Worst-case Realizations Research Articles

A distributed robust ADMM-based model for the energy management in local energy communities

Increasing the number of participants in energy communities leads to a new challenge in power systems, which is finding the optimal strategy for community members. Accordingly, this paper presents a distributed model for determining the optimal energy trading strategy of community participants such as buyers, sellers, and the community manager (CM). In the proposed model, the local day-ahead energy market, peer-to-peer (P2P) contracts, and the power grid are considered for trading energy between participants as well as compensating for power shortages/surpluses in the community. To model the uncertainty of PV generation, and selling/buying prices of the distribution network, the robust optimization (RO) approach is used. According to the defined budget of uncertainty, the optimal strategies of community members are determined based on the worst-case realizations of uncertain parameters. To decrease the solution time, the distributed optimization method is addressed. Accordingly, the augmented Lagrangian relaxation (ALR) and the alternating direction method of multipliers (ADMM) methods are used to decompose the optimization problem. The performance of the proposed model is evaluated through a case study. Simulation results demonstrate that the proposed model reduces the solution time of energy management problem in communities, significantly.

A novel adaptive robust model for scheduling distributed energy resources in local electricity and flexibility markets

Aggregators, as intermediaries between consumers, prosumers, and local market operators, manage their clients’ resources for participation in the multi-markets. Various resources of consumers and prosumers have different technical characteristics, which shall be considered in the scheduling strategy. The fast response and controllability of some resources may increase the profit of aggregators and improve the flexibility of distribution grids. Therefore, aggregators need a scheduling strategy to determine their participation level in different markets. However, the expected profit of aggregators can be affected by uncertainties in some resources such as PV generation or consumption of prosumers. This work provides a decomposed bi-level strategy approach for aggregators to deploy the resources of prosumers in the local energy and flexibility markets. The upper and lower sub-problems determine the optimal commitment status of the aggregator’s resources and the participation level in the local energy and flexibility markets, respectively. In the proposed model, the adaptive robust optimization (ARO) is addressed to model uncertainties of consumption and generation of renewable resources based on the worst-case realizations of uncertain parameters. In addition, impacts of consumption control programs such as load-shifting and load reduction are considered in the proposed model. Through a case study and different scenarios, the performance and efficiency of the proposed model are validated.

A Robust Model for Portfolio Management of Microgrid Operator in the Balancing Market

The stochastic nature of renewable energy resources and consumption has the potential to threaten the balance between generation and consumption as well as to cause instability in power systems. The microgrid operators (MGOs) are financially responsible for compensating for the imbalance of power within their portfolio. The imbalance of power can be supplied by rescheduling flexible resources or participating in the balancing market. This paper presents a robust optimization (RO)-based model to maintain the balance of a portfolio according to uncertainties in renewable power generation and consumption. Furthermore, load reduction (LR) and battery energy storage (BES) are considered flexible resources of the MGO on the consumption side. The model is formulated based on the minimax decision rule that determines the minimum cost of balancing based on the worst-case realizations of uncertain parameters. Through the strong duality theory and big-M theory, the proposed minimax model is transformed into a single-level linear maximization problem. The proposed model is tested on a six-node microgrid test system. The main contributions of the proposed model are presenting a robust model for portfolio management of MGO and using BES and LR to improve the flexibility of microgrid. Simulation results demonstrate that using LR and BES could decrease the balancing cost. However, the optimal portfolio management to compensate for the imbalance of power is highly dependent on the risk preferences of MGO.

Robust Data-Driven Control Barrier Functions for Unknown Continuous Control Affine Systems

In this letter, we introduce robust data-driven control barrier functions (CBF-DDs) to guarantee robust safety of unknown continuous control affine systems despite worst-case realizations of generalization errors from prior data under various continuity assumptions. To achieve this, we leverage non-parametric data-driven approaches for learning guaranteed upper and lower bounds of an unknown function from the data set to formulate/obtain a safe input set for a given state. By incorporating the safe input set into an optimization-based controller, the safety of the system can be ensured. Moreover, we present several complexity reduction approaches including providing subproblems that can be solved in parallel and downsampling strategies to improve computational performance.

