Probability Distribution Of Uncertainty Research Articles

Optimal capacity configuration of joint system considering uncertainty of wind and photovoltaic power and demand response

AbstractTo reduce phenomenon of abandoning wind and photovoltaic power, improve the limitations of traditional methods in dealing with uncertainty of wind and photovoltaic power and system planning, and improve the optimal configuration of resources, an optimal capacity configuration of joint system considering uncertainty of wind and photovoltaic power and demand response is proposed. Firstly, using probability distribution information of wind and photovoltaic power output, the distance between actual probability distribution and forecast probability distribution is constrained based on the 1‐norm and ∞‐norm. A fuzzy set considering uncertainty probability distribution is constructed, and a two‐stage distributed robust planning model is established. The first stage involves optimizing joint system capacity for scenarios with the lowest probability of wind and photovoltaic power; the second stage builds on capacity optimization scheme from the first stage and aims to minimize operating costs through simulation optimization. Secondly, column and constraint generation is used to solve the model. Finally, constructing an example based on actual data from a power grid in Northeast China for simulation and analysis, the results show that the method achieves a balanced optimization of robustness and economy, effectively reduces carbon emissions and improves ability of the system to consume wind and photovoltaic power.

Optimal demand response for a virtual power plant with a hierarchical operation framework

There are significant uncertainties and variations in demand response under different operating conditions and incentives. These challenges hinder virtual power plants (VPPs) from fully leveraging the regulatory capabilities of flexible resources. To address this issue, this study introduces an optimal demand response (DR) approach for VPPs, featuring a hierarchical operational framework that accounts for the uncertainty in user participation in DR. First, a refined DR model is developed, incorporating response characteristic parameters based on the DR mechanism. Second, a two-stage distributionally robust optimization model is formulated within a market trading framework for energy scheduling of VPPs. This model aims to minimize operational costs while considering uncertainties in DR. Additionally, the ambiguity set for the uncertainty probability distributions is constructed using a data-driven method. Finally, the study presents an optimal DR disaggregation method that aims to maximize aggregator satisfaction and accounts for the impact of load rebound on the optimization results. Simulation results demonstrate that the proposed method significantly reduces the deviation in demand response by 40 % and enhances the profit of the VPP by 20.7 %.

Research on distributionally robust energy storage capacity allocation for output fluctuations in high permeability wind and solar distribution networks.

This paper presents a novel approach to addressing the challenges associated with energy storage capacity allocation in high-permeability wind and solar distribution networks. The proposed method is a two-phase distributed robust energy storage capacity allocation method, which aims to regulate the stochasticity and volatility of net energy output. Firstly, an energy storage capacity allocation model is established, which considers energy storage's investment and operation costs to minimize the total cost. Then, a two-stage distributed robust energy storage capacity allocation model is established with the confidence set of uncertainty probability distribution constrained by 1-norm and ∞-norm. Finally, a Column and Constraint Generation (C&CG) algorithm is used to solve the problem. The validity of the proposed energy storage capacity allocation model is confirmed by examining different wind and solar penetration levels. Furthermore, the model's superiority is demonstrated by comparing it with deterministic and robust models.

Distributionally Robust Optimal Power Flow via Ellipsoidal Approximation

This paper proposes a distributionally robust joint chance-constrained AC optimal power flow to manage the risk of operational limits violations which are caused by uncertain renewable generation. By determining the dispatch schedule, this approach can help to decrease the risk of renewable curtailment, load shedding, or emergency redispatch in real-time. To model the proposed approach, the renewable uncertainty is first modeled as a distributionally robust ellipsoidal bound based on the Wasserstein metric. This bound is built upon a limited historical renewable forecast errors dataset without any assumption on the probability distribution of uncertainty. Then, this uncertainty bound is adopted within a semidefinite relaxation of the optimal power flow. The system responses to the renewable generation forecast errors are modeled as linear sensitivities, where all conventional generators are responsible for compensating the impacts of renewable forecast errors. Numerical experiments on IEEE 14-bus and 118-bus systems show the validity and scalability of the proposed method. Furthermore, the effectiveness of the proposed approach in terms of meeting probabilistic guarantees, cost-effectiveness and computational time is also demonstrated in the experiments.

