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Modeling, Prediction and Risk Management of Distribution System Voltages with Non-Gaussian Probability Distributions
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Abstract
High renewable energy penetration into power distribution systems causes a substantial risk of exceeding voltage security limits, which needs to be accurately assessed and properly managed. However, the existing methods usually rely on the joint probability models of power generation and loads provided by probabilistic prediction to quantify the voltage risks, where inaccurate prediction results could lead to over or under estimated risks. This paper proposes an uncertain voltage component (UVC) prediction method for assessing and managing voltage risks. First, we define the UVC to evaluate voltage variations caused by the uncertainties associated with power generation and loads. Second, we propose a Gaussian mixture model-based probabilistic UVC prediction method to depict the non-Gaussian distribution of voltage variations. Then, we derive the voltage risk indices, including value-at-risk (VaR) and conditional value-at-risk (CVaR), based on the probabilistic UVC prediction model. Third, we investigate the mechanism of UVC-based voltage risk management and establish the voltage risk management problems, which are reformulated into linear programming or mixed-integer linear programming for convenient solutions. The proposed method is tested on power distribution systems with actual photovoltaic power and load data and compared with those considering probabilistic prediction of nodal power injections. Numerical results show that the proposed method is computationally efficient in assessing voltage risks and outperforms existing methods in managing voltage risks. The deviation of voltage risks obtained by the proposed method is only 15% of that by the methods based on probabilistic prediction of nodal power injections.
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This paper proposes a tractable distributionally robust chance-constrained conservation voltage reduction (DRCC-CVR) method with enriched data-based ambiguity set in unbalanced three-phase distribution systems. The increasing penetration of distributed renewable generation not only brings clean power but also challenges the voltage regulation and energy-saving performance of CVR by introducing high uncertainties to distribution systems. In most cases, the conventional robust optimization methods for CVR only provide conservative solutions. To better consider the impacts of load and PV generation uncertainties on CVR implementation in distribution systems and provide less conservative solutions, this paper develops a data-based DRCC-CVR model with tractable reformulation and data enrichment method. Even though the uncertainties of load and photovoltaic (PV) can be captured by data, the availability of smart meters (SMs) and micro-phasor measurement units (PMUs) is restricted by cost budget. The limited data access may hinder the performance of the proposed DRCC-CVR. Thus, we further present a data enrichment method to statistically recover the high-resolution load and PV generation data from low-resolution data with Gaussian Process Regression (GPR) and Markov Chain (MC) models, which can be used to construct a data-based moment ambiguity set of uncertainty distributions for the proposed DRCC-CVR. Finally, the nonlinear power flow and voltage dependant load models and DRCC with moment-based ambiguity set are reformulated to be computationally tractable and tested on a real distribution feeder in Midwest U. S. to validate the effectiveness and robustness of the proposed method.
Scalable coordination of photovoltaic (PV) inverters, considering the uncertainty in PV and load in distribution networks (DNs), is challenging due to the lack of real-time communications. Decentralized PV inverter setpoints can be achieved to address this issue by capitalizing on the abundance of data from smart utility meters and the scalable architecture of artificial neural networks (ANNs). To this end, we first use an offline, centralized data-driven conservative convex approximation of chance-constrained optimal power flow (CVaR-OPF) in which conditional value-at-risk (CVaR) is used to compute reactive power setpoints of PV inverter, taking into account PV and load uncertainties in DNs. Following that, an artificial neural network (ANN) controller is trained for each PV inverter to emulate the optimal behavior of the centralized control set- points of PV inverter in a decentralized fashion. Additionally, the voltage regulation performance of the developed ANN controllers is compared with other decentralized designs (local controllers) developed using model-based learning (regression-based controller), optimization (affine feedback controller), and case-based learning (mapping) approaches. Numerical tests using real-world feeders corroborate the effectiveness of ANN controllers in voltage regulation and loss minimization.
