Stochastic Dual Dynamic Programming Approach Research Articles

Medium-Term Hydrothermal Scheduling of the Infiernillo Reservoir Using Stochastic Dual Dynamic Programming (SDDP): A Case Study in Mexico

This article aims to obtain and evaluate medium-term operating policies for the hydrothermal scheduling problem by using the stochastic dual dynamic programming (SDDP) approach. To this end, to feed the mathematical model and build the probability distribution functions that best fit each month of the actual inflow volume, monthly inflow data recorded from 1938 to 2018 for the Infiernillo reservoir located in Mexico were employed. Moreover, we simulated inflow volume scenarios using the Monte Carlo method for each month of a one-year planning period. The SDDP approach to solving the optimization problem consisted of the simulation of one forward scenario per iteration and the stabilization of the total operating cost as a convergence criterion, which results in an operating policy. We then assessed its quality by estimating the one-sided optimality gap. It is worth mentioning that the best operation policy required scenario trees of up to 17,000 inflow realizations per stage. Additionally, to study the evolution of the expected value along the planning horizon of the main variables involved in the medium-term hydrothermal scheduling problem, we simulated the best operation policy over 10,000 inflow scenarios. Finally, to show the practical value of the proposed approach, we report its computational complexity.

Hydropower Aggregation by Spatial Decomposition—An SDDP Approach

The balance between detailed technical description, representation of uncertainty and computational complexity is central in long-term scheduling models applied to hydro-dominated power system. The aggregation of complex hydropower systems into equivalent energy representations (EER) is a commonly used technique to reduce dimensionality and computation time in scheduling models. This work presents a method for coordinating the EERs with their detailed hydropower system representation within a model based on stochastic dual dynamic programming (SDDP). SDDP is applied to an EER representation of the hydropower system, where feasibility cuts derived from optimization of the detailed hydropower are used to constrain the flexibility of the EERs. These cuts can be computed either before or during the execution of the SDDP algorithm and allow system details to be captured within the SDDP strategies without compromising the convergence rate and state-space dimensionality. Results in terms of computational performance and system operation are reported from a test system comprising realistic hydropower watercourses.

A variant SDDP approach for periodic-review approximately optimal pricing of a slow-moving a item in a duopoly under price protection with end-of-life return and retail fixed markdown policy

In this paper, we examine a selling environment where a manufacturer-controlled retailer and an independent retailer sell a slow-moving A item. The manufacturer offers the independent retailer a price protection contract stipulating that the manufacturer reimburses the independent retailer in case of a reduction in the wholesale price. The price set by the independent retailer is assumed to be determined by Retail Fixed Markdown (RFM) policy. The manufacturer also offers the independent retailer a special discount rate for the replenishment orders and the retailers are assumed to follow (R, S) inventory replenishment policy. The manufacturer adopts a periodic-review pricing strategy and the mean demand observed by each retailer in a given period depends on the prices. We also take the customers choosing no-purchase option into account. We employ multinomial logit (MNL) models to forecast customers’ preferences based on retail prices. The retailers’ market shares are estimated by customized choice probability functions. We propose stochastic programming models to determine the manufacturer’s pricing strategy. Then, we propose a variant Stochastic Dual Dynamic Programming (SDDP) algorithm to determine the manufacturer’s approximately optimal pricing strategy by getting around three curses of dimensionality. Then, we move on to the observations on the impact of four critically important contractual parameters on the price, the market shares and the expected total net profits and finally discuss some possible approaches for the selection of the best compromise values of those contractual parameters.

A stochastic dual dynamic programming method for two-stage distributionally robust optimization problems

This paper studies a class of two-stage distributionally robust optimization (TDRO) problems which comes from many practical application fields. In order to set up some implementable solution method, we first transfer the TDRO problem to its equivalent robust counterpart (RC) by the duality theorem of optimization. The RC reformulation of TDRO is a semi-infinite stochastic programming. Then we construct a conditional value-at-risk-based sample average approximation model for the RC problem. Furthermore, we analyse the error bound of the approximation model and obtain the convergent results with respect to optimal value and optimal solution set. Finally, a so-called stochastic dual dynamic programming approach is proposed to solve the approximate model. Numerical results validate the solution approach of this paper.

Impact of Long-Term Water Inflow Uncertainty on Wholesale Electricity Prices in Markets with High Shares of Renewable Energies and Storages

Renewable energy shares in electricity markets are increasing and therefore also require an increase in flexibility options. Conventional electricity price modelling with optimisation models in thermally dominated markets is not appropriate in markets with high shares of renewable energies and storages because price structures are not adequately represented. Previous research has already identified the impact of uncertainty in renewable energy feed-in on investment and dispatch decisions. However, we are not aware of any work that investigates the influence of uncertainties on price structures by means of optimisation models. Appropriate modelling of electricity price structures is important for investment and policy decisions. We have investigated the influence of uncertainty concerning water inflow by applying a second stage stochastic dual dynamic programming approach in a linear optimisation model using Norway as an example. We found that the influence of uncertainty concerning water inflow combined with high shares of storages has a strong impact on the electricity price structures. The identified structures are highly influenced by seasonal water inflow, electricity demand, wind, and export profiles. Additionally, they are reinforced by seasonal primary energy source prices and import prices. Incorporating uncertainties in linear optimisation models improves the price modelling and provides, to a large extent, an explanation for the seasonal patterns of Norwegian electricity market prices. The paper explains the basic pricing mechanisms in markets with high shares of storages and renewable energies which are subject to uncertainty. To identify these fundamental mechanisms, we focused on uncertainty regarding water inflow, but the basic results hold true for uncertainties regarding other renewable energies as well.

