Joint Probabilistic Constraints Research Articles

A novel surrogate polytope method for day-ahead virtual power plant scheduling with joint probabilistic constraints

Virtual Power Plant (VPP) is an emerging concept that can effectively manage a large number of distributed energy resources (DERs). However, the inherent uncertainty of load and renewable generation poses a challenge to reliable VPP power scheduling. This manuscript proposes a novel method for day-ahead VPP scheduling with a joint probabilistic guarantee on its power availability without violating the DER constraints and network constraints. A surrogate polytope is first used to find the inner approximation of the VPP power, implicitly including the low-level DER power, DER constraints, and network constraints. Then, a multivariate Gaussian distribution is used to fit the random parameters of the surrogate polytope, after which the iterative supporting hyperplane algorithm is used to solve the VPP scheduling problem. Extensive case studies based on real-world renewable generation scenarios demonstrate the superior performance of the proposed method in out-of-sample cost and reliability, with a manageable computing complexity.

Uncertainty-Informed Operation Coordination in a Water-Energy Nexus

The widespread deployment of smart heterogeneous technologies and the growing complexity in our modern society calls for effective coordination of the interdependent lifeline networks. In particular, operation coordination of electric power and water infrastructures is urgently needed as the water system is one of the most energy-intensive networks, an interruption in which may quickly evolve into a dramatic societal concern. The closely-intertwined ecosystem of water and power infrastructures is commonly known as water-energy nexus. This paper develops a novel analytic for uncertainty-aware day-ahead operation optimization of the interconnected power and water systems (PaWS). Joint probabilistic constraint (JPC) programming is employed to capture the uncertainties in wind resources and water demand forecasts. The proposed integrated stochastic model is presented as a non-linear non-convex optimization problem, where the non-linear hydraulic constraints in the water network are linearized using piece-wise linearization technique, and the non-convexity is efficiently tackled with a Boolean solution methodology to convert the proposed model with JPCs to a tractable mixed-integer linear programming (MILP) formulation that can be quickly solved to optimality. The suggested framework is applied to a 15-node commercial-scale water network jointly operated with a power transmission system using a modified IEEE 57-bus test system. The numerical results demonstrate the of the proposed stochastic framework, resulting in cost reduction (13% on average when compared to the traditional setting) and energy saving of the integrated model under different realizations of uncertain renewable energy sources (RESs) and water demand scenarios. Additionally, the scalability of the proposed model is tested on a modified IEEE 118-bus test system connected to five water networks.

Solving linear programs with joint probabilistic constraints with dependent rows using a dynamical neural network

In this paper, we study joint probabilistic constrained linear programs with dependent rows. We take the special case where the dependence between the rows is driven by Gumbel–Hougaard copula. We first transform the problem into a deterministic biconvex optimization problem. Then, we solve the obtained problem using a dynamical neural network based on the partial KKT system. We show the stability and the convergence of the proposed neural network. The main feature of our approach is to solve the joint probabilistic constraints problem without the use of any convex approximations. We finally use a problem of profit maximization to evaluate the performances of our approach.

A neural network approach to solve geometric programs with joint probabilistic constraints

We study a dynamical neural network approach to solve nonconvex geometric programs with joint probabilistic constraints (GPPC for short) with normally distributed coefficients and independent matrix row vectors. The main feature of our framework is to solve the biconvex deterministic equivalent problem of GPPC without the use of any convex approximation unlike the state-of-the-art solving methods. The second feature of our approach is to solve the dynamical system without the use of any iterative method but only by solving an ordinary differential equation system. We prove the stability together with the convergence of our neural network in the sense of Lyapunov. We also prove the equivalence between the optimal solution of the deterministic equivalent problem of GPPC and the solution of the dynamical system. We provide numerical experiments to show the performances of our approach compared to the state-of-art.

