsoidal approximation for voltage regulation problems
... A more complex classifier would be a convex quadratic function. Such classifier has been shown to be effective for capturing chance constraints [15], which motivated us to chose the same option as the set F that we are trying to capture here is the feasible set of a chance constraint on p as well. While there may be more sophisticated options, such as multi-linear and neural networkbased classifiers, they are left for future research. ...
... where V (x) is the fixed volume of P(x). Consequently, under (16), problem (15) simplifies to the linear program ...
... Parameters (C, P, R) are the thermal capacitance, resistance, and power transfer of the house. They were drawn uniformly at random from [1.5, 2.5], [3,5], and [15,30], respectively. The temperature setpoint θ s was drawn uniformly at random from [24,26]. ...
Bundling a large number of distributed energy resources through a load aggregator has been advocated as an effective means to integrate such resources into whole-sale energy markets. To ease market clearing, system operators allow aggregators to submit bidding models of simple prespecified polytopic shapes. Aggregators need to carefully design and commit to a polytope that best captures their energy flexibility along a day-ahead scheduling horizon. This work puts forth a model-informed data-based optimal flexibility design for aggregators, which deals with the time-coupled, uncertain, and non-convex models of individual loads. The proposed solution first generates efficiently a labeled dataset of (non)-disaggregatable schedules. The feasible set of the aggregator is then approximated by an ellipsoid upon training a convex quadratic classifier using the labeled dataset. The ellipsoid is subsequently inner approximated by a polytope. Using Farkas lemma, the obtained polytope is finally inner approximated by the polytopic shape dictated by the market. Numerical tests show the effectiveness of the proposed flexibility design framework for designing the feasible sets of small- and large-sized aggregators coordinating solar photovoltaics, thermostatically-controlled loads, batteries, and electric vehicles. The tests further demonstrate that it is crucial for the aggregator to consider time-coupling and uncertainties in optimal flexibility design.
Bundling a large number of distributed energy resources through a load aggregator has been advocated as an effective means to integrate such resources into wholesale energy markets. To ease market clearing, system operators allow aggregators to submit bidding models of simple prespecified polytopic shapes. Aggregators need to carefully design and commit to a polytope that best captures their energy flexibility along a day-ahead scheduling horizon. This work puts forth a model-informed data-based optimal flexibility design for aggregators, which deals with the time-coupled, uncertain, and non-convex models of individual loads. The proposed solution first generates efficiently a labeled dataset of (in)-feasible aggregation schedules. The feasible set of the aggregator is then approximated by an ellipsoid upon training a convex quadratic classifier using the labeled dataset. The ellipsoid is subsequently inner approximated by a polytope. Using Farkas’ lemma, the obtained polytope is finally inner approximated by the polytopic shape dictated by the market. Numerical tests show the effectiveness of the proposed flexibility design framework for designing the feasible sets of small- and large-sized aggregators coordinating solar photovoltaics, thermostatically-controlled loads, batteries, and electric vehicles. The tests further demonstrate that it is crucial for the aggregator to consider time-coupling and uncertainties in optimal flexibility design.
Renewable resources are starting to constitute a growing portion of the total generation mix of the power system. A key difference between renewables and traditional generators is that many renewable resources are managed by individuals, especially in the distribution system. In this paper, we study the capacity investment and pricing problem, where multiple renewable producers compete in a decentralized market. It is known that most deterministic capacity games tend to result in very inefficient equilibria, even when there are a large number of similar players. In contrast, we show that due to the inherent randomness of renewable resources, the equilibria in our capacity game becomes efficient as the number of players grows and coincides with the centralized decision from the social planner's problem. This result provides a new perspective on how to look at the positive influence of randomness in a game framework as well as its contribution to resource planning, scheduling, and bidding. We validate our results by simulation studies using real world data.