Risk-averse stochastic dual dynamic programming approach for the operation of a hydro-dominated power system in the presence of wind uncertainty

Operation planning models for hydro-dominated power systems usually use low temporal resolutions due to the excessive computational burden, thus ignoring short-term characteristics of such systems. As a result, in systems coupled with wind energy, such models may fail to accurately capture wind variability, and may not appropriately take into account potential consequences of uncertainty on the system operation. This paper addresses this drawback by (i) “controlling” the cost associated with the operation of a hydro-dominated power system equipped with wind power and batteries via a risk-measure and (ii) formulating the short-term load balance as probabilistic constraints in order to hedge against potential extreme wind power scenarios. The risk-averse scheme is embedded in the stochastic dual dynamic programming framework. Simulation results for a case study on a real industrial setting show that hedging the system against the short-term volatility of wind power contributes to mitigating the risk of excessive operations costs or load curtailments, and that the consideration of the decision maker risk profile contributes to decreasing the variability of the solutions. In addition, the results of the application also illustrate the potential of the scheme to assess the energy situation of a country or a region under the penetration of wind energy and batteries deployment.

Stochastic Dual Dynamic Programming for Operation of DER Aggregators Under Multi-Dimensional Uncertainty

The operation of aggregators of distributed energy resources (DER) is highly complex, since it entails the optimal coordination of a diverse portfolio of DER under multiple sources of uncertainty. The large number of possible stochastic realizations that arise can lead to complex operational models that become problematic in real-time market environments. Previous stochastic programming approaches resort to two-stage uncertainty models and scenario reduction techniques to preserve the tractability of the problem. However, two-stage models cannot fully capture the evolution of uncertain processes and the a priori scenario selection can lead to suboptimal decisions. In this context, this paper develops a novel stochastic dual dynamic programming approach which does not require discretization of either the state space or the uncertain variables and can be efficiently applied to a multi-stage uncertainty model. Temporal dependencies of the uncertain variables as well as dependencies among different uncertain variables can be captured through the integration of any linear multidimensional stochastic model, and it is showcased for a p-order vector autoregressive model. The proposed approach is compared against a traditional scenario-tree-based approach through a Monte-Carlo validation process, and is demonstrated to achieve a better trade-off between solution efficiency and computational effort.

Two-stage stochastic dual dynamic programming for transmission expansion planning with significant renewable generation and N-k criterion

The large-scale integration of renewable energy sources (RES) is the global trend to deal with the energy crisis and greenhouse emissions. Due to the intermittent nature of RES together with the uncertainty of load demand, the problem of transmission expansion planning (TEP) is facing more and more challenges from uncertainties. In this paper, the TEP problem is modeled as a two-stage formulation, so as to minimize the total of investment costs and generation costs. To ensure the utilization level of the RES generation, the expansion plan is required to provide sufficient transmission capacity for the integration of RES. Also, N-k security criterion is considered into the model, so the expansion plan can meet the required security criteria. The stochastic dual dynamic programming (SDDP) approach is applied to consider the uncertainties, and the whole model is solved by Benders' decomposition technique. Two case studies are carried out to compare the performance of the SDDP approach and the deterministic approach. Results show that the expansion plan obtained by the SDDP approach has a better performance than that of the deterministic approach.

A Generic Storage Model Based on a Future Cost Piecewise-Linear Approximation

This paper presents a generic storage model (GSM) inspired by the scheduling of hydraulic reservoirs. The model for steady state short-term operational studies interlaces with the long-term (LT) energy scheduling through a piecewise-linear future cost function (FCF). Under the assumption that a stochastic dual dynamic programming approach has been used to solve the energy schedule for the LT, the FCF output from that study will be processed to obtain an equivalent marginal opportunity cost for the storage unit. The linear characteristic of a segment of the FCF will allow a linear modeling of the storage unit production cost. This formulation will help to coordinate the renewable resource along with storage facilities in order to find the optimal operation cost while meeting end-point conditions for the LT plan of the energy storage. The generic model will be implemented to represent a battery storage and a pumped-hydro storage. A stochastic unit commitment with the GSM will be formulated and tested to assess the day-ahead scheduling strategy of a virtual power plant facing uncertainties from production, consumption, and market prices.

Long-term optimal allocation of hydro generation for a price-maker company in a competitive market: latest developments and a stochastic dual dynamic programming approach

Since the liberalisation of the power industry, there has been a large amount of literature on the determination of optimal bidding decisions for price-maker energy producers. The vast majority of the work developed so far has focused on short-term horizons and may be viewed as successful approaches for systems whose operation is generally deterministic. In the case of price-maker hydro plants with significant storage capacity, however, the solution of the strategic bidding problem is more subtle. The reason is that hydro reservoirs allow the bidder to postpone energy production if future prices are expected to be higher than the current price. This demands the management of an energy-constrained resource and determines a time-coupling characteristic to the problem, implying that the bidding strategy should ideally take into account the following stages and consider the stochasticity of inflows. These aspects characterise the strategic bidding for price-maker hydro agents as a multi-stage stochastic programming problem, with significant computational challenges. The objective of this work is to present a new methodology for the strategic bidding problem of a price-maker hydropower-based company, taking into account several hydro plants, time-coupling and stochastic inflow scenarios. The proposed approach considers a deterministic residual demand curve and is based on stochastic dual dynamic programming (SDDP), which has been successfully applied to the least-cost hydrothermal scheduling problem. Since the technique requires the problem to be concave, a piecewise linear approximation of the expected future benefit function is proposed. The application of the methodology is exemplified with a real case study based on the hydrothermal system of El Salvador.