Stochastic Linear Problem Research Articles

Policy with guaranteed risk-adjusted performance for multistage stochastic linear problems

Policy with guaranteed risk-adjusted performance for multistage stochastic linear problems

Exact Quantization of Multistage Stochastic Linear Problems

Exact Quantization of Multistage Stochastic Linear Problems

Risk-constrained probabilistic coordination in coupled transmission and distribution system

High penetration of renewable energy has instigated stochastic power injection at interconnection between transmission system (TS) and distribution system (DS). This paper delves into the intricate collaborative risk based stochastic dispatching challenges due to contingencies and the integration of renewable sources in the TS and DS. A framework of risk constrained probabilistic problem (RCPC) is presented, which is divided into Probabilistic coordination (PC) and risk-based stochastic security-constrained unit commitment (RSSCUC). The PC is a stochastic linear problem and RSSCUC is a stochastic mixed integer non-linear problem. Therefore, the probabilistic analytical target cascading along with polynomial chaos expansion (PCE) has been utilized for solving the PC problem for coupled TS and DS. Here, PCE has provided the proposed algorithm with the needed capability to present stochastic shared variables as coefficients for PC. Further, the benders decomposition algorithm has been utilized for solving RSSCUC, by dividing it into master problem and sub problem to generate feasibility cuts. The proposed technique reduces the computational burden on the system, as it requires single stochastic coordination problem for all scenarios as an alternative of each coordination problem for each scenario and polynomial coefficients instead of multiple scenarios for each shared variable between TS and DS. Different case studies have been performed utilizing the 6-bus system and IEEE 118-bus system as TSs, 7-bus, 9-bus, 85-bus and 69-bus systems as DSs. Results depict the efficacy of the proposed method.

A contract-based trading of power flexibility between a variable renewable energy producer and an electricity retailer

Variable renewable energy producers and electricity retailers encounter several uncertainties in their decision-making problems, such as intermittency of renewable energy sources, variability of consumption, and market price volatility. To cope with these uncertainties, this paper presents a new contract-based trading mechanism of power flexibility (FlexCon) between two parties, a variable renewable energy producer and an electricity retailer. The proposed mechanism is managed by a new entity, named FlexCon operator, to oversee the energy and financial trades through the contract and coordinate the transactions with the system operator. Through the FlexCon, the parties are able to exchange their energy imbalances as a source of power flexibility to alleviate the negative impacts of uncertainties in their decision-making problems. To this end, two two-stage stochastic linear problems are introduced from each party’s point of view. In the first stage, the variable renewable energy producer and the electricity retailer submit their bids to sell and purchase in the day-ahead market, respectively. Following the day-ahead market clearing, closer to the delivery time, the parties submit their decisions on the contract to the introduced FlexCon operator. The operator allocates possible power flexibility transactions based on the surpluses or shortages of the parties. Assuming that the imbalances are not completely resolved with the FlexCon, the remaining deviations are settled in the balancing market. The parties’ decisions related to the balancing market and the FlexCon are modeled in the second stage of the stochastic problem. The uncertainties associated with prices, renewable generation, electricity consumption, and the maximum exchangeable power flexibility through the FlexCon are considered via scenarios. Meanwhile, the profit risk is considered by the Conditional Value at Risk measure. The numerical results show that FlexCon effectively diminishes the impacts of uncertainties on the parties’ profit.

Solving multistage stochastic linear programming via regularized linear decision rules: An application to hydrothermal dispatch planning

The solution of multistage stochastic linear problems (MSLPs) represents a challenge for many applications. Long-term hydrothermal dispatch planning (LHDP) materializes this challenge in a real-world problem that affects electricity markets, economies, and natural resources worldwide. No closed-form solutions are available for MSLP and the definition of non-anticipative policies with high-quality out-of-sample performance is crucial. Linear decision rules (LDRs) provide an interesting simulation-based framework for finding high-quality policies for MSLP through two-stage stochastic models. In practical applications, however, the number of parameters to be estimated when using an LDR may be close to or higher than the number of scenarios of the sample average approximation problem, thereby generating an in-sample overfit and poor performances in out-of-sample simulations. In this paper, we propose a novel regularized LDR to solve MSLP based on the AdaLASSO (adaptive least absolute shrinkage and selection operator). The goal is to use the parsimony principle, as largely studied in high-dimensional linear regression models, to obtain better out-of-sample performance for a LDR applied to MSLP. Computational experiments show that overfitting is a non-negligible threat when using the classical non-regularized LDR to solve the LHDP, one of the most studied MSLP with relevant applications in industry. Our analysis highlights the following benefits of the proposed framework in comparison to the non-regularized benchmark: 1) significant reductions in the number of non-zero coefficients (model parsimony), 2) substantial cost reductions in out-of-sample evaluations, and 3) improved spot-price profiles.