We propose a data-based method to solve a multi-stage stochastic optimal power flow (OPF) problem based on limited information about forecast error distributions. The framework explicitly combines multi-stage feedback policies with any forecasting method and historical forecast error data. The objective is to determine power scheduling policies for controllable devices in a power network to balance operational cost and conditional value-at-risk (CVaR) of device and network constraint violations. These decisions include both nominal power schedules and reserve policies, which specify planned reactions to forecast errors in order to accommodate fluctuating renewable energy sources. Instead of assuming the uncertainties across the networks follow prescribed probability distributions, we consider ambiguity sets of distributions centered around a finite training dataset. By utilizing the Wasserstein metric to quantify differences between the empirical data-based distribution and the real unknown data-generating distribution, we formulate a multi-stage distributionally robust OPF problem to compute optimal control policies that are robust to both forecast errors and sampling errors inherent in the dataset. Two specific data-based distributionally robust stochastic OPF problems are proposed for distribution networks and transmission systems.
Battery participants in performance-based frequency regulation markets must consider the cost of battery aging in their operating strategies to maximize market profits. In this paper we solve this problem by proposing an optimal control policy and an optimal bidding policy based on realistic market settings and an accurate battery aging model. The proposed control policy has a threshold structure and achieves near-optimal performance with respect to an offline controller that has complete future information. The proposed bidding policy considers the optimal control policy to maximize market profits while satisfying the market performance requirement through a chance-constraint. It factors the value of performance and supports a trade-off between higher profits and a lower risk of violating performance requirements. We demonstrate the optimality of both policies using simulations. A case study based on the PJM regulation market shows that our approach is effective at maximizing operating profits.
Distributed renewable energy, particularly photovoltaics (PV), has expanded rapidly over the past decade. Distributed PV is located behind the meter and is thus invisible to the retailers and the distribution system operator (DSO). This invisible generation thus injects additional uncertainty in the net load and makes it harder to forecast. This paper proposes a data-driven probabilistic net load forecasting method specifically designed to handle a high penetration of behind-the-meter (BtM) PV. The capacity of BtM PV is first estimated using a maximal information coefficient (MIC)-based correlation analysis and a grid search. The net load profile is then decomposed into three parts (PV output, actual load, and residual) which are forecast individually. Correlation analysis based on copula theory is conducted on the distributions and dependencies of the forecasting errors to generate a probabilistic net load forecast. Case studies based on ISO New England data demonstrate that the proposed method outperforms other approaches, particularly when the penetration of BtM PV is high.
This work considers the problem of design centering. Geometrically, this can be thought of as inscribing one shape in another. Theoretical approaches and reformulations from the literature are reviewed; many of these are inspired by the literature on generalized semi-infinite programming, a generalization of design centering. However, the motivation for this work relates more to engineering applications of robust design. Consequently, the focus is on specific forms of design spaces (inscribed shapes) and the case when the constraints of the problem may be implicitly defined, such as by the solution of a system of differential equations. This causes issues for many existing approaches, and so this work proposes two restriction-based approaches for solving robust design problems that are applicable to engineering problems. Another feasible-point method from the literature is investigated as well. The details of the numerical implementations of all these methods are discussed. The discussion of these implementations in the particular setting of robust design in engineering problems is new.
Demand response is designed to motivate electricity customers to modify their loads at critical time periods. The accurate estimation of impact of demand response signals to customers' consumption is central to any successful program. In practice, learning these response is nontrivial because operators can only send a limited number of signals. In addition, customer behavior also depends on a large number of exogenous covariates. These two features lead to a high dimensional inference problem with limited number of observations. In this paper, we formulate this problem by using a multivariate linear model and adopt an experimental design approach to estimate the impact of demand response signals. We show that randomized assignment, which is widely used to estimate the average treatment effect, is not efficient in reducing the variance of the estimator when a large number of covariates is present. In contrast, we present a tractable algorithm that strategically assigns demand response signals to customers. This algorithm achieves the optimal reduction in estimation variance, independent of the number of covariates. The results are validated from simulations on synthetic data.