Flexible operation of a CHP‐VPP considering the coordination of supply and demand based on a strengthened distributionally robust optimization

AbstractDue to the generation variability, the growing capacity of renewable energy has posed unprecedented challenges to ensure the security of power system operation. Here, a two‐stage strengthened distributionally robust optimization (DRO) scheme is proposed for theself‐scheduling of a combined heat and power virtual power plant (CHP‐VPP) over a coupled electric power network (EPN) and district heating network (DHN). The CHP‐VPP operator maximizes its profits in the day‐ahead market and minimizes its cost in the real‐time market under the worst‐case realization of the uncertainties. Instead of assuming that the uncertainties follow known probability distributions or confidence bounds, a strengthened ambiguity set based on moment information and Wasserstein metric is built to provide more accurate characterizations of the true probability distribution of uncertainties. In addition, in order to enhance the flexibility of the system, a HOMIE model considering indoor activities and outside temperatures of each building is built to satisfy the comfortable indoor temperature. To make the whole problem tractable, linearisation and duality theory are adopted, and then a tailored column‐and‐constraint generation algorithm is developed to solve the problem. The validity and applicability of the strengthened DRO scheme are verified by an IEEE 33‐bus EPN and 14‐node DHN.

Low carbon economic dispatch of integrated energy system considering extended electric heating demand response

Retrofitting coal-fired units with carbon capture plants is an important means to achieve low-carbon improvements in integrated energy systems (IES). However, the lack of flexible resources restricts the development of carbon capture power plants, and the uncertainty of IES affects the reliability of system. With this background, this study proposes a low-carbon economic dispatch strategy considering extended electric heating demand response. Firstly, the organic Rankine cycle waste heat power generation is introduced to decouple the “power determined by heat” mode of combined heat and power. Secondly, incentive-based electric load demand response, as well as comfort fuzziness-based heat load demand response are considered. Subsequently, a data-driven distributionally robust optimization model is developed to solve the source-load uncertainty problem, where the confidence set of the uncertainty probability distribution is constrained by both 1-norm and ∞-norm constraints. The column and constraint generation (C&CG) algorithm is used to solve the model and obtain the operating output configuration. Finally, the test system verifies the effectiveness of the proposed method.

Data-driven distributionally robust scheduling of community integrated energy systems with uncertain renewable generations considering integrated demand response

A community integrated energy system (CIES) is an important carrier of the energy internet and smart city in geographical and functional terms. Its emergence provides a new solution to the problems of energy utilization and environmental pollution. To coordinate the integrated demand response and uncertainty of renewable energy generation (RGs), a data-driven two-stage distributionally robust optimization (DRO) model is constructed. A comprehensive norm consisting of the 1-norm and ∞-norm is used as the uncertainty probability distribution information set, thereby avoiding complex probability density information. To address multiple uncertainties of RGs, a generative adversarial network based on the Wasserstein distance with gradient penalty is proposed to generate RG scenarios, which has wide applicability. To further tap the potential of the demand response, we take into account the ambiguity of human thermal comfort and the thermal inertia of buildings. Thus, an integrated demand response mechanism is developed that effectively promotes the consumption of renewable energy. The proposed method is simulated in an actual CIES in North China. In comparison with traditional stochastic programming and robust optimization, it is verified that the proposed DRO model properly balances the relationship between economical operation and robustness while exhibiting stronger adaptability. Furthermore, our approach outperforms other commonly used DRO methods with better operational economy, lower renewable power curtailment rate, and higher computational efficiency.

Data-driven crude oil scheduling optimization with a distributionally robust joint chance constraint under multiple uncertainties

Crude oil scheduling optimization is crucial for decreasing the production cost of refineries. However, the feasibility of the optimized schemes is challenged by uncertainties such as possible ship arrival delays and fluctuating crude demands. This study develops a novel data-driven continuous-time optimization model with a distributionally robust joint chance constraint to ensure the overall feasibility probability of the crude oil processing plan under multiple uncertainties. Industrial data are collected to build an ambiguity set using Wasserstein distance to include the potential joint probability distribution of uncertainties. The radius of the ambiguity set is chosen by cross-validation. The model is constructed considering the worst case in the ambiguity set. First, it is formulated as a conditional value-at-risk constrained optimization model. A big-M coefficient and additional binary variables are then utilized to convert the proposed model into a resolvable problem. The efficacy and reliability of the method are explored through case studies.