Planning energy economy and eco-environment nexus system under uncertainty: A copula-based stochastic multi-level programming method

Energy consumption, economic growth, and eco-environment protection of urban agglomeration is full of contradictions and continues to be challenges faced by decision makers. It is desired to develop effective methods to realize coordinated management of energy economy and eco-environment nexus (EEEN) system. In this study, a copula-based stochastic multi-level programming (CSMP) approach is first developed, where the conflicting objectives with multiple hierarchical levels and the uncertainty presented as random variables with different probability distributions can be tackled. CSMP can also reflect the complex interactions among random variables and examine the risk of violating joint-probabilistic constraints. The CSMP is then applied to planning EEEN system of Pearl River Delta (PRD) urban agglomeration (China) during the period of 2021–2035. Six scenarios based on joint and individual risk levels for violating energy consumption and GDP constraints are analyzed. Results reveal that the share of tertiary industry and high-tech industry would attain to 62.4% and 14.9% by 2035, the energy and CO2 intensity would decrease by 45.1% and 56.9%, and ecological land would expand to absorb more CO2, indicating that EEEN system would develop towards energy saving, economy and eco-environment friendly pattern. Results also disclose that decisions at high risk level would lead to high GDP, energy consumption and CO2 emissions, occupy ecological land and reduce CO2 absorption, and reduce the reliability of fulfilling system requirements.

Development of an integrated bi-level model for China’s multi-regional energy system planning under uncertainty

Climate change mitigation and renewable resources utilization are becoming particularly urgent for energy system management. In this study, a bi-level joint-probabilistic programming (BJPP) method is developed for planning multi-regional energy system under different mitigation policies and uncertainties. BJPP can handle leader–follower issues in decision-making process as well as examine the risk of violating joint-probabilistic constraints. Based on the BJPP method, a China’s multi-regional energy system (named as BJPP_CMES) model is formulated to provide optimal scheme for energy system planning of China over a long-term horizon (2021–2050) by synergistically minimizing carbon dioxide (CO2) emission and system cost. A series of scenarios associated with different carbon capture and storage (CCS) levels and violation risks of energy-demand constraints are examined. Results reveal that: (i) the share of non-fossil energy in China’s energy supply would keep increasing in 2021–2050, and the highest growth of the renewable supply would occur in Ningxia (rising 47.7%); (ii) Sichuan, Inner Mongolia, and Gansu would be the top three suppliers of renewable electricity; (iii) the CO2 emission of China would reach a peak of [44.3, 54.8] billion tonnes during the period of 2026–2030; Shandong, Inner Mongolia, and Shanxi would be main contributors of CO2 emission in the future; (iv) compared with the single-level model, the CO2 emission from the BJPP_CMES model would reduce by [2.7, 5.7]%; (v) among developed regions, the individual probability level of Jiangsu-Zhejiang-Shanghai is the most significant parameter for both CO2 emission and system cost. The findings are helpful for decision makers to optimize multi-regional energy system (MES) with a low-carbon and cost-effective manner, as well as to provide useful information for renewable energy utilization and regional sustainable development.

Uncertainty-Aware Deployment of Mobile Energy Storage Systems for Distribution Grid Resilience

With the spatial flexibility exchange across the network, mobile energy storage systems (MESSs) offer promising opportunities to elevate power distribution system resilience against emergencies. Despite the remarkable growth in integration of renewable energy sources (RESs) in power distribution systems (PDSs), most recovery and restoration strategies do not unlock the full potential in such resources due to their inherent uncertainty and stochasticity. This paper develops a novel restoration mechanism in PDSs for routing and scheduling of MESSs integrated with stochastic RESs to achieve agile system response and recovery in facing the aftermath of high-impact low-probability (HILP) incidents. The proposed integrated model is presented as a non-convex non-linear stochastic optimization formulation with joint probabilistic constraints (JPCs). The problem is equivalently reformulated to a tractable mixed-integer linear programming (MILP) model that can be solved by commercial off-the-shelf solvers. Case studies on the IEEE 33-node and 123-node test systems demonstrate the effectiveness and scalability of the proposed framework in boosting the system resilience. This is achieved via effective routing and scheduling of MESSs jointly managed with dynamic network reconfiguration in presence of stochastic RESs.