As high amounts of new energy and electric vehicle (EV) charging stations are connected to the distribution network, the voltage deviations are likely to occur, which will further affect the power quality. It is challenging to manage high quality voltage control of a distribution network only relying on the traditional reactive power control mode. If the reactive power regulation potentials of new energy and EVs can be tapped, it will greatly reduce the reactive power optimization pressure on the network. Keeping this in mind, our reasearch first adds EVs to the traditional distribution network model with new forms of energy, and then a multi-objective optimization model, with achieving the lowest line loss, voltage deviation, and the highest static voltage stability margin as its objectives, is constructed. Meanwihile, the corresponding model parameters are set under different climate and equipment conditions. Ultimately, the optimization model under specific scenarios is obtained. Furthermore, considering the supply and demand relationship of the network, an improved technique for order preference by similarity to an ideal solution decision method is proposed, which aims to judge the adaptability of different algorithms to the optimized model, so as to select a most suitable algorithm for the problem. Finally, a comparison is made between the constructed model and a model without new energy. The results reveal that the constructed model can provide a high quality reactive power regulation strategy.
The unsteady motion of the atmosphere incurs nonlinear and spatiotemporally coupled uncertainties in the wind power prediction (WPP) of multiple wind farms. This brings both opportunities and challenges to wind power probabilistic forecasting (WPPF) of a wind farm cluster or region, particularly when wind power is highly penetrated within the power system. This paper proposes an Improved Deep Mixture Density Network (IDMDN) for short-term WPPF of multiple wind farms and the entire region. In this respect, a deep multi-to-multi (m2m) mapping Neural Network model, which adopts the beta kernel as the mixture component to avoid the density leakage problem, is established to produce probabilistic forecasts in an end-to-end manner. A novel modified activation function and several general training procedures are then introduced to overcome the unstable behavior and NaN (Not a Number) loss issues of the beta kernel function. Verification of IDMDN is based on an open-source dataset collected from seven wind farms, and comparison results show that the proposed model improves the WPPF performance at both wind farm and regional levels. Furthermore, a laconic and accurate probabilistic expression of predicted power at each time step is produced by the proposed model.
When uncontrollable resources fluctuate, Optimum Power Flow (OPF), routinely
used by the electric power industry to re-dispatch hourly controllable
generation (coal, gas and hydro plants) over control areas of transmission
networks, can result in grid instability, and, potentially, cascading outages.
This risk arises because OPF dispatch is computed without awareness of major
uncertainty, in particular fluctuations in renewable output. As a result, grid
operation under OPF with renewable variability can lead to frequent conditions
where power line flow ratings are significantly exceeded. Such a condition,
which is borne by simulations of real grids, would likely resulting in
automatic line tripping to protect lines from thermal stress, a risky and
undesirable outcome which compromises stability. Smart grid goals include a
commitment to large penetration of highly fluctuating renewables, thus calling
to reconsider current practices, in particular the use of standard OPF. Our
Chance Constrained (CC) OPF corrects the problem and mitigates dangerous
renewable fluctuations with minimal changes in the current operational
procedure. Assuming availability of a reliable wind forecast parameterizing the
distribution function of the uncertain generation, our CC-OPF satisfies all the
constraints with high probability while simultaneously minimizing the cost of
economic re-dispatch. CC-OPF allows efficient implementation, e.g. solving a
typical instance over the 2746-bus Polish network in 20 seconds on a standard
laptop.
The MATLAB toolbox YALMIP is introduced. It is described how YALMIP can be used to model and solve optimization problems typically occurring in systems and control theory. In this paper, free MATLAB toolbox YALMIP, developed initially to model SDPs and solve these by interfacing eternal solvers. The toolbox makes development of optimization problems in general, and control oriented SDP problems in particular, extremely simple. In fact, learning 3 YALMIP commands is enough for most users to model and solve the optimization problems
The occurrence of power flow reversal and off-limit of the voltage on common bus becomes more frequent because of the increasing penetration of renewable energy sources (RES) in microgrids. To guarantee the safe and stable operation, adjusting the power output of RES-based inverters to avoid the off-limit voltage is necessary. Considering the apparent power characteristics of inverters, as well as the minimum participation of active power, a voltage control strategy based on stage division to be within the voltage limit is investigated in this paper. In the case of unknown demand and distribution of loads, the proposed control strategy is able to make full use of the apparent power to regulate voltage using simple calculations, while the performance in economical operation is satisfactory. Simulation results prove the effectiveness of the proposed method on the common bus off-limit voltage adjustment.