Wasserstein Distributionally Robust Motion Control for Collision Avoidance Using Conditional Value-at-Risk

In this article, a risk-aware motion control scheme is considered for mobile robots to avoid randomly moving obstacles when the true probability distribution of uncertainty is unknown. We propose a novel model-predictive control (MPC) method for limiting the risk of unsafety even when the true distribution of the obstacles’ movements deviates, within an <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">ambiguity set</i> , from the empirical distribution obtained using a limited amount of sample data. By choosing the ambiguity set as a statistical ball with its radius measured by the <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Wasserstein metric</i> , we achieve a probabilistic guarantee of the <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">out-of-sample risk</i> , evaluated using new sample data generated independently of the training data. To resolve the infinite-dimensionality issue inherent in the distributionally robust MPC problem, we reformulate it as a finite-dimensional nonlinear program using modern distributionally robust optimization techniques based on the Kantorovich duality principle. To find a globally optimal solution in the case of affine dynamics and output equations, a spatial branch-and-bound algorithm is designed using McCormick relaxation. The performance of the proposed method is demonstrated and analyzed through simulation studies using nonlinear dynamic and kinematic vehicle models and a linearized quadrotor model. The simulation results indicate that, even when the sample size is small, the proposed method can successfully avoid randomly moving obstacles with a guarantee of out-of-sample risk, while its sample average approximation counterpart fails to do so.

Distributionally Robust MPC for Nonlinear Systems

Classical stochastic model predictive control (SMPC) methods assume that the true probability distribution of uncertainties in controlled systems is provided in advance. However, in real-world systems, only partial distribution information can be acquired for SMPC. The discrepancy between the true distribution and the distribution assumed can result in sub-optimality or even infeasibility of the system. To address this, we present a novel distributionally robust data-driven MPC scheme to control stochastic nonlinear systems. We use distributionally robust constraints to bound the violation of the expected state-constraints under process disturbance. Sequential linearization is performed at each sampling time to guarantee that the system's states comply with constraints with respect to the worst-case distribution within the Wasserstein ball centered at the discrete empirical probability distribution. Under this distributionally robust MPC scheme, control laws can be efficiently derived by solving a conic program. The competence of this scheme for disturbed nonlinear systems is demonstrated through two case studies.

Data-Driven Distributionally Robust Co-Optimization of P2P Energy Trading and Network Operation for Interconnected Microgrids

This paper proposes a data-driven distributionally robust co-optimization model for the peer-to-peer (P2P) energy trading and network operation of interconnected microgrids (MGs). In particular, three-phase unbalanced MG networks are considered to account for the implementation practices, and the emerging soft open point (SOP) technology is used for the flexible connection of the multi-MGs. First, the energy management in individual MGs is modeled as a distributionally robust optimization (DRO) problem considering the P2P energy trading options and various operational constraints. Later, a novel decentralized and incentive-compatible pricing strategy is developed for P2P energy trading using the alternating direction method of multipliers (ADMM). Furthermore, the uncertainties in load consumption and renewable generation (RG) are taken into account and the Wasserstein metric (WM) is used to construct the ambiguity set of the uncertainty probability distributions (PDs). Consequently, only historical data is needed rather than prior knowledge about the actual PDs. Finally, the equivalent linear programming reformulations are derived for the DRO model to achieve computational tractability. Numerical tests on four interconnected MGs corroborate the advantages of the proposed P2P energy trading scheme and also demonstrate that the proposed DRO model is more effective in handling the uncertainties compared to the robust optimization (RO) and the stochastic programming (SP) models.

Analyzing uncertainties involved in estimating collapse risk with and without considering uncertainty probability distribution parameters

Analyzing uncertainties involved in estimating collapse risk with and without considering uncertainty probability distribution parameters

Data-Driven Distributionally Robust Hierarchical Coordination for Home Energy Management

Home energy management (HEM) is conventionally formulated using central control or local control methods assuming known accurate probability distributions of uncertainties. Without coordination between the central and the local control hierarchies, mutual impacts cannot be properly considered and addressed, which undermine operation efficiency of HEM. Moreover, the probability distributions always have errors, even in emerging data-based methods. To address these unsolved issues, firstly, this paper proposes a new hierarchical coordination method for HEM. On the central hierarchy, a central controller optimizes schedule of non-thermal loads to minimize daily operating cost. On the local hierarchy, automatic local controllers respond to real-time variations of their corresponding thermal zone temperatures to fulfill customer thermal comfort requirements. Secondly, using a finite training dataset, a data-driven distributionally robust optimization (DRO) method is proposed to guarantee solution robustness against worst probability distribution of multiple uncertainties. The hierarchical coordination method is formulated as a DRO model using the Wasserstein metric, which can be solved with high computing efficiency. Numerical simulations validate high solution robustness and overall optimum of the proposed method, fully addressing mutual impacts and uncertainties. Moreover, the proposed method could efficiently save the daily electricity bill by 11%, compared to conventional methods.