Uncertainty representation in investment planning of low-carbon power systems

Power system operators and planners are dealing both with the integration of unparalleled levels of variable renewable energy sources and deep uncertainties that originate from new technological developments, changing regulatory frameworks, unknown investment, operational costs of technologies, etc. An inadequate representation of the uncertainties may result in a substantial risk of deploying inflexible investment solutions incapable of adapting efficiently to evolving scenarios. In this context, this work studies the effects of increasing the granularity used to represent the long-term uncertainty by analysing its impact on the resulting optimal portfolios of new transmission lines, battery energy storage systems and pumped-hydro storage systems. The studies are conducted on an instance of the Australian power system described by the system operator for planning purposes, including four types of uncertainty granularity, namely deterministic representation, and 2-stage, 3-stage and 4-stage stochastic representations. To address the computational challenges associated with the large mixed-integer linear stochastic problems, the different instances are reformulated using Dantzig-Wolfe decomposition, enabling the use of a column generation approach to solve the investment problem. The case study applications show substantial adjustments in the investment portfolios as uncertainty granularity changes, with a clear tendency to increase battery storage investment as uncertainty is better represented.

Generalized adaptive partition-based method for two-stage stochastic linear programs: Geometric oracle and analysis

Adaptive Partition-based Methods (APM) are numerical methods that solve, in particular, two-stage stochastic linear problems (2SLP). We say that a partition of the uncertainty space is adapted to the current first stage control xˇ if we can aggregate scenarios while conserving the true value of the expected recourse cost at xˇ. The core idea of APM is to iteratively construct an adapted partition to all past tentative first stage controls. Relying on the normal fan of the dual admissible set, we give a necessary and sufficient condition for a partition to be adapted even for non-finite distribution, and provide a geometric method to obtain an adapted partition. Further, by showing the connection between APM and the L-shaped algorithm, we prove convergence and complexity bounds of the APM methods. The paper presents the fixed recourse case and ends with elements to forgo this assumption.

Some matheuristic algorithms for multistage stochastic optimization models with endogenous uncertainty and risk management

Two matheuristic decomposition algorithms are introduced. The first one is a Progressive Hedging type so-named Regularized scenario Cluster Progressive Algorithm. The second one is a Frank-Wolfe PH type so-named Regularized scenario Cluster Simplicial Decomposition Progressive Algorithm. An extension of endogenous Type III uncertainty is considered for representing the decision-dependent scenario probability and outlook. Its performance is tested in the time-consistent Expected Conditional Stochastic Dominance risk averse environment. As a result of the modeling, the typical risk neutral multistage mixed 0–1 linear stochastic problem under uncertainty is replaced with an enlarged model that is equivalent to the required mixed 0–1 bilinear model. Based on the special features of the problem, it is unrealistic to seek the optimal solution for large-scale instances. Feasible solutions and lower bounds on the solution value of the original model are provided. In total, 48 strategies are considered, each one consists of a combination of a regularization norm, a calibration type for the PH pseudo-gradient computation, and a set of value intervals of the influential variables on a representative endogenous uncertainty-based piecewise function in the scenarios. Computational results are reported for a large-scale extension of a well-known real-life pilot case for preparedness resource allocation planning aiming to natural disaster relief. The matheuristics outperform the plain use of a state-of-the-art solver.

A scalable parallel implementation of the Cluster Benders Decomposition algorithm

Benders Decomposition (BD) is a method used to solve stochastic linear problems via scenario analysis. Cluster BD (CBD) is one of its smart improvements that speed up the execution time, taking advantage of tighter feasible cuts found by grouping scenarios into clusters. In this paper, we propose a new design for CBD, one which takes into account the role played by optimal cuts in the solution. Besides, we propose a new parallel scheme for CBD to deal with large-scale two-stage stochastic linear problems. Moreover, we characterise the problems for which our proposal performs best. The results obtained show computational gains from our proposal compared with the plain use of CPLEX, serial BD, parallel BD, serial CBD and parallel CBD.

