Multistage Stochastic Linear Programming Research Articles

A multi-stage stochastic programming model for the unit commitment of conventional and virtual power plants bidding in the day-ahead and ancillary services markets

As more uncontrollable renewable energy sources are present in the power generation portfolio, the need of more detailed and reliable tools for the optimal operation of energy systems has increased in the last years. This work presents a multi-stage stochastic Mixed Integer Linear Program with binary recourse for optimizing the day-ahead unit commitment of power plants and virtual power plants operating in the day-ahead and ancillary services markets. Scenarios reproduce the uncertainty of the ancillary services market requests, and production of photovoltaic panels. A novel decomposition algorithm is proposed to tackle the challenging multistage stochastic program. The methodology is tested on three types of large power plants: a natural gas-fired combined cycle, a combined heat and power combined cycle with thermal storage, and a virtual power plant integrating a combined cycle with battery and photovoltaic fields. Compared to the typical deterministic unit commitment approach, the proposed stochastic optimization approach allows to increase the revenues of the conventional power plant up to 13.58% and, for the combined heat and power and virtual power plant case, it allows finding a feasible and efficient operational scheduling.

Compromise policy for multi-stage stochastic linear programming: Variance and bias reduction

This paper focuses on algorithms for multi-stage stochastic linear programming (MSLP). We propose an ensemble method named the “compromise policy”, which not only reduces the variance of the function approximation but also reduces the bias of the estimated optimal value. It provides a tight lower bound estimate with a confidence interval. By exploiting parallel computing, the compromise policy provides demonstrable advantages in performance and stability with marginally extra computational time. We further propose a meta-algorithm to solve the MSLP problems based on in-sample and out-of-sample optimality tests. Our meta-algorithm is incorporated within an SDDP-type algorithm for MSLP and significantly improves the reliability of the decisions suggested by SDDP. These advantages are demonstrated via extensive computations, which illustrate the effectiveness of our approach.

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.

Duality and sensitivity analysis of multistage linear stochastic programs

In this paper we investigate the dual of a Multistage Stochastic Linear Program (MSLP). By writing Dynamic Programming equations for the dual, we can employ an SDDP type method, called Dual SDDP, which solves these Dynamic Programming equations. allows us to compute a sequence of nonincreasing deterministic upper bounds for the optimal value of the problem. Since the Relatively Complete Recourse (RCR) condition may fail to hold for the dual (even for simple problems), we design two variants of Dual SDDP, namely Dual SDDP with penalizations and Dual SDDP with feasibility cuts, that converge to the optimal value of the dual (and therefore primal when there is no duality gap) problem under mild assumptions. We also show that optimal dual solutions can be obtained using dual information from Primal SDDP (applied to the original primal MSLP) subproblems.As a byproduct of the developed methodology we study sensitivity of the optimal value of the problem as a function of the involved parameters. For the sensitivity analysis we provide formulas for the derivatives of the value function with respect to the parameters and illustrate their application on an inventory problem. Since these formulas involve optimal dual solutions, we need an algorithm that computes such solutions to use them, i.e., we need to solve the dual problem.Finally, as a proof of concept of the tools developed, we present the results of numerical experiments computing the sensitivity of the optimal value of an inventory problem as a function of parameters of the demand process and compare Primal and Dual SDDP on the inventory and a hydro-thermal planning problems.

Stochastic optimization of trading strategies in sequential electricity markets

Quantity and price risks determine key uncertainties market participants face in electricity markets with increased volatility, for instance due to high shares of renewables. In the time from day-ahead until real-time, there lies a large variation in best available information, such as between forecasts and realizations of uncertain parameters like renewable feed-in and electricity prices. This uncertainty reflects on both the market outcomes and the quantity of renewable generation, making the determination of sound trading strategies across different market segments a complex task. The scope of the paper is to optimize day-ahead and intraday trading decisions jointly for a portfolio with controllable and intermittent renewable generation under consideration of risk. We include a reserve market, a day-ahead market and an intraday market in stochastic modeling and develop a multi-stage stochastic Mixed Integer Linear Program. We assess the profitability as well as the risk exposure, quantified by the conditional value at risk metric, of trading strategies following different risk preferences. We conclude that a risk-neutral trader mainly relies on the opportunity of higher expected profits in intraday trading, whereas risk can be hedged effectively by trading on the day-ahead. Finally, we show that reserve market participation implies various rationales, including the relation of expected reserve prices among each other, the relation of expected reserve prices to spot market prices, as well as the relation of the spot market prices among each other.