Biobjective robust simulation-based optimization for unconstrained problems

We propose a biobjective robust simulation-based optimization (BORSO) method to solve unconstrained problems involving implementation errors and parameter perturbations. We adopt the notion that a solution is robust efficient (RE) if the region that dominates its worst-case realizations of the biobjectives under uncertainty does not contain (all) the worst-case realizations of the biobjectives of any other solution under uncertainty. Our algorithm aims to efficiently find a set of RE solutions through a series of function evaluations or simulations. First, we design surrogate-model guided search mechanisms for the worst-case neighbors of the current iterate. Subsequently, we determine the iteration distance along an effective local move direction, which excludes the worst-case neighbors from the neighborhood of the new iterate. Depending on the practical need for solution diversity, multiple initial solutions can be specified in our algorithm, and the final iterates of these solutions generate a set of RE solutions. The test results of a synthetic biobjective nonconvex optimization problem show the effectiveness of the BORSO method and its superior performance against a sampling-based robust multiobjective optimization solver. Furthermore, when the proposed algorithm is applied to a real-world biobjective traffic signal timing problem, satisfactory solutions can be obtained under a limited computational budget. These results indicate that the proposed BORSO method can solve unconstrained biobjective simulation-based optimization problems under uncertainties.

A robust model for aggregated bidding of energy storages and wind resources in the joint energy and reserve markets

The high reliability and flexibility of Battery Energy Storage (BES) resources in comparison with other renewable technologies promote the development of this technology in smart grids. The fast response of BES to load variations could help the power system operators to maintain the balance of generation and consumption in real-time, and improve the flexibility of the smart grid, effectively. In this work, a new model is presented that determines the aggregated scheduling of BES and Wind Power Resource (WPR) in the joint energy and reserve markets. To evaluate the performance of BES in different markets, the proposed model is divided into day-ahead and real-time planning horizons. According to market prices, ramp rates, marginal costs, and technical constraints of units, the optimal participation levels in different markets are determined. The deployed power in real-time and wind power are considered as the uncertain parameters and the Robust Optimization (RO) framework is proposed to manage the related financial risk based on the worst-case realizations of uncertain parameters. The robust strategy is formulated based on the Mixed Integer Linear Programming (MILP) technique, which can be solved via the branch-and-bound method. Finally, the performance and effectiveness of the model are analyzed via different case studies. Simulation results show that the day-ahead and real-time markets are the best options for buying and selling the energy of BESs, and participation in the reserve market and regulation service increases their profit, significantly. Furthermore, the expected profit greatly depends on the risk preferences of decision-makers, and reducing the variation interval of wind generation by 40 % leads to an increase of 74.65 % in revenues.

Sensitivity-Assisted multistage nonlinear model predictive control: Robustness, stability and computational efficiency

Key requirements for robust nonlinear model predictive control (NMPC) are stability, efficient performance under uncertainty, constraint satisfaction and computational efficiency. Multistage NMPC, based on a scenario tree formulation for the uncertainty, has been shown to satisfy the first three objectives under plant-model mismatch. However, a limiting factor in multistage NMPC, is the exponential scaling of the scenarios with respect to uncertain parameters and the length of the robust horizon. To address this issue, we present an approximate sensitivity-assisted multistage NMPC (samNMPC) scheme that reduces the problem size by dividing the scenario set into critical and noncritical scenarios, with the former composed of the worst-case realizations of the uncertain parameters. In this approach, the optimization is sought explicitly over the critical scenarios, while noncritical scenarios are included implicitly through nonlinear programming (NLP) sensitivity-based approximations in the objective function. A key advantage of the proposed approach is that the problem size is independent of the number of constraints and scales only linearly with the robust horizon. This allows faster computations with longer robust horizons that rigorously account for future uncertainty. In this paper, we explore the samNMPC approach and discuss its robust stability properties in context of the robust horizon. We demonstrate the applicability of the approach for the continuous stirred tank reactor (CSTR) and the quadtank case studies for tracking setpoints, and show that samNMPC compares favorably in performance and robustness to ideal multistage NMPC, but with a significant reduction in computational cost.

Hybrid Modeling Based Co-Optimization of Crew Dispatch and Distribution System Restoration Considering Multiple Uncertainties

Natural disasters could lead to large-scale power outages by causing severe damages to distribution networks (DNs). Developing highly efficient outage management schemes is imperative for expediting service restoration. To this end, a novel multistage co-optimization model is proposed to seamlessly integrate the repair crew dispatch with the distribution system restoration. Furthermore, multiple sources of uncertainties are incorporated. To properly handle these uncertainties, a novel hybrid modeling approach is developed. In particular, the first stage is cast as a deterministic optimization problem to determine the crew route option prior to the realization of uncertainties. The second stage is to reconfigure the DN for forming multimicrogrids considering the uncertainties in crew travel time and repair time, which is formulated as a stochastic program. In the third stage, a robust optimization framework is applied to acquire a scheduling plan of various DERs that is immune to the worst-case realizations of electricity demand and photovoltaic generation. Finally, an advanced solution method is devised to achieve the computational tractability based on constraint and column generation and progressive hedging approaches. Numerical tests on the modified IEEE 37-bus DN show that the computation performance of the proposed solution method well fits the real-time application and also demonstrate that the highest demand curtailment is reduced by 3.3% and the variance of the restoration outcome is lowered by 40.5% compared with a recently proposed benchmark.