A copula-based stochastic fractional programming method for optimizing water-food-energy nexus system under uncertainty in the Aral Sea basin

Abstract A copula-based stochastic fractional programming (CSFP) method is developed for optimizing water-food-energy nexus (WFEN) system, which can tackle random variables under conflicting objectives and analyze the interrelationship between marginal benefit and system-failure risk. Then, a CSFP-WFEN model is formulated for the Aral Sea basin and multiple scenarios with individual- and joint-probabilistic risks related to water resources, arable land, and hydropower generation are examined. Synthesizing the results under all scenarios, hydropower generation would decrease by 11.9% and crop area would increase by 12.4% by 2035 under severe water scarcity. Agricultural water would decrease with time (from 69.1% in 2021 to 53.9% in 2035), and the share of wheat farmland would increase by 16.4% for food security. Compared with the traditional stochastic fractional programming (SFP) and copula-based stochastic programming (CSP) methods, the marginal benefit from CSFP would be improved 1.7% and water would be saved 9.2%. To support sustainable development of the study basin, some suggestions can be derived from the results: (i) since agriculture is still the largest water user in the future, it is desired to reduce its water consumption through improving irrigation efficiency and adjusting crop planting structure; (ii) effectively reducing Uzbekistan’s water consumption can improve water allocations to other countries and relieve regional conflicts caused by water shortage; (iii) compared to the single management pattern, the integrated optimization of water, food and energy nexus not only has higher efficiency of resources allocation and utilization, but also allows an increased robustness in controlling system-failure risk under joint probabilistic constraints.

Synergetic management of energy-water nexus system under uncertainty: An interval bi-level joint-probabilistic programming method

Synergic management of energy-water nexus (EWN) system is essential for coping with the dilemma of joint shortage of energy and water and supporting socio-economic sustainable development. The system is full of multiple uncertainties, making deterministic analysis methods infeasible. In this study, an interval bi-level joint-probabilistic programming (IBJP) approach is first developed through incorporating bi-level programming (BP) and interval joint-probabilistic programming (IJP) within a framework. IBJP has advantages in balancing the tradeoff between two-level decision makers under uncertainty, tackling uncertainties expressed as joint probabilities and interval values, and examining the risk of violating joint-probabilistic constraints. Then, the developed method is applied to planning China’s EWN system over a long-term planning horizon (2021–2050). Multiple scenarios related to different groups of constraint-violation levels for violating electricity demand and/or water availability constraints are examined. Results reveal that uncertainties associated with joint and individual probabilities have effects on the synergic management of EWN system. Results also disclose that limited water resource can promote electricity generation structure toward a low water-intensity, clean and sustainable pattern, in which the share of clean energy would increase to 66.25% by 2050 and the corresponding water withdrawal would save 41.20%.

Land use structure optimization based on uncertainty fractional joint probabilistic chance constraint programming

An uncertainty fractional joint probability chance constraint programming is developed to process land use structure optimization under uncertainty. The model integrate uncertainty programming into fractional programming, and the uncertainty programming include interval programming, fuzzy programming, stochastic programming and joint probability chance constraint programming. The results of the study are a series of land use policies in multiple scenarios with interval and deterministic numbers. The advantage of the model include it can (1) effectively integrate the two objectives of economic benefit maximization and pollution minimization by the fractional programming; (2) effectively process the uncertainty by the corresponding uncertainty programming; (3) reflect the impact of uncertainty on system benefit, pollutant discharge, and land use structure policy; and (4) develop a series of possible scenarios and corresponding feasible plans. The results of the study can help planners or decision makers develop flexible land use policy to address the multi-objective problems of maximum, minimum, and uncertainty. The proposed method is universal and can be extended to other cases.

Economic Reliability-Aware MPC-LPV for Operational Management of Flow-Based Water Networks Including Chance-Constraints Programming

This paper presents an economic reliability-aware model predictive control (MPC) for the management of drinking water transport networks (DWNs). The proposed controller includes a new goal to increase the system and components reliability based on a finite horizon stochastic optimization problem with joint probabilistic (chance) constraints. The proposed approach is based on a single-layer economic optimization problem with dynamic constraints. The inclusion of components and system reliability in the MPC model using an Linear Parameter Varying (LPV) modeling approach aims to maximize the availability of the system by estimating system reliability. On the other hand, the use of a LPV-MPC control approach allows the controller to consider nonlinearities in the model in a linear like way. Moreover, the resulting MPC optimization problem can be formulated as a Quadratic Programming (QP) problem at each sampling time reducing the computational burden/time compared to solving a nonlinear programming problem. The use of chance-constraint programming allows the computation of an optimal strategy with a pre-established risk acceptability levels to cope with the uncertainty of the demand forecast. Finally, the proposed approach is applied to a part of the water transport network of Barcelona for demonstrating its performance. The obtained results show that the system reliability of the DWN is maximized compared with the other approaches.

