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Chance-constrained regulation capacity offering for HVAC systems under non-Gaussian uncertainties with mixture-model-based convexification
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Abstract
Heating, ventilation, and air-conditioning (HVAC) systems are ideal demand-side flexible resources to provide regulation services. However, finding the best hourly regulation capacity offers for HVAC systems in a power market ahead of time is challenging because they are affected by non-Gaussian uncertainties from regulation signals. Moreover, since HVAC systems need to frequently regulate their power according to regulation signals, numerous thermodynamic constraints are introduced, leading to a huge computational burden. This paper proposes a tractable chance-constrained model to address these challenges. It first develops a temporal compression approach, in which the extreme indoor temperatures in the operating hour are estimated and restricted in the comfortable range so that the numerous thermodynamic constraints can be compressed into only a few ones. Then, a novel convexification method is proposed to handle the non-Gaussian uncertainties. This method leverages the Gaussian mixture model to reformulate the chance constraints with non-Gaussian uncertainties on the left-hand side into deterministic non-convex forms. We further prove that these non-convex forms can be approximately convexified by second-order cone constraints with marginal optimality loss. Therefore, the proposed model can be efficiently solved with guaranteed optimality. Numerical experiments are conducted to validate the superiority of the proposed method.
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With a large-scale integration of distributed energy resources (DERs), distribution systems are expected to be capable of providing capacity support for the transmission grid. To effectively harness the collective flexibility from massive DER devices, this paper studies distribution-level power aggregation strategies for transmission-distribution interaction. In particular, this paper proposes a method to model and quantify the aggregate power flexibility, i.e., the net power injection achievable at the substation, in unbalanced distribution systems over time. Incorporating the network constraints and multi-phase unbalanced modeling, the proposed method obtains an effective approximate feasible region of the net power injection. For any aggregate power trajectory within this region, it is proved that there exists a feasible disaggregation solution. In addition, a distributed model predictive control (MPC) framework is developed for the practical implementation of the transmission-distribution interaction. At last, we demonstrate the performances of the proposed method via numerical tests on a real-world distribution feeder with 126 multi-phase nodes.
Large-scale wind and solar power integration are likely to cause a short-term mismatch between generation and load demand because of the intermittent nature of the renewables. System frequency is therefore challenged. In recent years, it has been proposed that a part of the residential load can be controlled for frequency regulation with little impact on customer comfort. This paper proposes a thermostatic load control (TLC) strategy for primary and secondary frequency regulation, in particular, using heating, ventilation and air-conditioning (HVAC) units and electric water heaters (EWHs). First, daily demand profile modeling indicates that these two loads are complementary in the daytime and can provide a relatively stable frequency reserve. Second, the progressive load recovery is specifically considered in the control scheme. The random switching and cycle recovery (RS-CR) method is proposed for mitigating power rebound after switching the air conditioners on again. The proposed control strategy can organize a large population of thermostatic loads for the provision of a frequency reserve. Consequently, the requirement of a spinning reserve is reduced. Finally, the proposed control strategy is verified by the dynamic simulation of IEEE RTS 24-bus system.
Power imbalances from fluctuating renewable electricity generators are counteracted by often expensive flexibility services. Heating, cooling, and air-conditioning (HVAC) of buildings, or domestic power-to-heat (P2H), are end uses of electricity that allow flexible load patterns due to the inertia of an attached thermal storage while meeting their quality constraints. Compared to smart appliances or electric vehicle charging, P2H exhibits large and predictable capacities of demand response (DR), because buildings in many countries account for 30–40% of the final energy demand, a large part of which is thermal. Yet, its practical flexibility potential remains largely unknown: is DR from P2H a mature technology for mass usage; is it cost-efficient, socially attractive, and ready to make key contributions to flexibility comparable to backup generators or battery storage? In the present paper, we review recent international field studies that are paving the way from research to practice. These field trials include real customers but have a broader research focus and a wider outreach than rolling out a new DR tariff or program or a specific new technology for DR. Their experience mirrors the technology readiness beyond revenue or policy studies, optimization frameworks or laboratory-scale micro-grids. We analyze the adequacy of the pricing mechanisms deployed for incentivization and remuneration and review the coordination mechanisms for balancing on different timescales including fast ancillary services. We conclude that current control and information technology and economic and regulatory frameworks which have been field-tested do not yet meet the flexibility challenges of smart grids with a very high share (>50%) of intermittent renewable generation.