Demand Response (DR) is considered a promising approach to cope with the increasing variability in power grids due to the penetration of renewable energy sources. However, it still remains a challenge to manage the aggregation of a large number of heterogeneous loads to achieve a desired response, especially at a fast time scale. In this paper, we present a system-level modeling and control design approach for DR management in smart grids, by following a contract-based methodology. Given a set of flexible DR loads, capable of adapting their power consumption upon external requests, we find a strategy for an aggregator to optimally track a DR requirement by combining the individual contributions of the DR components. In our framework, both the design requirements and the DR components' interfaces are specified by assume-guarantee contracts expressed using mixed-integer linear constraints. Contracts are used to formulate an optimal control problem, which is solved repeatedly over time, in a receding horizon fashion. We illustrate the effectiveness of our methodology for a set of DR components including fans and pumps, showing that it enables modular development of non-disruptive DR control schemes.
High penetration of distributed energy resources presents several challenges
and opportunities for voltage regulation in power distribution systems. A local
reactive power (VAR) control framework will be developed that can fast respond
to voltage mismatch and address the robustness issues of (de-)centralized
approaches against communication delays and noises. Using local bus voltage
measurements, the proposed gradient-projection based schemes explicitly account
for the VAR limit of every bus, and are proven convergent to a surrogate
centralized problem with proper parameter choices. This optimality result
quantifies the capability of local VAR control without requiring any real-time
communications. The proposed framework and analysis generalize earlier results
on the droop VAR control design, which may suffer from under-utilization of VAR
resources in order to ensure stability. Numerical tests have demonstrated the
validity of our analytical results and the effectiveness of proposed approaches
implemented on realistic three-phase systems.
Electric vehicles (EVs) can be considered as flexible mobile battery storages in microgrids. For multiple microgrids in an area, coordinated scheduling on charging and discharging are required to avoid power exchange spikes between the multimicrogrid system and the main grid. In this paper, a two-stage integrated energy exchange scheduling strategy for multimicrogrid system is presented, which considers EVs as storage devices. Then, several dual variables, which are representative of the marginal cost of proper constraints, are utilized to form an updated price, thereby being a modification on the original electricity price. With this updated price signal, a price-based decentralized scheduling strategy is presented for the microgrid central controller. Simulation results show that the two-stage scheduling strategy reduces the electricity cost and avoids frequent transitions between battery charging/discharging states. With the proposed decentralized scheduling strategy, each microgrid only needs to solve its local problem and limits the total power exchange within the safe range.
This paper addresses the problem of voltage regulation in power distribution networks with deep-penetration of distributed energy resources, e.g., renewable-based generation, and storage-capable loads such as plug-in hybrid electric vehicles. We cast the problem as an optimization program, where the objective is to minimize the losses in the network subject to constraints on bus voltage magnitudes, limits on active and reactive power injections, transmission line thermal limits and losses. We provide sufficient conditions under which the optimization problem can be solved via its convex relaxation. Using data from existing networks, we show that these sufficient conditions are expected to be satisfied by most networks. We also provide an efficient distributed algorithm to solve the problem. The algorithm adheres to a communication topology described by a graph that is the same as the graph that describes the electrical network topology. We illustrate the operation of the algorithm, including its robustness against communication link failures, through several case studies involving 5-, 34-, and 123-bus power distribution systems.
Optimal Power Flow (OPF) dispatches controllable generation at minimum cost
subject to operational constraints on generation and transmission assets. The
uncertainty and variability of intermittent renewable generation is challenging
current deterministic OPF approaches. Recent formulations of OPF use chance
constraints to limit the risk from renewable generation uncertainty, however,
these new approaches typically assume the probability distributions which
characterize the uncertainty and variability are known exactly. We formulate a
Robust Chance Constrained (RCC) OPF that accounts for uncertainty in the
parameters of these probability distributions by allowing them to be within an
uncertainty set. The RCC OPF is solved using a cutting-plane algorithm that
scales to large power systems. We demonstrate the RRC OPF on a modified model
of the Bonneville Power Administration network, which includes 2209 buses and
176 controllable generators. Deterministic, chance constrained (CC), and RCC
OPF formulations are compared using several metrics including cost of
generation, area control error, ramping of controllable generators, and
occurrence of transmission line overloads as well as the respective
computational performance.