To achieve effective intraday dispatch of photovoltaic (PV) power generation systems, a reliable ultra-short-term power generation forecasting model is required. Based on a gradient boosting strategy and a dendritic network, this paper proposes a novel ensemble prediction model, named gradient boosting dendritic network (GBDD) model which can reduce the forecast error by learning the relationship between forecast residuals and meteorological factors during the training of sub-models by means of a greedy function approximation. Unlike other machine learning models, the GBDD proposed is able to make fuller use of all meteorological factor data and has a good model interpretation. In addition, based on the structure of GBDD, this paper proposes a strategy that can improve the prediction performance of other types of prediction models. The GBDD is trained by analyzing the relationship between prediction errors and meteorological factors for compensating the prediction results of other prediction models. The experimental results show that the GBDD proposed has the benefit of achieving a higher PV power prediction accuracy for PV power generation and can be used to improve the prediction performance of other prediction models.
The increasing use of distributed energy resources changes the way to manage the electricity system. Unlike the traditional centralized powered utility, many homes and businesses with local electricity generators have established their own microgrids, which increases the use of renewable energy while introducing a new challenge to the management of the microgrid system from the mismatch and unknown of renewable energy generations, load demands, and dynamic electricity prices. To address this challenge, a rank-based multiple-choice secretary algorithm (RMSA) was proposed for microgrid management, to reduce the microgrid operating cost. Rather than relying on the complete information of future dynamic variables or accurate predictive approaches, a lightweight solution was used to make real-time decisions under uncertainties. The RMSA enables a microgrid to reduce the operating cost by determining the best electricity purchase timing for each task under dynamic pricing. Extensive experiments were conducted on real-world data sets to prove the efficacy of our solution in complex and divergent real-world scenarios.
Advances in natural gas-fired technologies have deepened the coupling between electricity and gas networks, promoting the development of the integrated electricity-gas network (IEGN) and strengthening the interaction between the active-reactive power flow in the power distribution network (PDN) and the natural gas flow in the gas distribution network (GDN). This paper proposes a day-ahead active-reactive power scheduling model for the IEGN with multi-microgrids (MMGs) to minimize the total operating cost. Through the tight coupling relationship between the subsystems of the IEGN, the potentialities of the IEGN with MMGs toward multi-energy cooperative interaction is optimized. Important component models are elaborated in the PDN, GDN, and coupled MMGs. Besides, motivated by the non-negligible impact of the reactive power, optimal inverter dispatch (OID) is considered to optimize the active and reactive power capabilities of the inverters of distributed generators. Further, a second-order cone (SOC) relaxation technology is utilized to transform the proposed active-reactive power scheduling model into a convex optimization problem that the commercial solver can directly solve. A test system consisting of an IEEE-33 test system and a 7-node natural gas network is adopted to verify the effectiveness of the proposed scheduling method. The results show that the proposed scheduling method can effectively reduce the power losses of the PDN in the IEGN by 9.86%, increase the flexibility of the joint operation of the subsystems of the IEGN, reduce the total operation costs by $32.20, and effectively enhance the operation economy of the IEGN.
Probabilistic forecasting of photovoltaic (PV) power provides system operators with pertinent information on the uncertainty of PV power generation. This paper proposes a spatio-temporal probabilistic forecasting model based on monotone broad learning system (MBLS) and Copula theory. MBLS is a novel neural network structure for providing an efficient quantile regression solution. MBLS guarantees the monotonicity between quantiles and their probability for thoroughly avoiding the quantile crossing problem. The historical PV data are then clustered using the self-organizing map and samples in each cluster are used for Copula parameter estimations. The proposed approach provides an efficient spatio-temporal forecast of multiple PV plants by combining marginal distributions predicted by MBLS with Copula functions. The real-world data of PV plants in Australia and USA are used to the validate the superiority of the proposed method through detailed comparisons with existing methods using comprehensive evaluation criteria. The presented results demonstrate that the proposed method can provide high-quality probabilistic forecasts corresponding with PV power scenarios.
The Gaussian mixture model (GMM) is a powerful tool to establish the probability distributions of random variables in power system analyses. GMM can model arbitrary probability distributions by increasing the number of its Gaussian components, but the commonly used expectation-maximization (EM) algorithm fails to obtain accurate GMM for large component numbers, which limits the application of GMM to multivariate wind power modeling. In this letter, a parameter estimation method for GMM with large component numbers is proposed based on kernel density estimation (KDE) and the improved density-preserving hierarchical EM algorithm. Then, the closed-form solution to probabilistic power flow (PPF) calculation is derived based on piecewise linearization and GMM, which validates the importance of large component numbers. Finally, the proposed uncertainty modeling method is compared with EM-based GMM, Copula functions, KDE, k-nearest neighbors and block neural autoregressive flow on actual wind speed data to validate its superiority in describing the details of probability densities of wind power. PPF calculation is performed to show the efficiency and accuracy of the proposed uncertainty analysis method.