Controllable building loads have the potential to significantly increase the flexibility of power systems. A key step in developing effective and attainable load control policies is modeling the set of feasible building load profiles. In this paper, we consider buildings whose source of flexibility is their HVAC loads. We propose a data-driven method to empirically estimate a robust feasible region of the load using coarse data, that is, only total building load and average indoor temperatures. The proposed method uses easy-to-gather coarse data and can be adapted to buildings of any type. The resulting feasible region model is robust to temperature prediction errors and is described by linear constraints. The mathematical simplicity of these constraints makes the proposed model adaptable to many power system applications, for example, economic dispatch and optimal power flow. We validate our model using data from EnergyPlus and demonstrate its usefulness through a case study in which the flexibility provided by the building loads is used to balance errors of wind power forecasts. IEEE
Energy Storage (ES) provides great flexibility and large benefits to power system operations and control. When providing ancillary services (e.g. regulation, reserve, etc.), the real time deployment of ES is uncertain, and it is important to manage state of charge (SoC) accordingly. Aiming to improve the ES performance for providing energy and regulation service in the electricity market, we propose two data-driven Automatic Generation Control (AGC) signal models. The first one is a historical-data-driven AGC signal model, which is based on the analysis of the historical AGC signals, and is designed for ES participation in the day-ahead (DA) market. The second one is a prediction-data-driven AGC signal model, which is based on the prediction of the AGC signals, and is designed for ES participation in the real-time (RT) market. We also develop a deployment availability check model and solution algorithm. The proposed framework is applied to an ES bidding problem in the DA and RT markets. The results indicate that deployment availability and operational performance of the ES are improved with the proposed data-driven AGC models compared to traditional benchmarks.
This paper studies the behavior of a strategic aggregator offering regulation capacity on behalf of a group of distributed energy resources (DERs, e.g. plug-in electric vehicles) in a power market. Our objective is to maximize the aggregator's revenue while controlling the risk of penalties due to poor service delivery. To achieve this goal, we propose data-driven risk-averse strategies to effectively handle uncertainties in: 1) The DER parameters (e.g., load demands and flexibilities) and 2) sub-hourly regulation signals (to the accuracy of every few seconds). We design both the day-ahead and the hour-ahead strategies. In the day-ahead model, we develop a two-stage stochastic program to roughly model the above uncertainties, which achieves computational efficiency by leveraging novel aggregate models of both DER parameters and sub-hourly regulation signals. In the hour-ahead model, we formulate a data-driven distributionally robust chance-constrained program to explicitly model the aforementioned uncertainties. This program can effectively control the quality of regulation service based on the aggregator's risk aversion. Furthermore, it learns the distributions of the uncertain parameters from empirical data so that it outperforms existing techniques, (e.g. robust optimization or traditional chance-constrained programming) in both modelling accuracy and cost of robustness. Finally, we derive a conic safe approximation for it which can be efficiently solved by commercial solvers. Numerical experiments are conducted to validate the proposed method.
We consider stochastic programs where the distribution of the uncertain
parameters is only observable through a finite training dataset. Using the
Wasserstein metric, we construct a ball in the space of (multivariate and
non-discrete) probability distributions centered at the uniform distribution on
the training samples, and we seek decisions that perform best in view of the
worst-case distribution within this Wasserstein ball. The state-of-the-art
methods for solving the resulting distributionally robust optimization problems
rely on global optimization techniques, which quickly become computationally
excruciating. In this paper we demonstrate that, under mild assumptions, the
distributionally robust optimization problems over Wasserstein balls can in
fact be reformulated as finite convex programs---in many interesting cases even
as tractable linear programs. Leveraging recent measure concentration results,
we also show that their solutions enjoy powerful finite-sample performance
guarantees. Our theoretical results are exemplified in mean-risk portfolio
optimization as well as uncertainty quantification.
The last decade has seen an explosion in the availability of data for
operations research applications as part of the Big Data revolution. Motivated
by this growing availability, we propose a novel schema for utilizing data to
design uncertainty sets for robust optimization using statistical hypothesis
tests. The approach is flexible and widely applicable, and robust optimization
problems built from our new sets are computationally tractable, both
theoretically and practically. Furthermore, optimal solutions to these problems
enjoy a strong, finite-sample probabilistic guarantee. We also propose concrete
guidelines for practitioners and illustrate our approach with applications in
portfolio management and queueing. Computational evidence confirms that our
data-driven sets significantly outperform traditional robust optimization
techniques whenever data is available.