To effectively enhance the integration of distributed and renewable energy
sources in future smart microgrids, economical energy management accounting for
the principal challenge of the variable and non-dispatchable renewables is
indispensable and of significant importance. Day-ahead economic generation
dispatch with demand-side management for a microgrid in islanded mode is
considered in this paper. With the goal of limiting the risk of the
loss-of-load probability, a joint chance constrained optimization problem is
formulated for the optimal multi-period energy scheduling with multiple wind
farms. Bypassing the intractable spatio-temporal joint distribution of the wind
power generation, a primal-dual approach is used to obtain a suboptimal
solution efficiently. The method is based on first-order optimality conditions
and successive approximation of the probabilistic constraint by generation of
p-efficient points. Numerical results are reported to corroborate the merits of
this approach.
This paper proposes a day-ahead stochastic scheduling model in electricity markets. The model considers hourly forecast errors of system loads and variable renewable sources as well as random outages of power system components. A chance-constrained stochastic programming formulation with economic and reliability metrics is presented for the day-ahead scheduling. Reserve requirements and line flow limits are formulated as chance constraints in which power system reliability requirements are to be satisfied with a presumed level of high probability. The chance-constrained stochastic programming formulation is converted into a linear deterministic problem and a decomposition-based method is utilized to solve the day-ahead scheduling problem. Numerical tests are performed and the results are analyzed for a modified 31-bus system and an IEEE 118-bus system. The results show the viability of the proposed formulation for the day-ahead stochastic scheduling. Comparative evaluations of the proposed chance-constrained method and the Monte Carlo simulation (MCS) method are presented in the paper.
This paper presents a centralized control scheme to regulate distribution network voltages in the presence of high penetration of distributed generation. The approach is inspired of Model Predictive Control in order to compensate for modeling inaccuracies and measurement noise. The control actions, calculated from a multi-step optimization, are updated and corrected by real-time measurements. The proposed controller uses a linear model to predict the behavior of the system and the optimization is solved using quadratic programming. The proposed corrective control has been tested in a 11-kV distribution network including 75 nodes and hosting 22 distributed generating units.
In this paper, a chance-constrained optimization (CCO) is presented to handle uncertainty in control of transmission voltages. A control scheme is proposed using a steady-state system model to achieve the goal of online voltage control and preventing long-term voltage instability. In order to model steady-state system response, the long-term model of governors and Automatic Voltage Regulators are employed in the control scheme. The Nordic32 test system is selected to show the simulation results of the proposed technique.
In this paper, we propose an architecture for voltage regulation in distribution networks that relies on controlling reactive power injections provided by distributed energy resources (DERs). A local controller on each bus of the network monitors the bus voltage and, whenever there is a voltage violation, it uses locally available information to estimate the amount of reactive power that needs to be injected into the bus in order to correct the violation. If the DERs connected to the bus can collectively provide the reactive power estimated by the local controller, they are instructed to do so. Otherwise, the local controller initiates a request for additional reactive power support from other controllers at neighboring buses through a distributed algorithm that relies on a local exchange of information among neighboring controllers. We show that the proposed architecture helps prevent voltage violations and shapes the voltage profile in radial distribution networks, even in the presence of considerable penetration of variable generation and loads. We present several case studies involving 8-, 13-, and 123-bus distribution systems to illustrate the operation of the architecture.
We formulate the control of reactive power generation by photovoltaic
inverters in a power distribution circuit as a constrained optimization that
aims to minimize real power losses subject to finite inverter capacity and
upper and lower voltage limits at all nodes in the circuit. When voltage
variations along the circuit are small and losses of both real and reactive
powers are small compared to the respective flows, the resulting optimization
problem is convex. Moreover, the cost function is separable enabling a
distributed, on-line implementation with node-local computations using only
local measurements augmented with limited information from the neighboring
nodes communicated over cyber channels. Such an approach lies between the fully
centralized and local policy approaches previously considered. We explore
protocols based on the dual ascent method and on the Alternating Direction
Method of Multipliers (ADMM) and find that the ADMM protocol performs
significantly better.