Price Responsive Demand Management of an Industrial Buyer in Day-ahead Electricity Market

AbstractAn industrial buyer, who can reschedule its production plan, may develop strategic bid by shifting its demand from high price period to low price period. Based on this concept, a new optimization formulation has been proposed to develop bidding strategy of such buyers considering Price Responsive Demand Shifting (PRDS). Since the bidding strategy of the buyer depends upon the market clearing price, which is volatile/uncertain due to various factors like gaming behavior of participants, demand forecasting error, transmission congestion etc, the proposed optimization problem is formulated as a stochastic linear problem, comprising of two sub-problems. The first sub-problem represents the market clearing process by the System Operator, which is formulated to maximize the social welfare of the market participants, while the second sub-problem aims at maximizing the purchase cost saving of the industrial buyer. The optimal bidding strategy has been obtained by solving these two sub-problems considering hourly market clearing for 24-hour scheduling period. The effectiveness of the proposed method has been tested on a 5-bus system and modified IEEE 30-bus system. Results obtained with the demand shifting based bidding strategy have been compared with those obtained with a Conventional Price Quantity (CPQ) bid strategy. It has been observed that the proposed approach leads to enhancement in the purchase cost saving as compared to the CPQ and meets the energy consumption targets of the industrial buyer.

Stochastic elastic equation driven by fractional Brownian motion

In this paper, we consider the stochastic elastic equation driven by a cylindrical fractional Brownian motion. The regularities of the solution to the linear stochastic problem corresponding to the stochastic elastic equation are proved. Then, we obtain the existence of the solution using the Picard iteration.

Two-stage Stochastic Linear Programming by a Series of Monte-Carlo Estimators

In this paper a stochastic adaptive method has been developed to solve stochastic linear problems by a finite sequence of Monte-Carlo sampling estimators. The method is based on the adaptive regulation of the size of Monte-Carlo samples and a statistical termination procedure taking into consideration statistical modelling accuracy. Our approach distinguishes itself by the treatment of accuracy of the solution in a statistical manner, testing the hypothesis of optimality according to statistical criteria, and estimating confidence intervals of the objective and constraint functions. To avoid “jamming” or “zigzagging” solving a constraint problem we implement the e–feasible direction approach. The proposed adjustment of a sample size, when it is taken inversely proportional to the square of the norm of the Monte-Carlo estimate of the gradient, guarantees convergence a. s. at a linear rate. The numerical study and examples in practice corroborate theoretical conclusions and show that the developed procedures make it possible to solve stochastic problems with sufficient accuracy by the means of an acceptable size of computations. DOI: 10.15181/csat.v2i2.891

Local Polynomial Chaos Expansion for Linear Differential Equations with High Dimensional Random Inputs

In this paper we present a localized polynomial chaos expansion for partial differential equations (PDE) with random inputs. In particular, we focus on time independent linear stochastic problems with high dimensional random inputs, where the traditional polynomial chaos methods, and most of the existing methods, incur prohibitively high simulation cost. The local polynomial chaos method employs a domain decomposition technique to approximate the stochastic solution locally. In each subdomain, a subdomain problem is solved independently and, more importantly, in a much lower dimensional random space. In a postprocesing stage, accurate samples of the original stochastic problems are obtained from the samples of the local solutions by enforcing the correct stochastic structure of the random inputs and the coupling conditions at the interfaces of the subdomains. Overall, the method is able to solve stochastic PDEs in very large dimensions by solving a collection of low dimensional local problems and can be highly efficient. In this paper we present the general mathematical framework of the methodology and use numerical examples to demonstrate the properties of the method.

Smoothing Techniques and Augmented Lagrangian Method for Recourse Problem of Two-Stage Stochastic Linear Programming

The augmented Lagrangian method can be used for solving recourse problems and obtaining their normal solution in solving two-stage stochastic linear programming problems. The augmented Lagrangian objective function of a stochastic linear problem is not twice differentiable which precludes the use of a Newton method. In this paper, we apply the smoothing techniques and a fast Newton-Armijo algorithm for solving an unconstrained smooth reformulation of this problem. Computational results and comparisons are given to show the effectiveness and speed of the algorithm.

