The increasing share of volatile renewables in power systems requires more reserves to balance forecast errors in renewable generation and power fluctuations. In contrast, common reserves such as gas-fired power plants are phased out, impeding the procurement of sufficient reserves. Alternative reserves, particularly on the demand side, such as battery storage systems, also exhibit some degree of freedom to deviate from their scheduled operating point to supply or consume more or less power, thus providing a flexibility potential. However, demand-side flexibility potentials are generally subject to uncertainties, and so is the generation of volatile renewables. The challenge is incorporating the uncertainties on both sides to procure sufficient (uncertain) flexibility potential in advance. Considering uncertainty is important to avoid additional, drastic measures in real-time to balance generation and demand, such as curtailing renewable generation or load shedding. This work presents a stochastic flexibility calculus that provides an indicator for computing the risk of insufficient flexibility potentials or, conversely, guarantees for sufficient flexibility potentials. Thus, the stochastic flexibility calculus contributes to overcoming the challenge of procuring sufficient flexibility potentials in renewable-based systems. An evaluation based on real data is performed using an example of a renewable energy community consisting of households equipped with photovoltaic power plants and battery storage systems. The newly introduced stochastic flexibility calculus computes the number of households that must operate their battery storage systems flexibly to balance forecast errors locally. The results show that the forecast method significantly influences this number. Some numerical results appear unexpected, as too many flexibility-friendly households can negatively impact the aggregated household flexibility potential.

To manage a large amount of flexible distributed energy resources (DERs) in the distribution networks, the virtual power plant (VPP) is introduced into the industry. The VPP can optimally dispatch these resources in a cluster manner and provide flexibility for the power system operation as a whole. Most existing studies formulate the equivalent power flexibility of the aggregating DERs as deterministic optimization models without considering their uncertainties. In this paper, we introduce the stochastic power flexibility range (PFR) and time-coupling flexibility (TCF) to describe the power flexibility of VPP. In this model, both operational constraints and the randomness of the DERs' output are incorporated, and a combined model and data-driven solution is proposed to obtain the stochastic PFR, TCF, and cost function of VPP. The aggregating model can be easily incorporated into the optimization model for the power system operator or market bidding, considering uncertainties. Finally, a numerical test is performed. The results show that the proposed model not only has higher computational efficiency than the scenario-based methods but also achieves more economic benefits.

In order to unlock the flexibility potential of energy consumers and prosumers, the development of market mechanisms for flexibility planning and procurement is necessary. The authors propose a stochastic local flexibility market to solve grid issues such as voltage deviations and grid congestion in a distribution grid. Their proposed solution includes activation of flexibility assets at the consumers’ premises, using a stochastic local flexibility market design. They consider a pooled local flexibility market design under demand uncertainty and stochastic bidding process. Optimization models are used to determine flexibility demand and supply bids by the distribution system operator and the aggregator respectively. A stochastic AC-optimal power model to determine flexibility demand and a two-stage stochastic model to supply flexibility are implemented to simulate a stochastic local flexibility market. This allows to determine stochastic flexibility supply bid curves, and optimum flexibility supply dispatch. The analysis shows that the cost of grid operations is reduced when the system uses the local flexibility market. The proposed methodology is applicable for local flexibility market designs aiming to use potential end-user flexibility for grid operations.

The traveling deliveryman problem under uncertainty: Fundamentals for flexible supply chains

This study deals with two general solutions for a simply supported linear elastic Bernoulli–Euler beam with a stochastic bending flexibility, subjected to a deterministic loading. Two model problems are considered. The first problem is associated with a trapezoidally distributed load, whereas the second problem treats a sinusoidally distributed load. The importance of the solution for the trapezoidal load lies in its practicality. The derivation of stochastic characteristics for random beams under a sinusoidal load is useful due to the expandability to generally distributed loads by a Fourier sine series expansion. Numerical results are reported for various cases illustrating the effect of stochasticity of the beam’s properties on its flexural response.