Solution approaches to chance constrained program- ming (CCP) have been recently developed and applied in many areas for optimization under uncertainty. Due to the nonlinear model with multiple uncertain variables as well as multiple output constraints, CCP has not been directly applied to optimal power flow (OPF) under uncertainty. The objective of this paper is twofold. First, we introduce the CCP approach to OPF under un- certainty and analyze the computational complexity of the chance constrained OPF. Second, the effectiveness of implementing a back-mapping approach and a linear approximation of the non- linear model equations to solve the formulated CCP problem is investigated. Load power uncertainties are considered as multi- variate random variables with correlated normal distribution. Based on both the nonlinear and the linearized model, results of a five-bus system and the IEEE 30-bus test system are presented to demonstrate the scope of chance constrained OPF. Index Terms—Chance constrained programming, multivariate integration, optimal power flow, uncertainty.
We propose a branch flow model for the anal- ysis 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.
We are concerned with characterization of attraction domains for linear state-space systems. Considered are bounded linear state feedback control and saturated control. Attraction domains for such systems are described in terms of invariant ellipsoids using LMI-based techniques and semidefinite programming (SDP). For systems with saturated control, the ideology of absolute stability is adopted. An application to nonlinear systems is provided.
In this paper, fast recursive algorithms for the approximation of an n-dimensional convex polytope by means of an inscribed ellipsoid are presented. These algorithms consider at each step a single inequality describing the polytope and, under mild assumptions, they are guaranteed to converge in a finite number of steps. For their recursive nature, the proposed algorithms are better suited to treat a quite large number of constraints than standard off-line solutions, and have their natural application to problems where the set of constraints is iteratively updated, as on-line estimation problems, nonlinear convex optimization procedures and set membership identification.
We present a framework for margin based active learning of linear separators. We instantiate it for a few important cases, some of which have been previously considered in the literature. We analyze the effectiveness of our frame- work both in the realizable case and in a specific noisy setting related to the Tsy - bakov small noise condition.
We study approximations of optimization problems with probabilistic constraints in which the original distribution of the underlying random vector is replaced with an empirical distribution obtained from a random sample. We show that such a sample approximation problem with a risk level larger than the required risk level will yield a lower bound to the true optimal value with probability approaching one exponentially fast. This leads to an a priori estimate of the sample size required to have high confidence that the sample approximation will yield a lower bound. We then provide conditions under which solving a sample approximation problem with a risk level smaller than the required risk level will yield feasible solutions to the original problem with high probability. Once again, we obtain a priori estimates on the sample size required to obtain high confidence that the sample approximation problem will yield a feasible solution to the original problem. Finally, we present numerical illustrations of how these results can be used to obtain feasible solutions and optimality bounds for optimization problems with probabilistic constraints.
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 non-unity 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.
The paper presents a computational method that approximates feasible sets specified by linear or convex inequalities. This numerically efficient approach to power system optimization is based on computational geometry of multidimensional ellipsoids and is potentially applicable to problems with high dimensions, as it builds on recent advances in convex optimization. In an important application, it provides ranges in which nodal (generator) injections can vary without violating operational constraints in security analysis. The model is applied to two important problems in deregulated power systems: optimal economic dispatch (OED) and calculation of locational marginal prices (LMPs) in a day-ahead power market. Optimization problem with convex (ellipsoid-based) constraints is solved by a linear matrix inequality (LMI)-based procedure. The method is verified on the benchmark example with 68 buses, 16 generators, and 86 lines.