High penetration of renewable generation poses challenges to power system operation due to its uncertain nature. In droop-controlled microgrids, the voltage volatility induced by renewable uncertainties is aggravated by the high droop gains. This paper proposes a chance-constrained optimal power flow (CC-OPF) problem with power flow routers (PFRs) to better regulate the voltage profile in microgrids. PFR refers to a general type of network-side controller that brings more flexibility to the power network. Comparing with the normal CC-OPF that relies on node power flexibility only, the proposed model introduces a new dimension of control from power network to enhance system performance under renewable uncertainties. Adopting a partial linearization method and an iterative algorithm allows us to address the CC-OPF problem by iteratively solving a subproblem. Since the inclusion of PFRs complicates the subproblem and makes common solvers no longer applicable directly, a semidefinite programming relaxation is used to transform each subproblem into a convex form. The proposed method is verified on a modified IEEE 33-bus system and the results show that PFRs significantly reduce the standard deviations of voltage magnitudes and contribute to mitigating the voltage volatility, which makes the system operate in a more economic and secure way.
Convex relaxations of the AC Optimal Power Flow (OPF) problem are essential not only for identifying the globally optimal solution but also for enabling the use of OPF formulations in Bilevel Programming and Mathematical Programs with Equilibrium Constraints (MPEC), which are required for solving problems such as the coordination between transmission and distribution system operator (TSO/DSO) or optimal network investment. Focusing on active distribution grids and radial networks, this paper introduces a framework that collects and compares, for the first time to our knowledge, the performance of the most promising convex OPF formulations for practical applications. Our goal is to establish a solid basis that will inform the selection of the most appropriate algorithm for different applications. This paper (i) introduces a unified mathematical and simulation framework, (ii) extends existing methods to retain exactness in a wider number of cases and (iii) consider reactive power injections. We conduct simulations on the IEEE 34 and 123 radial test feeders with distributed energy resources (DERs), using yearly solar irradiation and load data.
This study proposes an optimization framework that coordinates the operations of a home energy management system (HEMS) in a low-voltage (LV) distribution network and Volt/VAR optimization (VVO) in a medium-voltage (MV) distribution network through flexible electricity consumption and production of prosumers. The proposed framework consists of a three-level optimization problem, which corresponds to the HEMS for a prosumer at the first level, HEMS aggregator at the second level, and VVO at the third level. The optimal operations of home appliances and distributed energy resources are scheduled in the HEMS according to the prosumer’s preferred appliance scheduling and comfort level. Given the optimal energy consumption schedules from multiple HEMSs, the HEMS aggregator recalculates them while interacting with the VVO, which monitors and controls the MV distribution network efficiently. Furthermore, to incorporate the uncertainty for the predicted errors of residential solar photovoltaic generation and outdoor temperature into the proposed framework, the deterministic optimization (DO)-based HEMS aggregator model is reformulated into a chance constrained optimization (CCO)-based model. Numerical examples tested in IEEE 13-node MV and CIGRE 18-node LV distribution systems show that, in contrast with a method without coordination of HEMS and VVO, the proposed DO-based approach reduces the total energy losses, active energy consumption, and reactive energy consumption by 21.03%,7.62%, and 115.51%, respectively, in the LV system, and 2.34%,1.36%, and 4.07%, respectively, in the MV system. In addition, the performance of the proposed CCO-based approach was validated in terms of probability level of chance constraints.
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.
This book opens up new ways to develop mathematical models and optimization methods for interdependent energy infrastructures, ranging from the electricity network, natural gas network, district heating network, and electrified transportation network. The authors provide methods to help analyze, design, and operate the integrated energy system more efficiently and reliably, and constitute a foundational basis for decision support tools for the next-generation energy network. Chapters present new operation models of the coupled energy infrastructure and the application of new methodologies including convex optimization, robust optimization, and equilibrium constrained optimization. This book provides theoretical foundation and technical applications for energy system integration. The interdisciplinary research presented in this book will be useful to readers in many fields including electrical engineering, civil engineering, and industrial engineering.