Fouling is a substantial economic, energy, and safety issue for all the process industry applications, heat transfer units in particular. Although this phenomenon can be mitigated, it cannot be avoided and proper cleaning cycle scheduling is the best way to deal with it. After thorough literature research about the most reliable fouling model description, cleaning procedures have been optimized by minimizing the Time Average Losses (TAL) under nominal operating conditions according to the well-established procedure. For this purpose, different cleaning actions, namely chemical and mechanical, have been accounted for. However, this procedure is strictly related to nominal operating conditions therefore perturbations, when present, could considerably compromise the process profitability due to unexpected shutdown or extraordinary maintenance operations. After a preliminary sensitivity analysis, the uncertain variables and the corresponding disturbance likelihood were estimated. Hence, cleaning cycles were rescheduled on the basis of a stochastic flexibility index for different probability distributions to show how the uncertainty characterization affects the optimal time and economic losses. A decisional algorithm was finally conceived in order to assess the best number of chemical cleaning cycles included in a cleaning supercycle. In conclusion, this study highlights how optimal scheduling is affected by external perturbations and provides an important tool to the decision-maker in order to make a more conscious design choice based on a robust multi-criteria optimization.

A mixed-integer conic programming formulation for computing the flexibility index under multivariate gaussian uncertainty

Due to recent technological achievements, stochastic optimization, which inherently captures the uncertainty of intermittent resources, is being used to capture the variability and uncertainty of wind and solar resources. However, due to persistent computational limitations, it is not practical to consider all possible variable generation scenarios. As a result, a reduced number of most likely scenarios is usually considered. While this helps reduce the computational burden, it also leaves the system operator vulnerable to some risk. In order to address this issue, this paper aims at providing insight into using an explicit reserve requirement in a stochastic modeling framework in order to provide system operators with greater confidence in stochastic dispatch solutions. This is accomplished by simulating a modified version of the IEEE 118 bus system in a fully stochastic, multi-timescale framework with flexibility reserve requirements. Results show that utilizing a stochastic flexibility reserve requirement within the stochastic modeling framework offers the most reliability benefit.

We present a novel failure analysis approach combining structural properties of stochastic Petri Nets and flexibility of fuzzy logic. Firstly, we develop a powerful fuzzy ranking technique. We analyze major drawbacks of existing ranking techniques. Then we demonstrate the capabilities of the presented algorithm to overcome such drawbacks. The approach considers weight, spread, and difference of x coordinate of the center of gravity (COG) point of each fuzzy number and is able to deal with a wide variety of fuzzy numbers. Using this technique, we utilize isomorphism between stochastic Petri Nets and their corresponding Markov chains and present a failure analysis algorithm incorporating some critical factors. This algorithm can be implemented in diverse industrial applications.

Synthesis and optimization of the recovery route for residual products under uncertain product demand

Abstract A new, unified theory and algorithms, based on multiparametric programming techniques, for the solution of flexibility analysis and design optimization problems in linear process systems are presented. They are used for the flexibility test and index problems in systems with deterministic parameters, and for the stochastic and expected stochastic flexibility evaluation problems in systems with stochastic parameters. They are computationally efficient and give the explicit dependence of various flexibility metrics on the values of the continuous design variables. The latter feature enables the easy and efficient comparison of design alternatives. It also allows for the compact formulation of design optimization problems that can be solved parametrically to yield the exact algebraic form of the trade‐off curve of economics against flexibility. Key features of the proposed approach are demonstrated through both mathematical and process examples.

New exact solutions for randomly loaded beams with stochastic flexibility

This paper presents developments toward a unified framework for incorporating both process flexibility and robustness (in terms of product quality) criteria in process optimization under uncertainty. A robust design methodology, known as the Taguchi approach, is discussed in the context of uncertainty, and some limitations are identified. Taguchi's method is then extended to take into account process constraints and probabilistic uncertainty. A framework for establishing the interactions and synergistic benefits between the two operability objectives is proposed, based on an expected measure, where product quality losses are taken into account explicitly. Tradeoffs between stochastic flexibility and a robust criterion are explored in order to depict optimal operating policies in the presence of uncertainty; extensions toward design optimization are also briefly discussed. A number of examples is presented to illustrate the applicability of the proposed framework.

Behavior of Stochastic Shear Beams Under Random Loading via Stochastic Variational Principles

Exact and approximate solutions, and variational principles for stochastic shear beams under deterministic loading

Flexibility and Robustness Issues in Process Optimization Under Uncertainty

Flexibility analysis and design of dynamic processes with stochastic parameters

Synthesis of flexible and reliable short-term batch production plans

A branch and bound procedure for the design of multipurpose batch plants with uncertain demands

Automatic identification techniques of reliability bottlenecks in process systems under uncertainty