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Effective quantification of the cost of flexibility and its contribution to the system is essential for activating the flexibility of the numerous distributed energy resources (DERs) in power systems operation. We first propose a unified flexibility cost quantification method for heterogonous DERs based on the adjustment ranges in both power and accumulated energy consumption. Compared with traditional power range-based approaches, the proposed method can reflect the potential inconvenience caused to different DER users by flexibility activation and directly capture the time-dependent characteristics of DER's flexibility regions. Extending to the aggregator level, we then propose a flexibility cost formulation aligned with the aggregated flexibility model. On this basis, a model for distribution system operators (DSOs) is proposed to coordinate the DER aggregators' participation in the transmission system operation for energy arbitrage and ancillary services provision. This model derives a method for quantifying the contribution of each aggregator's activated flexibility region to the DSO's revenue from the transmission system, which is used to allocate the revenue to each aggregator while ensuring that the DSO is non-profit. Numerical tests on the IEEE 33-node distribution system validate the proposed methods.

Uncertainty modeling has become increasingly important in power system decision-making. The widely-used tractable uncertainty modeling method-chance constraints with Conditional Value at Risk (CVaR) approximation, can be overconservative and even turn an originally feasible problem into an infeasible one. This paper proposes a new approximation method for multiple joint chance constraints (JCCs) to model the uncertainty in dispatch problems, which solves the conservativeness and potential infeasibility concerns of CVaR. The proposed method is also convenient for controlling the risk levels of different JCCs, which is necessary for power system applications since different resources may be affected by varying degrees of uncertainty or have different importance to the system. We then formulate a data-driven distributionally robust chance-constrained programming model for the power system multiperiod dispatch problem and leverage the proposed approximation method to solve it. In the numerical simulations, two small general examples clearly demonstrate the superiority of the proposed method, and the results of the multiperiod dispatch problem on IEEE test cases verify its practicality.

The DC network security constraints have been extensively studied in numerous power system problems, such as optimal power flow (OPF), security-constrained economic dispatch (SCED), and security-constrained unit commitment (SCUC). Linear shift factors, i.e., power transfer distribution factors (PTDFs), are widely applied to replace DC power flow constraints. However, the PTDF matrix is extremely dense, making it difficult to solve security-constraint optimization problems. This paper analyzes/investigates the computational inefficiency of PTDF-based security constraints from the sparse structure perspective of the primal-dual interior point method(IPM). Additionally, a matrix transformation method is proposed for restoring the sparsity of the linear system during IPM iterations. It turns out that the transformation method is equivalent to solving the original optimization problem expressed in pure voltage angle, which preserves the sparsity structure but introduces additional variables and constraints proportional to one to two times the total number of buses. The regular B-$\theta$ formulation is also a variant of the proposed transformation. Numerical studies show that sparsity rather than the size of variables and constraints is the key factor impacting the speed of solving convex quadratic problems (QP), i.e., OPF and SCED problems. In contrast, sparsity is less desirable when solving a mixed integer problem (MIP), such as the SCUC problem, where reoptimization techniques are significantly more critical and the dual simplex method is typically employed rather than IPM.

Concise and reliable modeling for aggregating power flexibility of distributed energy resources in active distribution networks (ADNs) is a crucial technique for coordinating transmission and distribution networks. Our recent research has successfully derived an explicit expression for the exact aggregation model (EAM) of power flexibility at the substation level under linearized distribution network constraints. The EAM, however, is impractical for decision-making purposes due to its exponential complexity. In this paper, we propose an inner approximation method for aggregating flexibility in ADNs that utilizes the properties of the EAM to improve performance. Specifically, the geometric prototype of the inner approximation model is defined according to a subset of the coefficient vector set of the EAM, which enhances the accuracy. On the other hand, the computation efficiency of the inner approximation is also significantly improved by exploiting the regularity of coefficient vectors in the EAM in the parameter calculation process. The inner approximated flexibility model of ADNs is further incorporated into the security-constrained unit commitment problem as an application. Numerical simulations verify the effectiveness of the proposed method., Comment: 10 pages

In the first part of the two-part series, the model to describe the exact aggregated feasible region (AFR) of multiple types of demand-side resources is derived. Based on a discrete-time unified individual model of heterogeneous resources, the calculation of AFR is, in fact, a feasible region projection problem. Therefore, the Fourier-Motzkin Elimination (FME) method is used for derivation. By analyzing the redundancy of all possible constraints in the FME process, the mathematical expression and calculation method for the exact AFR is proposed. The number of constraints is linear with the number of resources and is exponential with the number of time intervals, respectively. The computational complexity has been dramatically simplified compared with the original FME. However, the number of constraints in the model is still exponential and cannot be simplified anymore. Hence, In Part II of this paper, several approximation methods are proposed and analyzed in detail., Comment: 10 pages