• Provides theoretical foundation and technical applications for energy system integration;
• Contains new research problems and methodologies for researchers and practitioners in energy system integration;
• Presents in-depth discussion in a systematic fashion.
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.
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.
Volt/var control (VVC) is one of the primary functions of the distribution management system aiming at optimum operation of power distribution networks while respecting all of their operational and security constraints. However, the recent huge integration of highly stochastic distributed generation (DG) sources with the grid presents a significant challenge to the traditionalVVCschemes,whichassumethefuturetobeperfectly known. This paper presents a two-stage chance constrained optimization scheme to handle these nodal power uncertainties and guarantees that the operational and security constraints are respectedforalmostallrealizationsoftheuncertainty.Thechance constrained model is solved by collecting enough randomly chosen samples from the probability spaces of the uncertain parameters so that the class of the problem, a mixed integer second order cone program (MISOCP), is not elevated. The algorithm not only dispatches the optimum schedule for discrete controlling devices like transformers and shunt capacitors but also optimizes the predefined decision rules for reactive power control of DG sources, thus falling in line with the requirement laid down in the revised IEEE 1547 Standard. Numerical simulations on three different test systems show the superiority of the proposed algorithm over the traditional deterministic methods. IEEE
A novel probabilistic optimal power flow (P-OPF) model with chance constraints that considers the uncertainties of wind power generation (WPG) and load is proposed in this paper. An affine generation dispatch strategy is adopted to balance the system power uncertainty by several conventional generators, and thus the linear approximation of the cost function with respect to the power uncertainty is proposed to compute the quantile (which is also recognized as the value-at-risk) corresponding to a given probability value. The proposed model applies this quantile as the objective function and minimizes it to meet distinct probabilistic cost regulation purposes via properly selecting the given probability. In particular, the hedging effect due to the used affine generation dispatch is also thoroughly investigated. In addition, an analytical method to calculate probabilistic load flow (PLF) is developed with the probability density function of WPG, which is proposed to be approximated by a customized Gaussian mixture model whose parameters are easily obtained. Accordingly, it is successful to analytically compute the chance constraints on the transmission line power and the power outputs of conventional units. Numerical studies of two benchmark systems show the satisfactory accuracy of the PLF method, and the effectiveness of the proposed P-OPF model.
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.
The authors report a power-flow-minimum heuristic algorithm for
determining the minimum loss configuration of radial distribution
networks. The algorithm is based on the concept of optimum flow pattern
which is determined by solving the KVL and KCL (Kirchoff's voltage and
current laws) equations of the network. The optimum flow pattern of a
single loop formed by closing a normally open switch is found, and the
flow pattern is established in the radial network by opening a closed
switch. This process is repeated until the minimum loss configuration is
obtained. A simple, fast and approximate power flow method has also been
developed to assist the reconfiguration algorithm. The proposed
reconfiguration algorithm has been found to give better network
configuration than those obtained by some other methods
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
High-dimensional multivariate forecasting with low-rank Gaussian copula processes
- D Salinas
- M Bohlke-Schneider
- L Callot
- R Medico
- J Gasthaus
D. Salinas, M. Bohlke-Schneider, L. Callot, R. Medico, and J. Gasthaus,
"High-dimensional multivariate forecasting with low-rank Gaussian
copula processes," in Proc. 33rd conf. Adv. Neural Inf. Process. Syst.,
Vancouver, Canada, 2019, pp. 1-11, doi: 10.5555/3454287.3454900.
Distributionally robust chance-constrained optimization for soft open points operation in active distribution networks
- Q Hou
- G Chen
- N Dai
- H Zhang
Q. Hou, G. Chen, N. Dai and H. Zhang, "Distributionally robust
chance-constrained optimization for soft open points operation in active
distribution networks," CSEE J. Power Energy Syst., early access, 2023,
doi: 10.17775/CSEEJPES.2021.02110.
Darts: User-Friendly Modern Machine Learning for Time Series
- J Herzen
J. Herzen et al., "Darts: User-Friendly Modern Machine Learning for
Time Series," J. Mach. Learn. Res., vol. 23, no. 1, pp. 5442-5447, Jan.
2022.