Voltage control plays an important role in the operation of electricity distribution networks, especially with high penetration of distributed energy resources. These resources introduce significant and fast varying uncertainties. In this paper, we focus on reactive power compensation to control voltage in the presence of uncertainties. We adopt a chance constraint approach that accounts for arbitrary correlations between renewable resources at each of the buses. We show how the problem can be solved efficiently using historical samples analogously to the stochastic quasi-gradient methods. We also show that this optimization problem is convex for a wide variety of probabilistic distributions. Compared to conventional per-bus chance constraints, our formulation is more robust to uncertainty and more computationally tractable. We illustrate the results using standard IEEE distribution test feeders.
Renewable resources are starting to constitute a growing portion of the total generation mix of the power system. A key difference between renewables and traditional generators is that many renewable resources are managed by individuals, especially in the distribution system. In this paper, we study the capacity investment and pricing problem, where multiple renewable producers compete in a decentralized market. It is known that most deterministic capacity games tend to result in very inefficient equilibria, even when there are a large number of similar players. In contrast, we show that due to the inherent randomness of renewable resources, the equilibria in our capacity game becomes efficient as the number of players grows and coincides with the centralized decision from the social planner's problem. This result provides a new perspective on how to look at the positive influence of randomness in a game framework as well as its contribution to resource planning, scheduling, and bidding. We validate our results by simulation studies using real world data.
Demand response is designed to motivate electricity customers to modify their loads at critical time periods. The accurate estimation of impact of demand response signals to customers' consumption is central to any successful program. In practice, learning these response is nontrivial because operators can only send a limited number of signals. In addition, customer behavior also depends on a large number of exogenous covariates. These two features lead to a high dimensional inference problem with limited number of observations. In this paper, we formulate this problem by using a multivariate linear model and adopt an experimental design approach to estimate the impact of demand response signals. We show that randomized assignment, which is widely used to estimate the average treatment effect, is not efficient in reducing the variance of the estimator when a large number of covariates is present. In contrast, we present a tractable algorithm that strategically assigns demand response signals to customers. This algorithm achieves the optimal reduction in estimation variance, independent of the number of covariates. The results are validated from simulations on synthetic data.
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.
Evolving power systems with increasing levels of stochasticity call for a need to solve optimal power flow problems with large quantities of random variables. Weather forecasts, electricity prices, and shifting load patterns introduce higher levels of uncertainty and can yield optimization problems that are difficult to solve in an efficient manner. Efficient solution methods for single chance constraints in optimal power flow problems have been considered in the literature; however, joint chance constraints have predominantly been solved via scenario-based approaches or by utilizing the overly conservative Boole's inequality as an upper bound. In this paper, joint chance constraints are used to solve an AC optimal power flow problem which maintain desired levels of voltage magnitude in distribution grids under high penetrations of photovoltaic systems. A tighter version of Boole's inequality is derived and used to provide a new upper bound on the joint chance constraint, and simulation results are shown demonstrating the benefit of the proposed upper bound.
Voltage control plays an important role in the operation of electricity distribution networks, especially when there is a large penetration of renewable energy resources. In this paper, we focus on voltage control through reactive power compensation and study how different information structures affect the control performance. In particular, we first show that only using voltage measurements to determine reactive power compensation is insufficient to maintain voltage in the acceptable range. Then we proposes two fully decentralized algorithms by slightly adding additional information into the control design. The two algorithms are guaranteed to stabilize the voltage in the acceptable range regardless of the system operating condition. The one with higher complexity can further minimize a cost of reactive power compensation in a particular form. Both of the two algorithms use only local measurements and local variables and require no communication.
Distribution microgrids are being challenged by reverse power flows and
voltage fluctuations due to renewable generation, demand response, and electric
vehicles. Advances in photovoltaic (PV) inverters offer new opportunities for
reactive power management provided PV owners have the right investment
incentives. In this context, reactive power compensation is considered here as
an ancillary service. Accounting for the increasing time-variability of
distributed generation and demand, a stochastic reactive power compensation
scheme is developed. Given uncertain active power injections, an online
reactive control scheme is devised. Such scheme is distribution-free and relies
solely on power injection data. Reactive injections are updated using the
Lagrange multipliers of the second-order cone program obtained via a convex
relaxation. Numerical tests on an industrial 47-bus test feeder corroborate the
reactive power management efficiency of the novel stochastic scheme over its
deterministic alternative as well as its capability to track variations in
solar generation.
