For several decades, power system planning and operations has been one of the most active application areas for multistage stochastic optimization techniques. This is due to the combination of the following factors: (i) ever increasing social and economic importance of electricity supply for both developed and emerging countries; (ii) yearly investment and operation costs on the order of hundreds of billions of dollars; (iii) coupling of time stages due to storage, originally from hydro reservoirs and, more recently, batteries and other devices; (iv) uncertainties on several key parameters, such as the production variability of hydropower, wind and other renewables; annual demand growth; fuel costs; construction times; macro-weather effects such as El Niño, plus climate change; and short-term electricity prices. In this talk, I will describe some recent algorithmic and modeling advances on one widely applied technique, stochastic dual dynamic programming (SDDP): (a) implicit representation of the immediate cost function in each stage as a pre-calculated piecewise linear surface, which allows the detailed modeling of 730 hourly intervals in each stage with the same computational effort as an aggregate model with five blocks; (b) representation of uncertainties in the parameter values of wind and inflow stochastic models (this problem became relevant due to the shorter historical records of modern renewables); (c) an integrated Markov chain-AR model framework for the joint representation of uncertainties of renewable production, load growth, fuel costs, macro-weather and spot prices; (d) generation expansion planning with a risk-averse SDDP (CVaR on either operating costs or supply reliability); and (e) an extension of SDDP to produce generation expansion strategies, in which the investment decisions depend on the system state, for example past load growth rates. This allows the correct valuation of technologies with different construction times (e.g. six years for hydro, three for a gas turbine and two for wind) when planning under uncertainty. The application of the above algorithms will be illustrated for actual planning and operation studies in Latin America and Asia.

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SDDP implementations have long been used in hydrothermal systems to calculate optimal system scheduling and optimal expansion planning decisions in a deeply integrated way, in which Benders cuts resulting from the probabilistic operations module are used to solve the expansion MIP problem. In this paper, we study the modified SDDP formulation adopted by the Brazilian System Operator (ONS) since 2013, which introduces a nested CVaR mechanism to incorporate risk aversion in the hydrothermal scheduling decisions. In particular, we show that using the expected marginal costs from the probabilistic simulations of CVaR policy directly would lead to inconsistencies between the investment decisions and the preferences implied by the objective function in the operations module, highlighting a need to extend the optimal system planning approach. We propose an alternative implementation that calculates a weighted average of the Benders cut coefficients from the lower bound of the CVaR-SDDP scheme, showing that it results in consistent decisions for the two optimization problems. A contrast between planning studies carried out with and without risk-aversion is used to illustrate and discuss the implications of this methodology.

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The modelling of modern power markets requires the representation of the following main features: (i) a stochastic dynamic decision process, with uncertainties related to renewable production and fuel costs, among others; and (ii) a game-theoretic framework that represents the strategic behaviour of multiple agents, for example in daily price bids.

These features can be in theory represented as a stochastic dynamic programming recursion, where we have a Nash equilibrium for multiple agents. However, the resulting problem is very challenging to solve.

This work presents an iterative process to solve the above problem for realistic power systems. The proposed algorithm is consist of a fixed point algorithm, in which, each step is solved via stochastic dual dynamic programming method.

The application of the proposed algorithm are illustrated in case studies with the Panama systems.

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This paper compares three analytical approaches to represent risk aversion in stochastic hydrothermal scheduling: (i) convex combination of expected value and CVaR of operation costs in the objective function; we show that this approach is a special case of importance sampling on the inflow probability distributions and, from that, we derive a practical procedure to calculate the exact upper bound of the CVaR-based SDDP recursion; (ii) representation of unserved energy costs as a piecewise linear function, where the first segment represents the "economic" value of energy shortage whereas the second segment ensures a CVaR-based reliability target; this is calculated by a iterative line search / SDDP procedure; and (iii) use of feasibility cuts to represent a "risk aversion surface" for storage values at each stage that ensures a given supply reliability target. The feasibility cuts are calculated with a hybrid stochastic/robust optimization scheme.

These approaches are compared in terms of execution time, ease of calibration and other indices for a configuration of the Brazilian power system.