Stochastic Predictive Control of Multi-Microgrid Systems

In this paper, integrated operation management of cooperative microgrids is formulated in the framework of stochastic predictive control. In the proposed scheme, a joint probabilistic constraint on the microgrids power exchange with the main grid couples operation of individual microgrids. In order to tackle the coupling constraint, a cooperative energy management strategy is proposed in which based on the statistical characteristics of uncertain parameters, the deterministic counterpart of the problem is derived and an efficient solution strategy is achieved. The proposed strategy is evaluated for an illustrative test case including two microgrids based on modified CIGRE benchmark. Moreover, statistical analysis is conducted to evaluate robustness characteristics of the solution strategy.

On probabilistic capacity maximization in a stationary gas network

ABSTRACTThe question for the capacity of a given gas network appears as an essential question that network operators and political administrations are regularly faced with. In that context, we present a novel mathematical approach in order to assist gas network operators in managing increasing uncertainty with respect to customers gas nominations and in exposing free network capacities while reliability of transmission and supply is taken into account. The approach is based on the rigorous examination of optimization problems with nonlinear probabilistic constraints. As consequence we deal with solving a problem belonging to the class of probabilistic/robust optimization problems, which can be formulated with some joint probabilistic constraint over an infinite system of random inequalities. We will show that the inequality system can be reduced to a finite one in the situation of considering a tree network topology. A detailed study of the problem of maximizing bookable capacities in a stationary gas network is presented. The focus will be on both the theoretical and numerical side. The analytical part consists in introducing and validating a generalized version of the known rank two constraint qualification. The results are important in order to solve the capacity problem numerically.

Stochastic Distributed Predictive Tracking Control for Networks of Autonomous Systems With Coupling Constraints

In this paper, we propose a distributed model predictive control method with chance constraints for tracking reference signals. More specifically, considering a network of systems with independent dynamics, the proposed control scheme allows each subsystem to reach feasible setpoints, while respecting local and joint probabilistic constraints. Both a simulation example consisting of an academic temperature control problem and experimental results concerning the motion and coordination of moving agents are illustrated.

Relaxations for probabilistically constrained stochastic programming problems: review and extensions

We consider probabilistically constrained stochastic programming problems, in which the random variables are in the right-hand sides of the stochastic inequalities defining the joint chance constraints. Problems of that kind arise in a variety of contexts, and are particularly difficult to solve for random variables with continuous joint distributions, because the calculation of the cumulative distribution function and its gradient values involves numerical integration and/or simulation in large dimensional spaces. We revisit known and provide new relaxations extensions to various probability bounding schemes that permit to approximate the feasible set of joint probabilistic constraints. The derived mathematical formulations relax the requirement to handle large multivariate cumulative distribution functions and involve instead the computation of marginal and bivariate cumulative distribution functions. We analyze the convexity of and computational challenges posed by the inferred relaxations

Spatio-temporal variability of Arctic summer temperatures over the past 2 millennia

Abstract. In this article, the first spatially resolved and millennium-length summer (June–August) temperature reconstruction over the Arctic and sub-Arctic domain (north of 60° N) is presented. It is based on a set of 44 annually dated temperature-sensitive proxy archives of various types from the revised PAGES2k database supplemented with six new recently updated proxy records. As a major advance, an extension of the Bayesian BARCAST climate field (CF) reconstruction technique provides a means to treat climate archives with dating uncertainties. This results not only in a more precise reconstruction but additionally enables joint probabilistic constraints to be imposed on the chronologies of the used archives. The new seasonal CF reconstruction for the Arctic region can be shown to be skilful for the majority of the terrestrial nodes. The decrease in the proxy data density back in time, however, limits the analyses in the spatial domain to the period after 750 CE, while the spatially averaged reconstruction covers the entire time interval of 1–2002 CE.The centennial to millennial evolution of the reconstructed temperature is in good agreement with a general pattern that was inferred in recent studies for the Arctic and its subregions. In particular, the reconstruction shows a pronounced Medieval Climate Anomaly (MCA; here ca. 920–1060 CE), which was characterised by a sequence of extremely warm decades over the whole domain. The medieval warming was followed by a gradual cooling into the Little Ice Age (LIA), with 1766–1865 CE as the longest centennial-scale cold period, culminating around 1811–1820 CE for most of the target region.In total over 600 independent realisations of the temperature CF were generated. As showcased for local and regional trends and temperature anomalies, operating in a probabilistic framework directly results in comprehensive uncertainty estimates, even for complex analyses. For the presented multi-scale trend analysis, for example, the spread in different paths across the reconstruction ensemble prevents a robust analysis of features at timescales shorter than ca. 30 years. For the spatial reconstruction, the benefit of using the spatially resolved reconstruction ensemble is demonstrated by focusing on the regional expression of the recent warming and the MCA. While our analysis shows that the peak MCA summer temperatures were as high as in the late 20th and early 21st centuries, the spatial coherence of extreme years over the last decades of the reconstruction (1980s onwards) seems unprecedented at least back until 750 CE. However, statistical testing could not provide conclusive support of the contemporary warming to exceed the peak of the MCA in terms of the pan-Arctic mean summer temperatures: the reconstruction cannot be extended reliably past 2002 CE due to lack of proxy data and thus the most recent warming is not captured.