Thermostatically controlled load (TCL, such as heating, ventilation, and air conditioning system) is a desirable demand-side flexibility source in distribution networks. It can participate in regulation services and mitigate power imbalances from fluctuating distributed renewable generation. To effectively utilize the load flexibility from spatially and temporally distributed TCLs in a distribution network, it is necessary to consider power flow constraints to avoid possible voltage or current violations. Published works usually adopt optimal power flow models (OPF) to describe these constraints. However, these models require accurate topology of the distribution network that is often unobservable in practice. To bypass this challenge, this paper proposes a novel learning-based OPF to optimize TCLs for regulation services. This method trains three regression multi-layer perceptrons (MLPs) based on the distribution network's historical operation data to replicate its power flow constraints. The trained MLPs are further equivalently reformulated into linear constraints with binary variables so that the optimization problem becomes a mixed-integer linear program that can be effectively solved. Numerical experiments based on the IEEE 123-bus system validate that the proposed method can achieve better TCL power scheduling performance with guaranteed feasibility and optimality than other state-of-art models.
Heating, ventilation, and air-conditioning (HVAC) systems play an increasingly important role in the construction of smart cities because of their high energy consumption and available operational flexibility for power systems. To enhance energy efficiency and utilize their flexibility, strategic operation is indispensable. However, finding a desirable control policy for multi-zone HVAC systems is a challenging task because of unavoidable forecasting errors of ambient temperature and heat loads. This paper addresses this challenge by proposing a fast power dispatch model for multi-zone HVAC systems. A distributionally robust chance-constrained approach, which does not require the exact probability distributions of uncertainties, is employed to handle the uncertainties from forecasting errors. Both the uncertainty propagation among zones and accumulation over time are explicitly described based on the delicate indoor thermal model. Wasserstein distance is employed for the construction of ambiguity sets to improve the solution optimality. To overcome the computational intractability of Wasserstein-distance-based method, we first develop a time-efficient inner approximation for the objective function. A separation approach is then proposed to achieve the off-line calculation of uncertain parts in chance constraints. Numerical experiments prove that the proposed model can effectively achieve optimal power dispatch for HVAC systems with high computational efficiency.
This paper proposes a data-driven chance-constrained optimal gas-power flow (OGPF) calculation method without any prior assumption on the distribution of uncertainties of wind power generation. The Gaussian mixture model is employed to fit the uncertainty distribution, where the Bayesian nonparametric Dirichlet process is adopted to tune the component number. To facilitate the online application of the proposed methods, an online-offline double-track distribution construction approach is established, where the frequency of training the relatively time-consuming Dirichlet process Gaussian mixture model can be reduced. On account of the quadratic gas consumption expression of gas-fired generators as well as the linear decision rule based uncertainty mitigation mechanism, the chance constraints would become quadratic ones with quadratic terms of uncertainties, which makes the proposed model more intractable. An equivalent linear separable counterpart is then provided for the quadratic chance constraints, after which the intractable chance constraints could be converted into traditional linear ones. The convex-concave procedure is used to crack the nonconvex Weymouth equation in the gas network and the auxiliary quadratic equalities. Simulation results on two test systems validate the effectiveness of the proposed methods.
Flexibility in power systems is ability to provide supply-demand balance, maintain continuity in unexpected situations, and cope with uncertainty on supply-demand sides. The new method and management requirements to provide flexibility have emerged from the trend towards power systems increasing renewable energy penetration with generation uncertainty and availability. In this study, the historical development of power system flexibility concept, the flexible power system characteristics, flexibility sources, and evaluation parameters are presented as part of international literature. The impact of variable renewable energy sources penetration on power system transient stability, small-signal stability, and frequency stability are discussed; the studies are presented to the researchers for further studies. Moreover, flexibility measurement studies are investigated, and methods of providing flexibility are evaluated.
Managing uncertainty and variability in power injections has become a major concern for power system operators due to the increasing levels of fluctuating renewable energy connected to the grid. This work addresses this uncertainty via a joint chance-constrained formulation of the DC optimal power flow (OPF) problem, which satisfies all the constraints jointly with a pre-determined probability. The few existing approaches for solving joint chance-constrained OPF problems are typically either computationally intractable for large-scale problems or give overly conservative solutions that satisfy the constraints far more often than required, resulting in excessively costly operation. This paper proposes an algorithm for solving joint chance-constrained DC OPF problems by adopting an Sl1QP-type trust-region algorithm. This algorithm uses a sample-based approach that avoids making strong assumptions on the distribution of the uncertainties, scales favorably to large problems, and can be tuned to obtain less conservative results. We illustrate the performance of our method using several IEEE test cases. The results demonstrate the proposed algorithm's advantages in computational times and limited conservativeness of the solutions relative to other joint chance-constrained DC OPF algorithms.