The high penetration of converter-based renewable energy sources has brought challenges to the power system frequency control. It is essential to consider the frequency security constraints and frequency control reserve requirements in unit commitment (UC). Considering that the risk of frequency insecurity varies under the changeable operational condition, we propose to optimize the PFC droop gains and reserve capacities in the UC model to provide diverse control efforts in different risk levels adaptively. Copula theory is used to establish the joint distribution model among frequency control performance, secondary frequency control (SFC) reserve capacities, and power fluctuations. Then the distributionally robust optimization technique is utilized in the SFC reserve requirement determination to handle the possible error in the probability model. The UC simulation is conducted on IEEE 118-bus system to test the proposed optimal PFC droop gain strategy and SFC reserve requirement quantification method. Simulation results show that the proposed optimal PFC droop gain strategy is better than the traditional fixed PFC droop gain setting on economic efficiency and operational flexibility. Besides, the SFC reserve requirement calculated by the proposed method is more appropriate than the actual SFC reserve capacity in the historical operation., Comment: This paper has been submitted to Journal of Modern Power Systems and Clean Energy

The rapid development of renewable energy has increased the peak to valley difference of the netload, making the netload following being a new challenge to the power system. Electric boiler with thermal storage (EBTS) occupies a non-negligible part of the load in the winter season in Northern China. EBTS operation optimization can not only save its own energy cost but also reduce the peak shaving and valley filling pressure of the system. To this end, the operation optimization of EBTS for providing the power balancing service is studied in this paper, which mainly includes three parts: First, the joint probability distribution between the predicted and actual temperatures is built by utilizing the Copula theory; Secondly, the actual temperatures are sampled based on the predicted temperatures of the next day, and the scenario set is generated by clustering these samples, where K-means clustering method are used; Thirdly, the stochastic operation optimization model of EBTS considering the uncertainty of outdoor temperature is constructed. Through the case study, it is found that the proposed method can save the total operation cost of the EBTS compared with the deterministic EBTS operation optimization model.

Large-scale integration of converter-based renewable energy sources (RESs) into the power system will lead to a higher risk of frequency nadir limit violation and even frequency instability after the large power disturbance. Therefore, it is essential to consider the frequency nadir constraint (FNC) in power system scheduling. Nevertheless, the FNC is highly nonlinear and non-convex. The state-of-the-art method to simplify the constraint is to construct a low-order frequency response model at first, and then linearize the frequency nadir equation. In this letter, an extreme learning machine (ELM)-based network is built to de-rive the linear formulation of FNC, where the two-step fitting process is integrated into one training process and more details about the physical model of the generator are considered to reduce the fitting error. Simulation results show the superiority of the proposed method on the fitting accuracy., Comment: This paper has been submitted to the CSEE Journal of Power and Energy Systems

The highly fluctuated renewable generations and electric vehicles have undergone tremendous growth in recent years. The majority of them are connected to the grid via power electronic devices, resulting in wide variation ranges for several key parameters in the frequency response model (FRM) such as system inertia and load damping factor. In this paper, an automatic generation control (AGC) method considering the uncertainties of these key parameters is proposed. First, the historical power system operation data following large power disturbances are used to identify the FRM key parameters offline. Second, the offline identification results and the normal operation data prior to the occurrence of the disturbance are used to train the online probability estimation model of the FRM key parameters. Third, the online estimation results of the FRM key parameters are used as the input, and the model predictive-based AGC signal optimization method is developed based on distributionally robust optimization (DRO) technology. Case studies conducted on the IEEE 118-Bus System show that the proposed AGC method outperforms the widely utilized PI-based control method in terms of performance and efficiency., Comment: This manuscript has been submitted to IEEE Transactions on Power Systems

With the large-scale integration of renewable power generation, frequency regulation resources (FRRs) are required to have larger capacities and faster ramp rates, which increases the cost of the frequency regulation ancillary service. Therefore, it is necessary to consider the frequency regulation cost and constraint along with real-time economic dispatch (RTED). In this paper, a data-driven distributionally robust optimization (DRO) method for RTED considering automatic generation control (AGC) is proposed. First, a Copula-based AGC signal model is developed to reflect the correlations among the AGC signal, load power and renewable generation variations. Secondly, samples of the AGC signal are taken from its conditional probability distribution under the forecasted load power and renewable generation variations. Thirdly, a distributionally robust RTED model considering the frequency regulation cost and constraint is built and transformed into a linear programming problem by leveraging the Wasserstein metric-based DRO technique. Simulation results show that the proposed method can reduce the total cost of power generation and frequency regulation., Comment: This paper has been accepted by IEEE Transactions on Power Systems