Combining Polyhedral Approaches and Stochastic Dual Dynamic Integer Programming for Solving the Uncapacitated Lot-Sizing Problem Under Uncertainty

We study the uncapacitated lot-sizing problem with uncertain demand and costs. The problem is modeled as a multistage stochastic mixed-integer linear program in which the evolution of the uncertain parameters is represented by a scenario tree. To solve this problem, we propose a new extension of the stochastic dual dynamic integer programming algorithm (SDDiP). This extension aims at being more computationally efficient in the management of the expected cost-to-go functions involved in the model, in particular by reducing their number and by exploiting the current knowledge on the polyhedral structure of the stochastic uncapacitated lot-sizing problem. The algorithm is based on a partial decomposition of the problem into a set of stochastic subproblems, each one involving a subset of nodes forming a subtree of the initial scenario tree. We then introduce a cutting plane–generation procedure that iteratively strengthens the linear relaxation of these subproblems and enables the generation of an additional strengthened Benders’ cut, which improves the convergence of the method. We carry out extensive computational experiments on randomly generated large-size instances. Our numerical results show that the proposed algorithm significantly outperforms the SDDiP algorithm at providing good-quality solutions within the computation time limit. Summary of Contribution: This paper investigates a combinatorial optimization problem called the uncapacitated lot-sizing problem. This problem has been widely studied in the operations research literature as it appears as a core subproblem in many industrial production planning problems. We consider a stochastic extension in which the input parameters are subject to uncertainty and model the resulting stochastic optimization problem as a multistage stochastic integer program. To solve this stochastic problem, we propose a novel extension of the recently published stochastic dual dynamic integer programming (SDDiP) algorithm. The proposed extension relies on two main ideas: the use of a partial decomposition of the scenario tree and the exploitation of existing knowledge on the polyhedral structure of the stochastic uncapacitated lot-sizing problem. We provide the results of extensive computational experiments carried out on large-size randomly generated instances. These results show that the proposed extended algorithm significantly outperforms the SDDiP at providing good-quality solutions for the stochastic uncapacitated lot-sizing problem. Although the paper focuses on a basic lot-sizing problem, the proposed algorithmic framework may be useful to solve more complex practical production planning problems.

Adaptive partition-based SDDP algorithms for multistage stochastic linear programming with fixed recourse

Adaptive partition-based SDDP algorithms for multistage stochastic linear programming with fixed recourse

Production planning of a furniture manufacturing company with random demand and production capacity using stochastic programming.

In this article two multi-stage stochastic linear programming models are developed, one applying the stochastic programming solver integrated by Lingo 17.0 optimization software that utilizes an approximation using an identical conditional sampling and Latin-hyper-square techniques to reduce the sample variance, associating the probability distributions to normal distributions with defined mean and standard deviation; and a second proposed model with a discrete distribution with 3 values and their respective probabilities of occurrence. In both cases, a scenario tree is generated. The models developed are applied to an aggregate production plan (APP) for a furniture manufacturing company located in the state of Hidalgo, Mexico, which has important clients throughout the country. Production capacity and demand are defined as random variables of the model. The main purpose of this research is to determine a feasible solution to the aggregate production plan in a reasonable computational time. The developed models were compared and analyzed. Moreover, this work was complemented with a sensitivity analysis; varying the percentage of service level, also, varying the stochastic parameters (mean and standard deviation) to test how these variations impact in the solution and decision variables.

A time-consistent Benders decomposition method for multistage distributionally robust stochastic optimization with a scenario tree structure

A computational method is developed for solving time consistent distributionally robust multistage stochastic linear programs with discrete distribution. The stochastic structure of the uncertain parameters is described by a scenario tree. At each node of this tree, an ambiguity set is defined by conditional moment constraints to guarantee time consistency. This method employs the idea of nested Benders decomposition that incorporates forward and backward steps. The backward steps solve some conic programming problems to approximate the cost-to-go function at each node, while the forward steps are used to generate additional trial points. A new framework of convergence analysis is developed to establish the global convergence of the approximation procedure, which does not depend on the assumption of polyhedral structure of the original problem. Numerical results of a practical inventory model are reported to demonstrate the effectiveness of the proposed method.

Single cut and multicut stochastic dual dynamic programming with cut selection for multistage stochastic linear programs: convergence proof and numerical experiments

We introduce a variant of Multicut Decomposition Algorithms, called CuSMuDA (Cut Selection for Multicut Decomposition Algorithms), for solving multistage stochastic linear programs that incorporates a class of cut selection strategies to choose the most relevant cuts of the approximate recourse functions. This class contains Level 1 (Philpott et al. in J Comput Appl Math 290:196–208, 2015) and Limited Memory Level 1 (Guigues in Eur J Oper Res 258:47–57, 2017) cut selection strategies, initially introduced for respectively Stochastic Dual Dynamic Programming and Dual Dynamic Programming. We prove the almost sure convergence of the method in a finite number of iterations and obtain as a by-product the almost sure convergence in a finite number of iterations of Stochastic Dual Dynamic Programming combined with our class of cut selection strategies. We compare the performance of Multicut Decomposition Algorithms, Stochastic Dual Dynamic Programming, and their variants with cut selection (using Level 1 and Limited Memory Level 1) on several instances of a portfolio problem. On these experiments, in general, Stochastic Dual Dynamic Programming is quicker (i.e., satisfies the stopping criterion quicker) than Multicut Decomposition Algorithms and cut selection allows us to decrease the computational bulk with Limited Memory Level 1 being more efficient (sometimes much more) than Level 1.

