Probability Distribution Of Uncertain Variables Research Articles

Day-ahead optimal peer-to-peer energy trading strategy for multi-microgrids based on Nash bargaining game with data-driven chance constraints

This paper proposes an optimization model to obtain the day-ahead optimal peer-to-peer (P2P) trading strategy for multi-microgrids (MMG). Firstly, a joint economic energy and reserve scheduling model of a microgrid (MG) is established while considering specific network constraints. Then, to mitigate the impact of renewable energy and load forecasting errors, chance constraints are introduced for the reserve capacity and buses voltage limitations of an individual microgrid. Additionally, a versatile distribution method is adopted to capture the probability distribution of uncertain variables in a data-driven manner, avoiding any prior assumptions. Finally, Nash bargaining theory is employed to deal with the P2P energy trading problem among MMG. The problem is equivalently transformed into two sequential subproblems for solving. Moreover, the alternating direction method of multipliers algorithm is used to solve the subproblems in a distributed manner for privacy concerns. The proposed model not only enables effective P2P energy trading for MMG, but also ensures compliance with the internal network constraints of each MG. Furthermore, the scheduling strategy exhibits robustness in handling forecast errors related to renewable energy and load. In case studies, MMG containing three interconnected MGs is constructed based on the IEEE-123 bus distribution network, and simulation results show that the cost of the MMG is reduced by 9.92 % while all the MGs can benefit from P2P trading. In addition, the risk-averse scheduling results of energy and reserve are obtained, and the conservativeness can be controlled by changing the confidence level, which verifies the effectiveness of the proposed model.

Two-stage robust optimization of thermal-ESS units scheduling under wind uncertainty

The uncertainties of wind turbine (WT) output have large impacts on the optimal operation of integrated energy systems. This paper investigates the two-stage robust optimization (RO) of hybrid energy integration regarding wind volatility. In RO method, the WT output is modeled with an uncertainty budget, which represents the degree of conservatism. Simulation results validate that when the uncertainty budget rises to a maximal value, ESS degradation cost increases 25.7%, and total cost decreases 2.0%. Additionally, demand response program (DRP) exhibits elastic and voluntary merits. With the implementation of DRP, 6139.9$ total savings are achieved compared with the ordinary RO method. Meanwhile, the merits of stochastic optimization (SO) have also been studied. Despite the ESS degradation cost and total cost in SO being lower than those in RO, SO requires accurate probability distribution of uncertain variables in real situations. RO can shortly provide worst-case schemes which are highly appreciated by power utility companies.

IGDT-Based Medium-Term Optimal Cascade Hydropower Operation in Multimarket with Hydrologic and Economic Uncertainties

Hydrologic uncertainty and economic uncertainty both affect market-oriented hydropower operation. For medium-term cascade hydropower operation, the uncertainties of daily reservoir inflow and market clearing power price can pose risks to operation, especially when power stations participate in multiple power markets. It is difficult to obtain the fluctuation range of uncertain variables for different expected power sale profits, and to plan the operation range of cascade hydropower stations. For a cascade hydropower system dispatching power to two markets, this work developed a risk assessment method for medium-term optimal cascade hydropower operation based on information gap decision-making theory (IGDT). We established a bilevel stochastic IGDT model for the main hydrologic and economic uncertainties in hydropower market participation. The lower-level model solves the minimum robustness profit or maximum opportunity profit from power sale, and the upper-level model solves the maximum robustness fluctuation range or minimum opportunity fluctuation range of the uncertain variables for different expected profits. Then we converted the bilevel model to two equivalent single-level models, and provided solutions. The proposed method does not require the probability distribution of uncertain variables, which is often unavailable; moreover, it reduces computational complexity while ensuring accuracy of results. The proposed method can provide decision makers with reasonable power dispatch options. This work used a cascade hydropower system in Southwest China as a case study. The proposed method provided acceptable fluctuation range of uncertain reservoir inflow and power price under different expected profits, and provided the operation range to guarantee the expected profit. Decision makers with different risk preferences can evaluate various strategies to optimize medium-term cascade hydropower operation to ensure the expected profit.

Wasserstein Distributionally Robust Stochastic Control: A Data-Driven Approach

Standard stochastic control methods assume that the probability distribution of uncertain variables is available. Unfortunately, in practice, obtaining accurate distribution information is a challenging task. To resolve this issue, in this article we investigate the problem of designing a control policy that is robust against errors in the empirical distribution obtained from data. This problem can be formulated as a two-player zero-sum dynamic game problem, where the action space of the adversarial player is a Wasserstein ball centered at the empirical distribution. A dynamic programming solution is provided exploiting the reformulation techniques for Wasserstein distributionally robust optimization. We show that the contraction property of associated Bellman operators extends a single-stage out-of-sample performance guarantee, obtained using a measure concentration inequality, to the corresponding multistage guarantee without any degradation in the confidence level. Furthermore, we characterize an explicit form of the optimal control policy and the worst-case distribution policy for linear-quadratic problems with Wasserstein penalty.

Distributionally robust optimization of home energy management system based on receding horizon optimization

This paper investigates the scheduling strategy of schedulable load in home energy management system (HEMS) under uncertain environment by proposing a distributionally robust optimization (DRO) method based on receding horizon optimization (RHO-DRO). First, the optimization model of HEMS, which contains uncertain variable outdoor temperature and hot water demand, is established and the scheduling problem is developed into a mixed integer linear programming (MILP) by using the DRO method based on the ambiguity sets of the probability distribution of uncertain variables. Combined with RHO, the MILP is solved in a rolling fashion using the latest update data related to uncertain variables. The simulation results demonstrate that the scheduling results are robust under uncertain environment while satisfying all operating constraints with little violation of user thermal comfort. Furthermore, compared with the robust optimization (RO) method, the RHO-DRO method proposed in this paper has a lower conservation and can save more electricity for users.