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One of the attractive characteristics in the application of SDDP to stochastic hydrothermal scheduling has been the ability to represent inflow uncertainty analytically as a multivariate autoregressive model, for example, AR-3. This analytical representation was later extended to other renewables such as wind and to random fluctuations in hourly load. However, two other important sources of uncertainty, which are fuel costs and annual load, were usually represented as a scenario tree with (weekly or monthly) stages and annual “splitting” of scenarios. In this approach, a separate SDDP backward recursion is carried out for the intra-year stages in each branch of the tree, followed by the construction of a combined future cost function in the yearly joining nodes. In this paper, we describe an alternative Markov/AR modeling of these uncertainties, where the Markov states represent the incremental rates with respect to the previous year values (for example, a 3% load growth and a 5% decrease in fuel cost), and the autoregressive models represent variations along the stages. In the case of fuel costs, there is an additional dualization/McCormick step that allows their representation in the problem RHS instead of the objective function. This approach is illustrated with a case study of a realistic Asian power system.

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The optimal scheduling of hydrothermal systems requires the representation of uncertainties in future streamflows to devise a cost-effective operations policy. Stochastic optimization has been widely used as a powerful tool to solve this problem but results will necessarily depend on the stochastic model used to generate future scenarios for streamflows. Periodic autoregressive (PAR) models have been widely used in this task. However, its parameters are typically unknown and must be estimated from historical data, incorporating a natural estimation error. Furthermore, the model is just a linear approximation of the real stochastic process. The consequence is that the operator will be uncertain about the correct linear model that should be used at each period.

The objective of this work is to assess the impacts of incorporating the uncertainty of the parameters of the PAR models into a stochastic hydrothermal scheduling model. It will be shown that when the uncertainty of the parameters is ignored, the policies given by the stochastic optimization tend to be too optimistic.

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Many capacity planning models used today are based on a Benders decomposition scheme composed of: (i) a MIP-based “investment module” which determines a trial expansion plan; (ii) a SDDP-based “operation module” which calculates the expected operation costs for the trial plan; and (iii) Benders cuts from the operation to the investment module, whose coefficients are calculated from the expected marginal costs of the capacity constraints in the operation module at the optimal solution.

Although this “traditional” planning model has been successfully applied in many countries, it has an inherent limitation, which has become more significant with the penetration of renewables with short construction times, such as solar: the optimal expansion plan is “static”, i.e. investment decisions do not change as the system state evolves (e.g. load growth is lower than expected, a very rainy season occurs etc.). As a consequence, there is a growing interest in the calculation of an integrated stochastic investment & operations strategy.

This paper describes an extension of the SDDP algorithm that allows the calculation of this integrated strategy. The first (and obvious) step of this extension is to represent investment decisions as state variables in the SDDP recursion. The second step is to represent the construction time of each candidate project in the recursion; this requires an efficient modeling of time delays in the update of state variables. The final step is to represent the integrality of investment decisions in the multistage stochastic optimization scheme. This is done by applying a customized Lagrangian scheme to the scheduling/investment subproblem of each stage and scenario that produces the strongest possible convex cut to the previous stage’s future cost function.

The application of the proposed algorithm will be illustrated in realistic capacity planning studies of the Central America system.

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The increasing penetration of renewable generation combined with the development of effective short-term energy storage batteries demand scheduling decisions of power systems to be represented on an hourly basis or even in smaller time intervals in scheduling models. This problem is compounded by the presence of hydropower in some systems, which is stochastic by nature. The need to represent the intermittency of renewable resources with the stochasticity of hydropower result in a multistage stochastic optimization scheduling problem that in such reduced time resolution would implicate in the significant increase of computational effort, possibly leading to the infeasibility to solve such problems.

In this thesis, we propose a method that is able to take into consideration such small time intervals while avoiding the considerable increase of computational times for hydrothermal scheduling problems. This method consists in calculating the analytical representation of the immediate cost function that will be applied in the context of stochastic dual dynamic programming (SDDP). The function represents immediate operation costs as a function of the total hydroelectric generation optimal decision. As the immediate cost function is piecewise linear, it leads to a structure very similar to the one used to approximate the future cost function (cut set). Results of the application of the method in real power systems are presented.

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