Power flow optimization is generally nonlinear and non-convex, and a second-order cone relaxation has been pro-posed recently for convexification. We prove several sufficient conditions under which the relaxation is exact. One of these conditions seems particularly realistic and suggests guidelines on integrating distributed generations.
Let [math not displayed] be a convex set. We assume that ||x||infinity ≤ 1 for all x ε C, and that C contains a ball of radius 1/R. For x ε Rn, r ε R, and B an n × n symmetric positive definite matrix, let E(x, B, r) = {y|(y - x)TB(y - x)≤ r2}. A β-rounding of C is an ellipsoid E(x, B, r) such that [math not displayed]. In the case that C is characterized by a separation oracle, it is well known that an O(n3/2)-rounding of C can be obtained using the shallow cut ellipsoid method in O(n3 ln(nR)) oracle calls. We show that a modification of the volumetric cutting plane method obtains an O(n3/2)-rounding of C in O(n2 ln(nR)) oracle calls. We also consider the problem of obtaining an O(n)-rounding of C when C has an explicit polyhedral description. Our analysis uses a new characterization of circumscribing ellipsoids centered at, or near, the volumetric center of a polyhedral set.
This paper address the problem of voltage regulation in power distribution
networks with deep penetration of distributed energy resources (DERs) without
any explicit communication between the buses in the network. We cast the
problem as an optimization problem with the objective of minimizing the
distance between the bus voltage magnitudes and some reference voltage profile.
We present an iterative algorithm where each bus updates the reactive power
injection provided by their DER. The update at a bus only depends on the
voltage magnitude at that bus, and for this reason, we call the algorithm a
local control algorithm. We provide sufficient conditions that guarantee the
convergence of the algorithm and these conditions can be checked a priori for a
set of feasible power injections. We also provide necessary conditions
establishing that longer and more heavily loaded networks are inherently more
difficult to control. We illustrate the operation of the algorithm through case
studies involving 8-,34- and 123-bus test distribution systems.
Support vector machines have met with significant success in numerous real-world learning tasks. However, like most machine learning algorithms, they are generally applied using a randomly selected training set classified in advance. In many settings, we also have the option of using pool-based active learning . Instead of using a randomly selected training set, the learner has access to a pool of unlabeled instances and can request the labels for some number of them. We introduce a new algorithm for performing active learning with support vector machines, i.e., an algorithm for choosing which instances to request next. We provide a theoretical motivation for the algorithm using the notion of a version space . We present experimental results showing that employing our active learning method can significantly reduce the need for labeled training instances in both the standard inductive and transductive settings.
Convex optimization problems arise frequently in many different fields. A comprehensive introduction to the subject, this book shows in detail how such problems can be solved numerically with great efficiency. The focus is on recognizing convex optimization problems and then finding the most appropriate technique for solving them. The text contains many worked examples and homework exercises and will appeal to students, researchers and practitioners in fields such as engineering, computer science, mathematics, statistics, finance, and economics.
The problem of best approximation of a convex compact set in a finite-dimensional space by ellipsoids with respect to a special
measure of deviation of an ellipsoid from a compact set is considered. An analytic description of ellipsoids of best approximation
is given.
This paper addresses the problem of voltage regulation in power distribution
networks with deep-penetration of distributed energy resources (DERs), e.g.,
renewable-based generation, and storage-capable loads such as plug-in hybrid
electric vehicles. We cast the problem as an optimization program, where the
objective is to minimize the losses in the network subject to constraints on
bus voltage magnitudes, limits on active and reactive power injections,
transmission line thermal limits and losses. We provide sufficient conditions
under which the optimization problem can be solved via its convex relaxation.