Comparative study of different risk measures for robust optimization of oil production under the market uncertainty: a regret-based insight

Model-based production optimization relies on a dynamic model that simulates the fluid flow in the oil reservoirs, and an economic objective function that assigns an economic measure to the recoverable oil reserves. An optimization algorithm utilizes the dynamic model to find the production scenario which maximizes the economic measure of profit. However, due to incompleteness and doubtfulness of available data, the reservoir model describing the complex subsurface geology is quite uncertain. Moreover, the definition of the economic objective functions such as net present value (NPV) requires economic variables such as oil price, interest rate, and production costs which unpredictably vary with time. In recent years, robust optimization (RO) has been widely used as an appropriate tool for handling the uncertainties in production optimization problems. However, previous works on robust optimization paid less attention to economic uncertainties arising from market volatility. Instead, they are mostly focused on geological uncertainties. This paper is devoted to production optimization under oil market uncertainty. To narrow down the range of economic uncertainties, a Bayesian framework for oil price history matching and forecasting has been developed which allows generating more reliable realizations of oil price future trend. It is common to include a measure of risk-averse in the objective function of RO problems. However, the quality of the solutions depends directly on the used risk measure. In the oil industry, risk measures such as worst-case scenario and CVaR (Conditional Value at Risk) have been used to mitigate the risk of low-profit realizations. These risk measures are appropriate in many cases for measuring the robustness. Though, they are inadequate in evaluating robustness in a relative sense in cases where the worst-case realizations have an undue effect on the final decisions. The risk measure defined based on the minimax regret approach takes into account all realizations instead of just considering the worst-case realizations. In this research, RO has been performed to maximize NPV using the minimax regret approach. In addition, the results are compared with the common risk measures used in the oil industry including expected profit, CVaR, and worst-case. Results show that while worst-case scenario and CVaR perform better than other risk measures in lower-profit realizations, they give inappropriate results for other scenarios. In contrast, regret-based approach and expected profit give nearly optimum solutions for all realizations. In this paper, the minimax regret approach was compared with other risk measures in the presence of oil price uncertainty. However, the results might be extended to optimization under geological uncertainty.

A stochastic-robust coordinated optimization model for CCHP micro-grid considering multi-energy operation and power trading with electricity markets under uncertainties

The combined cooling, heating and power (CCHP) micro-grids are getting increasing attentions due to the realizations of higher energy efficiency etc. The deepening of electricity market (EM) reforms provides an opportunity for CCHP micro-grid to interact with upstream power grid through participating in EMs while being operated. Hence, the multi-energy operation optimization of CCHP micro-grid should be extended to further involve the strategy of power trading with EMs. This paper proposes a stochastic-robust coordinated optimization model for CCHP micro-grid considering multi-energy operation and power trading with EMs, from the perspective of the day-ahead stage. By depicting the uncertain day-ahead and real-time clearing prices as stochastic scenarios, the objective function is constructed based on stochastic optimization and conditional value-at-risk, which is to minimize both the expected operational cost and potential risk of cost increase related to scenarios. By modeling the uncertain renewable generations as uncertainty set, the operational constraints are established based on robust optimization, which is to guarantee safe and stable operation under the worst-case realizations within uncertainty set. Through the abovementioned combination of multiple methods, the proposed model reasonably keeps both the conservativeness and computational complexity at relatively low levels. Simulations have verified the effectiveness of this model in terms of cost minimization, computational time, renewable power accommodation and risk control compared with existing methods. Furthermore, simulation results also reveal that additional benefits are brought for CCHP micro-grid by EM participations.

Tube-based Anticipative Robust MPC for Systems with Multiplicative Uncertainty

Many control problems contain time-varying or parameter-varying dynamics. With model predictive control (MPC), it is possible to include known plant variations into the controller for improving control performance. Unfortunately, perfect knowledge of the plant is rarely available and the accurateness of models may change over time and operating points. Robust control approaches consider worst-case realizations with a static model which ensure constraint satisfaction and stability but may yield conservative performance. The control algorithm presented in this paper uses anticipative information about future uncertainties and varying models to improve control performance while ensuring stability and feasibility. Possible system trajectories are bounded by polytopic tubes and recursive feasibility is achieved by the use of a terminal set. The controller properties are evaluated in a numerical example and compared to a similar control algorithm.