To capture the stochastic characteristics of renewable energy generation output, the chance-constrained unit commitment (CCUC) model is widely used. Conventionally, analytical solution for CCUC is usually based on simplified probability assumption or neglecting some operational constraints, otherwise scenar-io-based methods are used to approximate probability with heavy computation burden. In this paper, Gaussian mixture model (GMM) is employed to characterize the correlation between wind farms and probability distribution of their forecast errors. In our model, chance constraints including reserve sufficiency and branch power flow bounds are ensured to be satisfied with predetermined probability. To solve this CCUC problem, we propose a Newton method based procedure to acquire the quan-tiles and transform chance constraints into deterministic con-straints. Therefore, the CCUC model is efficiently solved as a mixed-integer quadratic programming problem. Numerical tests are performed on several systems to illustrate efficiency and scalability of the proposed method.
In this paper, we propose a data-driven algorithm for probabilistic optimal power flow (POPF). In particular, we develop a nonparametric Bayesian framework based on Dirichlet process mixture model (DPMM) and variational Bayesian inference (VBI) to establish a probabilistic model for capturing the uncertainties involved with wind generation and load power in power systems. In the proposed setup, the number of components in the mixture model can be automatically and analytically obtained from the consistently updated data. Moreover, we develop an efficient quasi-Monte Carlo sampling method to draw samples from the obtained DPMM, then propose the dynamic data-driven POPF algorithm. Performance of uncertainty modeling framework on publicly available datasets is examined by extensive numerical simulations. Furthermore, the proposed POPF algorithm is verified on multiple IEEE benchmark power systems. Numerical results show the feasibility and superiority of the proposed DPMM-based POPF algorithm for better informed decision-making in power systems with high level of uncertainties.
Uncertainties from deepening penetration of renewable energy resources have posed critical challenges to the secure and reliable operations of future electric grids. Among various approaches for decision making in uncertain environments, this paper focuses on chance-constrained optimization, which provides explicit probabilistic guarantees on the feasibility of optimal solutions. Although quite a few methods have been proposed to solve chance-constrained optimization problems, there is a lack of comprehensive review and comparative analysis of the proposed methods. We first review three categories of existing methods to chance-constrained optimization: (1) scenario approach; (2) sample average approximation; and (3) robust optimization based methods. Data-driven methods, which are not constrained by any particular distributions of the underlying uncertainties, are of particular interest. Key results of the analytical reformulation approach for specific distributions are briefly discussed. We then provide a comprehensive review on the applications of chance-constrained optimization in power systems. Finally, this paper provides a critical comparison of existing methods based on numerical simulations, which are conducted on standard power system test cases.
This paper presents a chance constrained stochastic conic program model for networked microgrids planning. Under a two-stage optimization framework, we integrate a multi-site microgrids investment problem and two sets of operational problems that correspond to the grid-connected and islanding modes, respectively. To handle the uncertain nature of renewable energy generation and load variation, as well as the contingent islanding caused by external disruptions, stochastic scenarios are employed to capture randomness and a joint chance constraint is introduced to control the operational risks. A second-order conic program (SOCP) formulation is also utilized to accurately describe the AC optimal power flow (OPF) in operational problems. As the resulting mixed integer SOCP model is computationally difficult, we customize the bilinear Benders decomposition with non-trivial enhancement techniques to deal with practical instances. Numerical results on 5-and 69-bus networked microgrids demonstrate the effectiveness of the proposed planning model and the superior performance of our solution algorithm.
Higher levels of renewable electricity generation increase uncertainty in power system operation. To ensure secure system operation, new tools that account for this uncertainty are required. In this paper, we formulate a chance-constrained AC optimal power flow problem, which guarantees that generation, power flows and voltages remain within their bounds with a pre-defined probability. We then propose an accurate, yet tractable analytical reformulation of the chance constraints. The reformulation maintains the full, non-linear AC power flow equations for the forecasted operating point, and models the impact of uncertainty through a linearization around this point. We discuss different solution algorithms, including one-shot optimization with and without recourse, and an iterative algorithm which enables scalable implementations. We further discuss how more general chance constraint reformulations can be incorporated within the iterative solution algorithm. In a case study based on four different IEEE systems, we compare the performance of the solution algorithms, and demonstrate scalability of the iterative scheme. We further show that the analytical reformulation accurately and efficiently enforces chance constraints in both in- and out-of-sample tests, and that the analytical approach outperforms two alternative, sample based chance constraint reformulations.