Using data from existing networks, we show that these sufficient conditions are
expected to be satisfied by most networks. We also provide an efficient
distributed algorithm to solve the problem. The algorithm adheres to a
communication topology described by a graph that is the same as the graph that
describes the electrical network topology. We illustrate the operation of the
algorithm, including its robustness against communication link failures,
through several case studies involving 5-, 34- and 123-bus power distribution
systems.
We prove a lower bound of Ω((1/ɛ)ln(1/δ)+VCdim(C)/ɛ) on the number of random examples required for distribution-free learning of a concept class C, where VCdim(C) is the Vapnik-Chervonenkis dimension and ɛ and δ are the accuracy and confidence parameters. This improves the previous best lower bound of Ω((1/ɛ)ln(1/δ)+VCdim(C)) and comes close to the known general upper bound of O((1/ɛ)ln(1/δ)+(VCdim(C)/ɛ)ln(1/ɛ)) for consistent algorithms. We show that for many interesting concept classes, including kCNF and kDNF, our bound is actually tight to within a constant factor.
A new technique for polytope approximation of the feasible region
for a design is presented. This method is computationally less expensive
than the simplicial approximation method. Results on several circuits
are presented. It is shown that the quality of the polytope
approximation is substantially better than an ellipsoidal approximation
We state and analyze the first active learning algorithm that finds an E-optimal hypothesis in any hypothesis class, when the underlying distribution has arbitrary forms of noise. The algorithm, A(2) (for Agnostic Active), relies only upon the assumption that it has access to a stream of unlabeled examples drawn i.i.d. from a fixed distribution. We show achieves an exponential improvement (i.e., requires only O(ln1/epsilon) samples to an E-optimal classifier) over the usual sample complexity of supervised learning, for several settings considered before in the realizable case. These include learning threshold classifiers and learning homogeneous linear separators with respect to an input distribution which is uniform over the unit sphere. (c) 2008 Elsevier Inc. All rights reserved
We characterize the sample complexity of active learning problems in terms of a parameter which takes into account the distribution over the input space, the specific target hypothesis, and the desired accuracy.
High penetration levels of distributed photovoltaic(PV) generation on an
electrical distribution circuit present several challenges and opportunities
for distribution utilities. Rapidly varying irradiance conditions may cause
voltage sags and swells that cannot be compensated by slowly responding utility
equipment resulting in a degradation of power quality. Although not permitted
under current standards for interconnection of distributed generation,
fast-reacting, VAR-capable PV inverters may provide the necessary reactive
power injection or consumption to maintain voltage regulation under difficult
transient conditions. As side benefit, the control of reactive power injection
at each PV inverter provides an opportunity and a new tool for distribution
utilities to optimize the performance of distribution circuits, e.g. by
minimizing thermal losses. We discuss and compare via simulation various design
options for control systems to manage the reactive power generated by these
inverters. An important design decision that weighs on the speed and quality of
communication required is whether the control should be centralized or
distributed (i.e. local). In general, we find that local control schemes are
capable for maintaining voltage within acceptable bounds. We consider the
benefits of choosing different local variables on which to control and how the
control system can be continuously tuned between robust voltage control,
suitable for daytime operation when circuit conditions can change rapidly, and
loss minimization better suited for nighttime operation.
The problem of capacitor placement on a radial distribution system
is formulated and a solution algorithm is proposed. The location, type,
and size of capacitors, voltage constraints, and load variations are
considered. The objective of capacitor placement is peak power and
energy loss reduction, taking into account the cost of the capacitors.
The problem is formulated as a mixed integer programming problem. The
power flows in the system are explicitly represented, and the voltage
constraints are incorporated. A solution method has been implemented
that decomposes the problem into a master problem and a slave problem.
The master problem is used to determine the location of the capacitors.
The slave problem is used by the master problem to determine the type
and size of the capacitors placed on the system. In solving the slave
problem, and efficient phase I-phase II algorithm is used