Rolling-horizon dispatch of advanced adiabatic compressed air energy storage based energy hub via data-driven stochastic dynamic programming

Faced with environmental pollution and energy crisis, energy hub yields an improvement on efficiency and flexibility of multi-energy supply. Advanced adiabatic compressed air energy storage (AA-CAES) is a promising large-scale energy storage technology and is attracting increasing attention due to its heat-electricity co-storage potentials. This paper investigates the external characteristics of advanced adiabatic compressed air energy storage and exploits its ability to implement an energy hub. First, a dual state-of-charge (SoC) model of advanced adiabatic compressed air energy storage is presented, taking into account the system off-design features and the impact of ambient temperature. The state-of-charge of the air storage tank depends on the mass of stored air, whose mass flow rate affects the charging and discharging electric power. The state-of-charge of the high-temperature thermal energy storage depends on the mass of heat transfer oil, whose mass flow rate determines the reserving and releasing heating power. Adjusting the mass flow rates of air and oil offers flexible control on the power and thermal outputs. An energy hub is built based upon the advanced adiabatic compressed air energy storage. To address the daily self-dispatch of the energy hub facing the uncertainties of load and ambient temperature, a data-driven stochastic dynamic programming model is proposed which allows a rolling horizon implementation. The Kernel regression is employed to estimate the conditional probability distribution of uncertainties. The cost-to-go functions in the Bellman equation are approximated via sampling and interpolation. Case studies validate the effectiveness of the proposed approach. The results indicate that: 1) The proposed dynamic programming method outperforms model predictive control in computational efficiency. 2) Neglecting the temperature effect on compressed air energy storage operation leads to 4.5%, 7.8%, and 9.2% regulation errors of charging, discharging and heating power, respectively.

Sensitivity analysis of an electrophysiology model for the left ventricle.

Modelling the cardiac electrophysiology entails dealing with the uncertainties related to the input parameters such as the heart geometry and the electrical conductivities of the tissues, thus calling for an uncertainty quantification (UQ) of the results. Since the chambers of the heart have different shapes and tissues, in order to make the problem affordable, here we focus on the left ventricle with the aim of identifying which of the uncertain inputs mostly affect its electrophysiology. In a first phase, the uncertainty of the input parameters is evaluated using data available from the literature and the output quantities of interest (QoIs) of the problem are defined. According to the polynomial chaos expansion, a training dataset is then created by sampling the parameter space using a quasi-Monte Carlo method whereas a smaller independent dataset is used for the validation of the resulting metamodel. The latter is exploited to run a global sensitivity analysis with nonlinear variance-based indices and thus reduce the input parameter space accordingly. Thereafter, the uncertainty probability distribution of the QoIs are evaluated using a direct UQ strategy on a larger dataset and the results discussed in the light of the medical knowledge.

Risk-Based Distributionally Robust Real-Time Dispatch Considering Voltage Security

A risk-based distributionally robust real-time dispatch approach is proposed to strike a balance between the operational costs and risk as well as considering the nodal voltage security even when the distribution of uncertainties cannot be precisely estimated. To model uncertainties, a data-drive ambiguity set is developed based on the imprecise probability theory without any presumption on the probability distribution of uncertainties. The obtained ambiguity set is based on the confidence interval, which can be constructed according to actual needs by choosing the point-wise or family-wise confidence interval. Two strategies for wind farms to provide voltage support are incorporated in the proposed approach so that the nodal voltage security can be ensured. By incorporating the risk tractable estimation method and sequential convex optimization method, an efficient algorithm is developed to release the computational burden. Numerical results show that compared with the regular robust optimization, the proposed approach reduces the total operational cost to achieve statistically optimal dispatch decisions. The proposed approach also outperformances the stochastic programming in term of the operational risk reliability. Meanwhile, the nodal voltage security issues can be mitigated with the voltage support from wind farms. In conclusion, under the uncertainty of probability distributions, the proposed approach can efficiently strike a balance between the operational cost and risk while ensuring the nodal voltage security.

Distributionally Robust Chance-Constrained AC-OPF for Integrating Wind Energy Through Multi-Terminal VSC-HVDC

Using the multi-terminal VSC-HVdc (MTDC) system to collect far-distance offshore wind power into the main onshore ac grid has become one of the ideal transmission approaches for renewable energy integration. However, with traditional fixed dc droop control in MTDC system, the lack of the capability of regulating power flow might pose a great challenge to the system operation. Accordingly, a novel MTDC model has been established in this paper which establishes that the output power of onshore converters can be formulated as a linear function of the control variables, namely power initial set point of each onshore converter. Based on this model, considering the uncertainty from wind power forecast error, a distributionally robust chance-constrained AC-OPF model is proposed for the hybrid AC/MTDC system. Without any presumption on the probability distribution of uncertainty, this model ensures the secure operation by enforcing chance constraints under the worst-case probability distribution over an ambiguity set based on the Wasserstein-Moment metric. Aiming at obtaining a tractable chance-constraint reformulation, we first employ the partial linearization of ac power flow equations to yield a linear model which represents the system response under uncertainties. We then develop an efficient and scalable approach to derive the linear analytical reformulation of distributionally robust chance constraints. The performance of the proposed model is firstly tested on the 14-bus system for an illustrative purpose, and then on the 1354-bus system for demonstrating scalability.