Algorithmic Approaches to Inventory Management Optimization

An inventory management problem is addressed for a make-to-order supply chain that has inventory holding and/or manufacturing locations at each node. The lead times between nodes and production capacity limits are heterogeneous across the network. This study focuses on a single product, a multi-period centralized system in which a retailer is subject to an uncertain stationary consumer demand at each time period. Two sales scenarios are considered for any unfulfilled demand: backlogging or lost sales. The daily inventory replenishment requests from immediate suppliers throughout the network are modeled and optimized using three different approaches: (1) deterministic linear programming, (2) multi-stage stochastic linear programming, and (3) reinforcement learning. The performance of the three methods is compared and contrasted in terms of profit (reward), service level, and inventory profiles throughout the supply chain. The proposed optimization strategies are tested in a stochastic simulation environment that was built upon the open-source OR-Gym Python package. The results indicate that, of the three approaches, stochastic modeling yields the largest increase in profit, whereas reinforcement learning creates more balanced inventory policies that would potentially respond well to network disruptions. Furthermore, deterministic models perform well in determining dynamic reorder policies that are comparable to reinforcement learning in terms of their profitability.

Stochastic Dynamic Linear Programming: A Sequential Sampling Algorithm for Multistage Stochastic Linear Programming

Multistage stochastic programming deals with operational and planning problems that involve a sequence of decisions over time while responding to an uncertain future. Algorithms designed to address...

Time Variant Interval Linear Programming for Environmental Management Systems

Optimization technology is widely applied to maximize economic profit under ecology constrains in environmental management systems. To tackle the inherent uncertainties, inexact optimization methods have been proposed. Interval linear program- ming (ILP) model has drawn increasing scholarly attention. ILP model describe uncertainty by one coarse scaled stochastic process. However, uncertainty often involves multiple stochastic processes when zooming into high resolution. ILP model may not satisfied fine scale constraints. A time variant interval linear programming (TVILP) model is developed to implement temporal downscaling, and likewise, a heuristic algorithm integrating dynamic programming is proposed for Markov chained TVILP. Dynamic programming can converts time complexity exponential to polynomial. In the current paper, the performance of TVILP model is analyzed based on the following three metrics: maximal profit (M_profit), constraint violation risk (CVR), and maximal profit path risk (MPR). The perfor- mance of TVILP is further compared with the performance of Best and Worst method, the classic ILP model, Interval linear programming contractor, and Interval-parameter multi-stage stochastic linear programming. Experimental results reveal that TVILP provides refined solutions on a smaller granularity whose decision space contracts based on the most possible transition paths. Unable to obtain the maxi- mum profit, though, TVILP does pose decreased constraint violation risk and maximal profit path risk, facilitating more feasible and reliable decision-making on environmental management.

Multivariate probabilistic forecasting and its performance’s impacts on long-term dispatch of hydro-wind hybrid systems

There are two difficulties in long-term optimal dispatch of hydro-wind hybrid systems. First, monthly runoffs and wind speeds have the dynamic characteristics such as variability, instability, seasonality, heteroscedasticity, linear and nonlinear dynamic correlations. Second, hydro-wind hybrid systems have highly non-convex nonlinear constraints. To overcome the problem, this research develops a novel X-12 seasonal adjustment, vector autoregressive integrated moving average (VARIMA), component generalized autoregressive conditional heteroscedasticity (C-GARCH) and dynamic copula mixed model to estimate the joint probability distribution of runoffs and wind speeds. And then, this paper builds a multistage stochastic mixed-integer linear programming (MILP) with the help of several linearization methods. Finally, the paper compares several probabilistic forecasting models’ performances and analyzes their impacts on the dispatch of the hydro-wind hybrid system under different hydrological years. A hydro-wind hybrid system in southwest China is taken as an example. The case study leads to the following conclusions: 1) the more sufficient to capture the dynamic characteristics of variables, the higher benefit will be; 2) it is necessary to increase the scale of scenario tree to reduce the electricity shortfall during the dry year; 3) serious spilled water can be caused by insufficient interregional transmission capacity under the wet year and it is the most appropriate to expand the capacity to 8000 MW; 4) the model proposed in this paper can increase the economic benefit by 0.466×109CNY, 1.775×109CNY and 0.400×109CNY during the normal, dry and wet year, respectively.