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., Comment: arXiv admin note: text overlap with arXiv:1706.04267

This is the second part of a two-part paper on data-based distributionally robust stochastic optimal power flow (OPF). The general problem formulation and methodology have been presented in Part I [1]. Here, we present extensive numerical experiments in both distribution and transmission networks to illustrate the effectiveness and flexibility of the proposed methodology for balancing efficiency, constraint violation risk, and out-of-sample performance. On the distribution side, the method mitigates overvoltages due to high photovoltaic penetration using local energy storage devices. On the transmission side, the method reduces N-1 security line flow constraint risks due to high wind penetration using reserve policies for controllable generators. In both cases, the data-based distributionally robust model predictive control (MPC) algorithm explicitly utilizes forecast error training datasets, which can be updated online. The numerical results illustrate inherent tradeoffs between the operational costs, risks of constraints violations, and out-of-sample performance, offering systematic techniques for system operators to balance these objectives.

Recharging a plug-in electric vehicle is more time-consuming than refueling an internal combustion engine vehicle. As a result, charging stations may face serious congestion problems during peak traffic hours in the near future with the rapid growth of plug-in electric vehicle population. Considering that drivers' time costs are usually expensive, charging congestion will be a dominant factor that affect a charging station's quality of service. Hence, it is indispensable to conduct adequate congestion analysis when designing charging stations in order to guarantee acceptable quality of service in the future. This paper proposes a data-driven approach for charging congestion analysis of plug-in electric vehicle charging stations. Based on a data-driven plug-in electric vehicle charging station planning model, we adopt the queuing theory to model and analyze the charging congestion phenomenon in these planning results. We simulate and analyze the proposed method for charging stations servicing shared-use electric taxis in the central area of Beijing leveraging real-world taxi travel data.

Plug-in electric bus (PEB) is an environmentally friendly mode of public transportation and plug-in electric bus fast charging stations (PEBFCSs) play an essential role in the operation of PEBs. Under effective control, deploying an energy storage system (ESS) within a PEBFCS can reduce the peak charging loads and the electricity purchase costs. To deal with the (integrated) scheduling problem of (PEBs charging and) ESS charging and discharging, in this study, we propose an optimal real-time coordinated charging and discharging strategy for a PEBFCS with ESS to achieve maximum economic benefits. According to whether the PEB charging loads are controllable, the corresponding mathematical models are respectively established under two scenarios, i.e., coordinated PEB charging scenario and uncoordinated PEB charging scenario. The price and lifespan of ESS, the capacity charge of PEBFCS and the electricity price arbitrage are considered in the models. Further, under the coordinated PEB charging scenario, a heuristics-based method is developed to get the approximately optimal strategy with computation efficiency dramatically enhanced. Finally, we validate the effectiveness of the proposed strategies, interpret the effect of ESS prices on the usage of ESS, and provide the sensitivity analysis of ESS capacity through the case studies.

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.

Plug-in electric vehicles (PEVs) can contribute to energy and environmental challenges. Among different types of PEVs, plug-in hybrid electric taxis (PHETs) go in advance. In this study, we provide a spatial and temporal PHET charging demand forecasting method based on one-month global positioning system (GPS)-based taxi travel data in Beijing. Then, using the charging demand forecasting results, a mixed integer linear programming (MILP) model is formulated to plan PHET charging stations in the central area of Beijing. The model minimizes both investment and operation costs of all the PHET charging stations and takes into account the service radius of charging stations, charging demand satisfaction and rational occupation rates of chargers. At last, the test of the planning method is carried out numerically through simulations and the analysis is complemented according to the results.

Increasing penetration of distribution generation (DG) and electric vehicles (EVs) calls for an effective way to estimate the achievable capacity connected to the distribution systems, but the exogenous uncertainties of DG outputs and EV charging loads make it challengeable. This study provides a joint capacity evaluation method with a risk threshold setting function for photovoltaic (PV) generation and EV charging stations (EVCSs). The method is mathematically formulated as a distributionally robust joint chance constrained programming model. And the worst-case conditional value at risk (WC-CVaR) approximation and an iterative algorithm based on semidefinite program (SDP) are used to solve the model. Finally, the method test is carried out numerically on IEEE 33-bus radial distribution system.