A risk-based modeling approach for radiation therapy treatment planning under tumor shrinkage uncertainty

Robust optimization approaches have been widely used to address uncertainties in radiation therapy treatment planning problems. Because of the unknown probability distribution of uncertainties, robust bounds may not be correctly chosen, and a risk of undesirable effects from worst-case realizations may exist. In this study, we developed a risk-based robust approach, embedded within the conditional value-at-risk representation of the dose-volume constraint, to deal with tumor shrinkage uncertainty during radiation therapy. The objective of our proposed model is to reduce dose variability in the worst-case scenarios as well as the total delivered dose to healthy tissues and target dose deviations from the prescribed dose, especially, in underdosed scenarios. We also took advantage of adaptive radiation therapy in our treatment planning approach. This fractionation technique considers the response of the tumor to treatment up to a particular point in time and reoptimizes the treatment plan using an estimate of tumor shrinkage. The benefits of our model were tested in a clinical lung cancer case. Four plans were generated and compared: static, nominal-adaptive, robust-adaptive, and conventional robust (worst-case) optimization. Our results showed that the robust-adaptive model, which is a risk-based model, achieved less dose variability and more control on the worst-case scenarios while delivering the prescribed dose to the tumor target and sparing organs at risk. This model also outperformed other models in terms of tumor dose homogeneity and plan robustness.

Primal-Dual Learning Based Risk-Averse Optimal Integrated Allocation of Hybrid Energy Generation Plants under Uncertainty

A groundswell of opinion in utilizing environmentally friendly energy technologies has been put forth worldwide. In this paper, we consider an energy generation plant distribution and allocation problem under uncertainty to get the utmost out of available developments, as well as to control costs and greenhouse emissions. Different clean and traditional energy technologies are considered in this paper. In particular, we present a risk-averse stochastic mixed-integer linear programming (MILP) model to minimize the total expected costs and control the risk of CO2 emissions exceeding a certain budget. We employ the conditional value-at-risk (CVaR) model to represent risk preference and risk constraint of emissions. We prove that our risk-averse model can be equivalent to the traditional risk-neutral model under certain conditions. Moreover, we suggest that the risk-averse model can provide solutions generating less CO2 than traditional models. To handle the computational difficulty in uncertain scenarios, we propose a Lagrange primal-dual learning algorithm to solve the model. We show that the algorithm allows the probability distribution of uncertainty to be unknown, and that desirable approximation can be achieved by utilizing historical data. Finally, an experiment is presented to demonstrate the performance of our method. The risk-averse model encourages the expansion of clean energy plants over traditional models for the reduction CO2 emissions.

A Distributionally Robust Reactive Power Optimization Model for Active Distribution Network Considering Reactive Power Support of DG and Switch Reconfiguration

Abstract In recent years, the rapid development of the Distributed Generation (DG) has brought new challenges to the reactive power optimization in the active distribution network. The traditional stochastic optimization and robust optimization methods may result in over-conservative or over-risky decisions when dealing with the uncertainty of renewable energies. Therefore, this paper proposes a distributionally robust reactive power optimization model for active distribution network, which regards the network loss as the its optimization target, and considers the reactive power support of and the switch reconfiguration. This model includes not only switching, on load tap changer (OLTC), discrete and continuous reactive power compensation devices usually contained in traditional reactive power optimization, but also the detailed model of capacity curve of Double-Fed Induction Generator and Micro-Gas Turbine. After the reactive power support is integrated, the linearization method of active and reactive power output coupling characteristics constraint is specially designed. Then, based on the typical scenario data of wind and photovoltaic power, and the regulation characteristics for different decision variables, a data-driven based two-stage distributionally robust reactive power optimization model is set up, in which the uncertainty probability distribution confidence set is simultaneously constrained by 1-norm and ∞-norm. Finally, by the Columns and Constraints Generation algorithm, the IEEE 33-bus system is applied to verify the effectiveness of proposed.