Probabilistic Bounds on Vehicle Trajectory Prediction Using Scenario Approach

The automotive industry concerns about improving road safety. One of the major challenges is to assess road risk and react accordingly in order to avoid accidents. This requires predicting the evolution of the surrounding vehicle trajectories. However, the prediction involves uncertainties from driver operations and ground situations. It is critical to obtain the vehicle trajectory prediction with probabilistic-guarantee bounds. This contribution paper proposes a novel approach to obtain probabilistic ellipsoidal bounds for vehicle trajectory prediction. The vehicle dynamics model adopts a classical bicycle model. The uncertainty of the future trajectory is from the driver’s intend and road condition which can be simplified by setting some parameters of the vehicle dynamics model as a stochastic model. Then, a stochastic optimization problem is formulated to obtain the probabilistic ellipsoidal bounds on the future vehicle trajectories. The proposed approach is validated in a numerical simulation which shows the relationship between the computation complexity and the conservatism of the probabilistic ellipsoidal bounds. The proposed method can be generally used for a physics-based motion method, maneuver-based motion method, and interaction-aware motion method by defining the probability distribution of uncertain variables differently.

Distributionally robust optimization model of active distribution network considering uncertainties of source and load

To ensure the safety and reliability of the distribution network and adapt to the uncertain development of renewable energy sources and loads, a two-stage distributionally robust optimization model is proposed for the active distribution network (ADN) optimization problem considering the uncertainties of the source and load in this paper. By establishing an ambiguity set to capture the uncertainties of the photovoltaic (PV) power, wind power and load, the piecewise-linear function and auxiliary parameters are introduced to help characterize the probability distribution of uncertain variables. The optimization goal of the model is to minimize the total expected cost under the worst-case distribution in the ambiguity set. The first-stage expected cost is obtained based on the predicted value of the uncertainty variable. The second-stage expected cost is based on the actual value of the uncertainty variable to solve the first-stage decision. The generalized linear decision rule approximates the two-stage optimization model, and the affine function is introduced to provide a closer approximation to the second-stage optimization model. Finally, the improved IEEE 33-node and IEEE 118-node systems are simulated and analyzed with deterministic methods, stochastic programming, and robust optimization methods to verify the feasibility and superiority of the proposed model and algorithm.

Crossing weighted uncertainty scenarios assisted distribution-free metamodel-based robust simulation optimization

In practice, computer simulations cannot be perfectly controlled because of the inherent uncertainty caused by variability in the environment (e.g., demand rate in the inventory management). Ignoring this source of variability may result in sub-optimality or infeasibility of optimal solutions. This paper aims at proposing a new method for simulation–optimization when limited knowledge on the probability distribution of uncertain variables is available and also limited budget for computation is allowed. The proposed method uses the Taguchi robust terminology and the crossed array design when its statistical techniques are replaced by design and analysis of computer experiments and Kriging. This method offers a new approach for weighting uncertainty scenarios for such a case when probability distributions of uncertain variables are unknown without available historical data. We apply a particular bootstrapping technique when the number of simulation runs is much less compared to the common bootstrapping techniques. In this case, bootstrapping is undertaken by employing original (i.e., non-bootstrapped) data, and thus, it does not result in a computationally expensive task. The applicability of the proposed method is illustrated through the Economic Order Quantity (EOQ) inventory problem, according to uncertainty in the demand rate and holding cost.

Choosing ecosystem service investments that are robust to uncertainty across multiple parameters

Info-gap decision theory facilitates decision making for problems in which uncertainty is large and probability distributions of uncertain variables are unknown. The info-gap framework allows the decision maker to maximize robustness to failure in the presence of uncertainty, where uncertainty is in the parameters of the model and failure is defined as the model output falling below some minimally acceptable performance threshold. Info-gap theory has found particular application to problems in conservation biology and ecological economics. In this study, we applied info-gap theory to an ecosystem services tradeoff case study in which a decision maker aiming to maximize ecosystem service investment returns must choose between two alternative land uses: native vegetation conservation or the establishment of an exotic timber plantation. The uncertain variables are the carbon price and the water price. With a "no-information" uncertainty model that assumes equal relative uncertainty across both variables, info-gap theory identifies a minimally acceptable reward threshold above which conservation is preferred, but below which plantation establishment is preferred. However, with an uncertainty model that allows the carbon price to be substantially more uncertain than the water price, conservation of native vegetation becomes an economically more robust investment option than establishing alien pine plantations. We explored the sensitivity of the results to the use of alternative uncertainty models, including asymmetric uncertainty in individual variables. We emphasize the general finding that the results of info-gap analyses can be sensitive to the choice of uncertainty model and that, therefore, future applications to ecological problems should be careful to incorporate all available qualitative and quantitative information relating to uncertainties or should at least justify the no-information uncertainty model.

Naturally Fractured Carbonate Oil Reservoir: Reserve Estimation Study

A naturally fractured carbonate reservoir is chosen for implementing uncertainty assessment on estimation of original oil in place. The probability distribution of uncertain variables is defined in reserve estimation and evaluating probable reserve by using Monte Carlo simulation. The reserves are estimated for Karabogaz, Kbb-C member of Karababa and Derdere formations. The reserve estimation in fracture is based on available logs and tests about fracture parameters like density and width. The production data is used for verifying the calculated fracture porosity distributions. The mean of reserve distributions are close to deterministic ones. The sensitivity analysis on input parameters showed that matrix porosity, net thickness, and fracture porosity are significant in Karabogaz and Kbb-C formation reserve estimation while water saturation and fracture porosity are most significant in estimation of Derdere formation reserves.