A chance-constrained two-stage stochastic programming model for humanitarian relief network design

We consider a stochastic pre-disaster relief network design problem, which mainly determines the capacities and locations of the response facilities and their inventory levels of the relief supplies in the presence of uncertainty in post-disaster demands and transportation network conditions. In contrast to the traditional humanitarian logistics literature, we develop a chance-constrained two-stage mean-risk stochastic programming model. This risk-averse model features a mean-risk objective, where the conditional value-at-risk (CVaR) is specified as the risk measure, and enforces a joint probabilistic constraint on the feasibility of the second-stage problem concerned with distributing the relief supplies to the affected areas in case of a disaster. To solve this computationally challenging stochastic optimization model, we employ an exact Benders decomposition-based branch-and-cut algorithm. We develop three variants of the proposed algorithm by using alternative representations of CVaR. We illustrate the application of our model and solution methods on a case study concerning the threat of hurricanes in the Southeastern part of the United States. An extensive computational study provides practical insights about the proposed modeling approach and demonstrates the computational effectiveness of the solution framework.

A joint model of probabilistic/robust constraints for gas transport management in stationary networks

We present a novel mathematical algorithm to assist gas network operators in managing uncertainty, while increasing reliability of transmission and supply. As a result, we solve an optimization problem with a joint probabilistic constraint over an infinite system of random inequalities. Such models arise in the presence of uncertain parameters having partially stochastic and partially non-stochastic character. The application that drives this new approach is a stationary network with uncertain demand (which are stochastic due to the possibility of fitting statistical distributions based on historical measurements) and with uncertain roughness coefficients in the pipes (which are uncertain but non-stochastic due to a lack of attainable measurements). We study the sensitivity of local uncertainties in the roughness coefficients and their impact on a highly reliable network operation. In particular, we are going to answer the question, what is the maximum uncertainty that is allowed (shaping a ’maximal’ uncertainty set) around nominal roughness coefficients, such that random demands in a stationary gas network can be satisfied at given high probability level for no matter which realization of true roughness coefficients within the uncertainty set. One ends up with a constraint, which is probabilistic with respect to the load of gas and robust with respect to the roughness coefficients. We demonstrate how such constraints can be dealt with in the framework of the so-called spheric-radial decomposition of multivariate Gaussian distributions. The numerical solution of a corresponding optimization problem is illustrated. The results might assist the network operator with the implementation of cost-intensive roughness measurements.

Stochastic geometric programming with joint probabilistic constraints

This paper discusses the geometric programs with joint probabilistic constraints. When the coefficients are normally distributed and independent of each other, we approximate the problem by using piecewise linear function and transform the approximation problem into a geometric program. We prove that this approximation method provides a lower bound, and we use Bonferroni approximation to find an upper bound.

Stochastic geometric optimization with joint probabilistic constraints

This paper discusses geometric programs with joint probabilistic constraints. When the stochastic parameters are normally distributed and independent of each other, we approximate the problem by using piecewise linear functions, and transform the approximation problem into a convex geometric program. We prove that this approximation method provides a lower bound. Then, we design a sequential convex optimization algorithm to find an upper bound. Finally, numerical tests are carried out on a stochastic shape optimization problem.