In this paper, we address the energy storage management problem in distribution networks from the perspective of an independent energy storage manager (IESM) who aims to realize optimal energy storage sharing with multi-objective optimization, i.e., optimizing the system peak loads and the electricity purchase costs of the distribution company (DisCo) and its customers. To achieve the goal of the IESM, an energy storage sharing strategy is therefore proposed, which allows DisCo and customers to control the assigned energy storage. The strategy is updated day by day according to the system information change. The problem is formulated as a bi-level mathematical model where the upper level model (ULM) seeks for optimal division of energy storage among Disco and customers, and the lower level models (LLMs) represent the minimizations of the electricity purchase costs of DisCo and customers. Further, in order to enhance the computation efficiency, we transform the bi-level model into a single-level mathematical program with equilibrium constraints (MPEC) model and linearize it. Finally, we validate the effectiveness of the strategy and complement our analysis through case studies.

We propose a data-driven method to solve a stochastic optimal power flow (OPF) problem based on limited information about forecast error distributions. The objective is to determine power schedules for controllable devices in a power network to balance operation cost and conditional value-at-risk (CVaR) of device and network constraint violations. These decisions include scheduled power output adjustments 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 assume the distributions are only observable through a finite training dataset. By utilizing the Wasserstein metric to quantify differences between the empirical data-based distribution and the real data-generating distribution, we formulate a distributionally robust optimization OPF problem to search for power schedules and reserve policies that are robust to sampling errors inherent in the dataset. A simple numerical example illustrates inherent tradeoffs between operation cost and risk of constraint violation, and we show how our proposed method offers a data-driven framework to balance these objectives.

Integration of plug-in electric vehicles (PEVs) with distributed renewable resources will decrease PEVs' well-to-wheels greenhouse gas emissions, promote renewable power adoption and defer power system investments. This paper proposes a multidisciplinary approach to jointly planning PEV fast-charging stations and distributed photovoltaic (PV) power plants on coupled transportation and power networks. First, we develop models of 1) PEV fast-charging stations; 2) highway transportation networks under PEV driving range constraints; 3) PV power plants with reactive power control. Then, we formulate a two-stage stochastic mixed integer second order cone program (MISOCP) to determine the sites and sizes of 1) PEV fast-charging stations; 2) PV power plants. To address the uncertainty of future scenarios, a significant number of future typical load, traffic flow and PV generation curves are adopted. This makes the problem large scale. We design a Generalized Benders Decomposition Algorithm to efficiently solve it. To the authors' knowledge, this work is the first that jointly plans both PEV fast-charging stations and PV plants with consideration for PEV driving range limits and reactive PV power control. We conduct numerical experiments to illustrate the effectiveness of the proposed method, and validate the benefits of the joint planning and adopting advanced PV reactive power control.

This paper studies siting and sizing of plug-in electric vehicle (PEV) fast-charging stations on coupled transportation and power networks. We develop a closed-form service rate model of highway PEV charging stations' service abilities, which considers heterogeneous PEV driving ranges and charging demands.We utilize a modified capacitated flow refueling location model (CFRLM) to explicitly capture time-varying PEV charging demands on the transportation network under driving range constraints. We explore extra constraints of the CFRLM to enhance model accuracy and computational efficiency.We then propose a stochastic mixed-integer second order cone programming (SOCP) model for PEV fast-charging station planning. The model considers the transportation network constraints of CFRLM and the power network constraints with AC power flow. Numerical experiments are conducted to illustrate the effectiveness of the proposed method.

In this paper, we study unit commitment (UC) problems considering the uncertainty of load and wind power generation. UC problem is formulated as a chance-constrained two-stage stochastic programming problem where the chance constraint is used to restrict the probability of load imbalance. In addition to the conventional mixed integer linear programming formulation using Big-M, we present the bilinear mixed integer formulation of chance constraint, and then derive its linear counterpart using McCormick linearization method. Then, we develop a bilinear variant of Benders decomposition method, which is an easy-to-implement algorithm, to solve the resulting large-scale linear counterpart. Our results on typical IEEE systems demonstrate that (i) the bilinear mixed integer programming formula-tion is stronger than the conventional one; (ii) the proposed Benders decomposition algorithm is generally an order of magnitude faster than using a professional solver to directly compute both linear and bilinear chance-constrained UC models., Comment: 10 pages, 2 figures