Accommodating uncertain wind power investment and coal-fired unit retirement by robust energy storage system planning

Increasing wind power integration and coal-fired unit retirement strain the power system's spinning reserve and increase the pressure on peak regulation. With the ability to stock extra power generation and supply the peak load, the energy storage system (ESS) can alleviate the rising demand on the spinning reserve and plays an increasingly important role in the power system. In this paper, a tri-level robust ESS planning model is proposed to accommodate uncertain wind power investment as well as coal-fired unit retirement. The upper-level of this model is to determine the planning scheme of ESSs, which iteratively takes the worst-case of wind power investment and coal-fired unit retirement into consideration. The middle-level and lower-level of this model are to make the optimal daily economic dispatch under the worst-case realizations of uncertainties. We derive an equivalent reformulation of the proposed robust ESS planning model and solve it with a dual column-and-constraint generation algorithm. Case studies are conducted on the IEEE RTS-79 system. The results demonstrate the superiority of the proposed planning method in comparison with other methods. Furthermore, the effects of the capital cost of ESS, the expected proportion of wind power, and the uncertainty budget on the development of ESS are studied. Taking the uncertainties of unit retirement and wind power investment into consideration achieves a better trade-off between the ESS investment cost and the operation cost.

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.

Adaptive robust optimization framework for day-ahead microgrid scheduling

This paper proposes a framework to address day-ahead scheduling of a microgrid (MG) with wind, solar and micro-turbine distributed generators as well as batteries, considering the associated uncertainties and AC power flow constraints. The problem uncertainties are wind and solar generations, demand, and grid electricity price. The uncertainties are modelled using bounded intervals within adaptive robust optimization method. Unlike previous robust models presented for MG scheduling, our proposed adaptive robust model optimizes both wait-and-see and here-and-now decisions simultaneously which leads to obtaining a better operating point. In order to make wait-and-see and here-and-now decisions and immunize the model against worst case realizations, the proposed MG scheduling model has been formulated as a tri-level optimization framework. In the first level, here-and-now commitment decisions are made (these are decisions made before realization of uncertainties). In the second stage, the worst case realization of the uncertain parameters is determined. In the third level, the wait-and-see recourse decisions are made (these are decisions made after realization of uncertainties). Since there are before uncertainty-realization decisions, worst uncertainty-realization determination, and after uncertainty-realization decisions, the proposed model becomes a tri-level optimization framework. The proposed adaptive robust tri-level MG scheduling model, the proposed solution approach including AdptRob and ConvAC problems, and the proposed out-of-sample analysis are the main contributions of this paper. A convexified AC power flow model is used in the proposed framework for modeling the distribution network. The 69-bus distribution test system is used to test the proposed adaptive robust model as well as the proposed solution approach which converges very fast. Moreover, the results of the out-of-sample analysis shows that the proposed adaptive robust model results in lower expected cost and lower cost variance.

Adaptive Robust Optimization-Based Optimal Operation of Microgrids Considering Uncertainties in Arrival and Departure Times of Electric Vehicles

The optimal operation of microgrids is challenging due to the presence of various uncertain factors, i.e., renewable energy sources, loads, market price signals, and arrival and departure times of electric vehicles (EVs). In order to incorporate these uncertainties into the operation model of microgrids, an adaptive robust optimization-based operation method is proposed in this paper. In particular, the focus is on the uncertainties in arrival and departure times of EVs. The optimization problem is divided into inner and outer problems and is solved iteratively by introducing column and constraint cuts. The unit commitment status of dispatchable generators is determined in the outer problem. Then, the worst-case realizations of all the uncertain factors are determined in the inner problem. Based on the values of uncertain factors, the generation amount of dispatchable generators, the amount of power trading with the utility grid, and the charging/discharging amount of storage elements are determined. The performance of the proposed method is evaluated using three different cases, and sensitivity analysis is carried out by varying the number of EVs and the budget of uncertainty. The impact of the budget of uncertainty and number of EVs on the operation cost of the microgrid is also evaluated considering uncertainties in arrival and departure times of EVs.

Robust portfolio selection for index tracking

We develop a robust portfolio selection model for tracking a market index using a subset of its assets. The model is a 0–1 integer program that seeks to maximize similarity between selected assets and the assets of the target index. We allow uncertainty in the objective function by using a computationally tractable robust framework that can control the conservativeness of the solution. This protects against worst-case realizations of potential estimation errors and other deviations. Out-of-sample experiments using the S&P 100 demonstrate the advantages of the robust model. Compared to portfolios constructed with the nominal model, moderately conservative robust portfolios are shown to have lower tracking error and risk profiles that are more similar to the target index.

Differential games with integral constraints on disturbances

Abstract A positional differential game for a linear system is considered. This game differs from games previously studied in the literature in that it does not assume geometrical constraints on the controls and the disturbances, but instead imposes integral constraints on the samples of the disturbance. The existence of an optimal strategy is established and a method of construction is suggested. Counterstrategies are constructed that generate the worst-case realizations of the disturbance. It is established that this differential game has a value and a saddle point.