SDDP – Stochastic hydrothermal dispatch with network restrictions

SDDP is a hydrothermal dispatch model with representation of the transmission network and used for short, medium and long term operation studies. The model calculates the least-cost stochastic operating policy of a hydrothermal system, taking into account the following aspects:

• Operational details of hydro plants (water balance, limits on storage and turbined outflow, spillage, filtration etc.);
• Detailed thermal plant modeling (unit commitment, generation constraints due to "take or pay" fuel contracts, concave and convex efficiency curves, fuel consumption constraints, bi-fuel plants etc.);
• Representation of spot markets and supply contracts;
• Hydrological uncertainty: it is possible to use stochastic inflow models that represent the system hydrological characteristics (seasonality, time and space dependence, severe droughts etc.) and the effect of specific climatic phenomena such as the El Niño;
• Detailed transmission network: Kirchhoff laws, power flow limit in each circuit, losses, security constraints, export and import limits for each electrical area etc;
• Load variation per load level and per bus, with monthly or weekly stages (medium or long term studies) or hourly stages (short term studies).

In addition to the least-cost operating policy, the model calculates several economical indexes such as the spot price (per submarket and per bus), wheeling rates and transmission congestion costs, water values for each hydro plant, marginal costs of fuel supply constraints and others.

Current uses of SDDP

SDDP has been used in studies for valuation of companies, international interconnections and analysis of new hydroelectric and thermal projects in Brazil for PSR clients, which include almost all international and local investors in the energy sector, and in several other projects outside Brazil. It has also been used in operations studies in more than 30 countries, including:

• all countries in South and Central America;
• USA and Canada;
• Austria, Spain, Norway, the Balkan region (10 countries) and Turkey;
• New Zealand, South China and Shangai province.

The model has also been used in the dispatch centers of Bolivia, Colombia, Chile, Guatemala, El Salvador, Ecuador, Panama and Venezuela.

Methodology

Because hydro plants have no direct operating costs, one could think that they would come first in the dispatch order. Note, however, that the hydro plant has the option of generating the energy today or storing the water for future use. For instance, suppose that the water stored in a given hydro electric can produce 1 MWh. Also, suppose that the current spot price is US$18/MWh, increasing to US$25/MWh by next week. Because the objective is to maximize economic efficiency, it is better to store the water until next week. In other words, although hydro plants do not have direct operation costs, they have an opportunity cost that reflects the benefit resulting from energy production in the future.

In the simple example above, where the future price is higher than current price, the best decision is obvious. In real situations, however, there is an uncertainty regarding future prices, which can be either higher or lower than the current one. Therefore, the decision of storing or using the water depends on an analysis of the consequences of each decision for all the future price scenarios. Unfortunately, the number of combinations of price scenarios grows exponentially along time. For instance, suppose that every week there are two price scenarios. At the end of one year, the number of combinations would be 2⁵², more than a quadrillion, which obviously makes it infeasible to apply any method of exhaustive enumeration. In addition, the transference of energy from one week to the other modifies the spot prices, because it decreases supply in the current week and increases it in the following. In summary, the dispatch of a hydrothermal system is a large scale stochastic optimization, whose solution is quite complex.

The solution methodology traditionally used to solve this dispatch problem is known as stochastic dynamic programming (SDP). Traditional SDP methods require the discretization of reservoir storage levels (100%, 95%, 90% etc.). When there are two reservoirs, it is necessary to enumerate all combinations of pairs of levels (100% and 100%; 100% and 95%;...; 95% and 100%; 95% and 95% etc.); and so on. As a consequence, the computational effort of SDP grows exponentially with the number of reservoirs, which constrains the application of traditional SDP to systems with only two or three reservoirs.

The SDDP model uses a new solution methodology called stochastic dual dynamic programming, developed by PSR. This methodology represents the future cost function of traditional SDP as a piecewise linear function. Because of this feature, it is not necessary to enumerate the combinations of reservoirs levels, which allows the determination of the stochastic optimal solution for systems with a large number of hydro plants.

Model results

All the detailed results of the model SDDP are written to *. csv format files. These files are managed by a graphic interface (the GRAF program) which produces Excel files with the desired results. The main SDDP results are:

• operative statistics: hydro and thermal generation, thermal operation costs, energy interchange, fuel consumption, deficit risks and energy not supplied;
• short run marginal costs (spot prices) for each submarket and for each bus;
• marginal capacity benefits: measure of the operational benefit of reinforcing the installed capacity of a thermal plant, the turbine limit of a hydro plant or the storage capacity of a reservoir. These indices are used to determine cost-effective system reinforcements.

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