Stochastic Lipschitz dynamic programming

We propose a new algorithm for solving multistage stochastic mixed integer linear programming (MILP) problems with complete continuous recourse. In a similar way to cutting plane methods, we construct nonlinear Lipschitz cuts to build lower approximations for the non-convex cost-to-go functions. An example of such a class of cuts are those derived using Augmented Lagrangian Duality for MILPs. The family of Lipschitz cuts we use is MILP representable, so that the introduction of these cuts does not change the class of the original stochastic optimization problem. We illustrate the application of this algorithm on two case studies, comparing our approach with the convex relaxation of the problems, for which we can apply SDDP, and for a discretized approximation, applying SDDiP.

Regularized stochastic dual dynamic programming for convex nonlinear optimization problems

We define a regularized variant of the dual dynamic programming algorithm called DDP-REG to solve nonlinear dynamic programming equations. We extend the algorithm to solve nonlinear stochastic dynamic programming equations. The corresponding algorithm, called SDDP-REG, can be seen as an extension of a regularization of the stochastic dual dynamic programming (SDDP) algorithm recently introduced which was studied for linear problems only and with less general prox-centers. We show the convergence of DDP-REG and SDDP-REG. We assess the performance of DDP-REG and SDDP-REG on portfolio models with direct transaction and market impact costs. In particular, we propose a risk-neutral portfolio selection model which can be cast as a multistage stochastic second-order cone program. The formulation is motivated by the impact of market impact costs on large portfolio rebalancing operations. Numerical simulations show that DDP-REG is much quicker than DDP on all problem instances considered (up to 184 times quicker than DDP) and that SDDP-REG is quicker on the instances of portfolio selection problems with market impact costs tested and much faster on the instance of risk-neutral multistage stochastic linear program implemented (8.2 times faster).

Sustainable biomass portfolio sourcing plan using multi-stage stochastic programming

Selecting the most sustainable biomass sources for biofuel production can overcome sever obstacles in the reliable supply of biomass feedstock, such as its seasonal availability. Developing a biomass portfolio can generally improve the sustainability of future biofuel systems. Bioethanol is an attractive biofuel having the potential for enhancing energy security and environmental safety over fossil fuels. The decisions to be made comprise the optimum amount of bioethanol feedstocks, and the investments to make in alternative activities. Uncertainty is assumed for biomass conversion rate to bioethanol, which is described by a set of scenarios. Therefore, for the entire planning horizon, a tree of scenarios is organized. The problem is formulated as a multi-stage stochastic linear programming problem. The goal is to maximize the total financial benefit. The risk associated with biomass availability affects the viability of biomass utilization and, therefore, should be taken into account when evaluating and analyzing biomass portfolio. In addition, the conditional value at risk (CVaR) is adopted to consider the risk hedge. Using a real case study, computational experiments are presented, showing that the stochastic approach is worth considering in these types of problems. Case study results demonstrate the validity of the proposed model.

Distributionally Robust Stochastic Dual Dynamic Programming

We consider a multistage stochastic linear program that lends itself to solution by stochastic dual dynamic programming (SDDP). In this context, we consider a distributionally robust variant of the...

The inventory routing problem under uncertainty with perishable products: an application in the agri-food supply chain

In this paper, we propose a dynamic and stochastic approach for an inventory routing problem in which products with a high perishability must be delivered from a supplier to a set of customers. This problem falls within the agri-food supply chain ( $${\mathcal {ASC}}$$ ) management field, which includes all the activities from production to distribution. The need for high-quality products that are subject to perishability is a critical issue to consider in the $${\mathcal {ASC}}$$ optimization. Moreover, the demand uncertainty makes the problem very challenging. In order to effectively manage all these features, a rolling horizon approach based on a multistage stochastic linear program is proposed. Computational experiments over medium-size instances designed on the basis of the real data provided by an agri-food company operating in Southern Italy show the effectiveness of the proposed approach.

Energy-intense production-inventory planning with participation in sequential energy markets

To support the uprise of demand response, especially in the context of industrial processes, we propose a new approach to integrally determine the production-inventory plan and the cost-minimizing bids to participate in sequential reserve and energy-only markets. In particular, our approach considers time-coupling constraints which occur in the context of a production-inventory planning problem. We extend this problem with a comprehensive bidding formulation, which allows evaluating revenues and potential cost from the market participation, considering price uncertainties and uncertain activations of committed reserve capacity. This results in a multistage stochastic mixed-integer linear program, which explicitly considers the stage-wise revelation of information in our setup. To illustrate the capabilities of our approach, we apply our model to a real-world case study in which we investigate the participation of a cement plant in the German energy-only and reserve markets. The results of our case study indicate significant revenues for flexible industrial processes when participating in German spot and reserve markets.