This paper focuses on distribution systems featuring renewable energy sources (RESs) and energy storage systems, and presents an AC optimal power flow (OPF) approach to optimize system-level performance objectives while coping with uncertainty in both RES generation and loads. The proposed method hinges on a chance-constrained AC OPF formulation where probabilistic constraints are utilized to enforce voltage regulation with prescribed probability. A computationally more affordable convex reformulation is developed by resorting to suitable linear approximations of the AC power-flow equations as well as convex approximations of the chance constraints. The approximate chance constraints provide conservative bounds that hold for arbitrary distributions of the forecasting errors. An adaptive strategy is then obtained by embedding the proposed AC OPF task into a model predictive control framework. Finally, a distributed solver is developed to strategically distribute the solution of the optimization problems across utility and customers.
Renewable energy becomes a key contributor to our modern society, but their integration to power grid poses significant technical challenges. Power quality is an important aspect of renewable energy integration. The major power quality concerns are: 1) Voltage and frequency fluctuations, which are caused by non-controllable variability of renewable energy resources. The intermittent nature of renewable energy resources due to ever-changing weather conditions leads to voltage and frequency fluctuations at the interconnected power grid. 2) Harmonics, which are introduced by power electronic devices utilized in renewable energy generation. When penetration level of renewable energy is high, the influence of harmonics could be significant. In this paper, an extensive literature review is conducted on emerging power quality challenges due to renewable energy integration. The paper consists of two sections: 1) Power quality problem definition. Wind turbines and solar photovoltaic (PV) systems and their power quality issues are summarized. 2) Existing approaches to improve power quality. Various methods are reviewed, and the control technology based power quality improvement is the major focus of the paper. The future research directions for emerging power quality challenges for renewable energy integration are recommended.
Coordinated aggregation of a large population of thermostatically controlled loads (TCLs) presents a great potential to provide various ancillary services to the grid. One of the key challenges of integrating TCLs into system-level operation and control is developing a simple and portable model to accurately capture their aggregate flexibility. In this paper, we propose a geometric approach to model the aggregate flexibility of TCLs. We show that the set of admissible power profiles of an individual TCL is a polytope, and their aggregate flexibility is the Minkowski sum of the individual polytopes. In order to represent their aggregate flexibility in an intuitive way and achieve a tractable approximation, we develop optimization-based algorithms to approximate the polytopes by the homothets of a given convex set. As a special application, this set is chosen as a
virtual battery model
, and the corresponding optimal approximations are solved efficiently by equivalent linear programming problems. Numerical results show that our algorithms yield significant improvement in characterizing the aggregate flexibility over existing modeling methods. We also conduct case studies to demonstrate the efficacy of our approaches by coordinating TCLs to track a frequency regulation signal from the Pennsylvania–New Jersey–Maryland Interconnection.
Frequency control reserves are a fundamental mechanism in the current electric power system to preserve the balance between demand and generation of active power at all times. The traditional approach is to exploit conventional power plants for frequency reserves. However, due to the increasing penetration of renewable energies, the demand side is expected to play an essential role in the near future.This paper illustrates the potential of commercial buildings as frequency reserves providers through an experimental demonstration conducted in a multi-zone university building. The proposed control methodology, that represents a first step to the full accomplishment of this task, is presented in details,including the control architecture, the controller design, model identification and hardware description. Finally, the effectiveness of the presented approach is tested by means of simulations and experiments in a very controlled environment.
Electric vehicles (EVs) have the potential to provide frequency regulation service to an independent system operator (ISO) by changing their real-time charging or discharging power according to an automatic generation control (AGC) signal. Recently, the Federal Energy Regulatory Commission has issued Order 755 to ISOs to introduce a performance-based compensation scheme in the frequency regulation market. The goal is to provide economic incentives for fast ramping resources such as EVs to participate in the market. In this paper, we model the EV frequency regulation service under the performance-based compensation scheme. Thereby, a robust optimization framework is adopted for the formulation of a frequency regulation capacity scheduling problem. Our problem formulation takes into account the performance-based compensation scheme, the random AGC signal, and the dynamic arrival and departure times of the EVs. We propose an efficient algorithm to solve the formulated problem. Simulation results show that the proposed algorithm improves the revenue under the performance-based compensation scheme compared with a benchmark algorithm.
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Time-efficient strategic power dispatch for district cooling systems considering the spatial-temporal evolution of cooling load uncertainties
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spatial-temporal evolution of cooling load uncertainties," CSEE J. Power
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Data-driven distributionally robust optimization for real-time economic dispatch considering secondary frequency regulation cost
- L Liu
- Z Hu
- X Duan
- N Pathak
L. Liu, Z. Hu, X. Duan, and N. Pathak, "Data-driven distributionally
robust optimization for real-time economic dispatch considering secondary frequency regulation cost," IEEE Trans. Power Syst., vol. 36,
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Chance constrained programs with mixture distributions
- Z Hu
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