A so-called Cluster Benders Decomposition approach for solving two-stage stochastic linear problems

The optimization of stochastic linear problems, via scenario analysis, based on Benders decomposition requires appending feasibility and/or optimality cuts to the master problem until the iterative procedure reaches the optimal solution. The cuts are identified by solving the auxiliary submodels attached to the scenarios. In this work, we propose the algorithm named scenario Cluster Benders Decomposition (CBD) for dealing with the feasibility cut identification in the Benders method for solving large-scale two-stage stochastic linear problems. The scenario tree is decomposed into a set of scenario clusters and tighter feasibility cuts are obtained by solving the auxiliary submodel for each cluster instead of each individual scenario. Then, the scenario cluster based scheme allows to identify tighter feasibility cuts that yield feasible second stage decisions in reasonable computing time. Some computational experience is reported by using CPLEX as the solver of choice for the auxiliary LP submodels at each iteration of the algorithm CBD. The results that are reported show the favorable performance of the new approach over the traditional single scenario based Benders decomposition; it also outperforms the plain use of CPLEX for medium-large and large size instances.

Stochastic Burgers' equation driven by fractional Brownian motion

In this paper, we consider the stochastic Burgers' equation driven by a genuine cylindrical fractional Brownian motion with Hurst parameter H > 1 4 . We first prove the regularities of the solution to the linear stochastic problem corresponding to the stochastic Burgers' equation. Then we obtain the local and global existence and uniqueness results for the stochastic Burgers' equation.

Epsilon-projection method for two-stage SLP

In this paper, a stochastic adaptive method with the ε-projection for stochastic gradient has been developed to solve two-stage stochastic linear problems. The method is based on the adaptive regulation of the size of Monte-Carlo samples and a statistical termination procedure taking into consideration the statistical modelling accuracy. To avoid “jamming” or “zigzagging” in solving the constraint problem the procedure for stochasticgradient epsilon-projectionhas been implemented. Four algorithms of the epsilonprojection are tested by the computer simulation. The recommendationsto solve two-stage stochasticlinear problems are made under the numerical study and results of solving the test examples.

Statistics of Entropy Production in Linearized Stochastic Systems

We consider a wide class of linear stochastic problems driven off the equilibrium by a multiplicative asymmetric force. The force breaks detailed balance, maintained otherwise, thus producing entropy. The large deviation function of the entropy production in the system is calculated explicitly. The general result is illustrated using an example of a polymer immersed in a gradient flow and subject to thermal fluctuations.

Application of statistical criteria to optimality testing in stochastic programming

Abstract In this paper the stochastic adaptive method has been developed to solve stochastic linear problems by a finite sequence of Monte‐Carlo sampling estimators. The method is grounded on adaptive regulation of the size of Monte‐Carlo samples and the statistical termination procedure, taking into consideration the statistical modeling error. Our approach distinguishes itself by treatment of the accuracy of the solution in a statistical manner, testing the hypothesis of optimality according to statistical criteria, and estimating confidence intervals of the objective and constraint functions. The adjustment of sample size, when it is taken inversely proportional to the square of the norm of the Monte‐Carlo estimate of the gradient, guarantees the convergence a. s. at a linear rate. We examine four estimators for stochastic gradient: by the differentiation of the integral with respect to x, the finite difference approach, the Simulated Perturbation Stochastic Approximation approach, the Likelihood Ratio...

APPLICATION OF STATISTICAL CRITERIA TO OPTIMALITY TESTING IN STOCHASTIC PROGRAMMING

In this paper the stochastic adaptive method has been developed to solve stochastic linear problems by a finite sequence of Monte‐Carlo sampling estimators. The method is grounded on adaptive regulation of the size of Monte‐Carlo samples and the statistical termination procedure, taking into consideration the statistical modeling error. Our approach distinguishes itself by treatment of the accuracy of the solution in a statistical manner, testing the hypothesis of optimality according to statistical criteria, and estimating confidence intervals of the objective and constraint functions. The adjustment of sample size, when it is taken inversely proportional to the square of the norm of the Monte‐Carlo estimate of the gradient, guarantees the convergence a. s. at a linear rate. We examine four estimators for stochastic gradient: by the differentiation of the integral with respect to x, the finite difference approach, the Simulated Perturbation Stochastic Approximation approach, the Likelihood Ratio approach. The numerical study and examples in practice corroborate the theoretical conclusions and show that the procedures developed make it possible to solve stochastic problems with a sufficient agreeable accuracy by means of the acceptable amount of computations.