By Debora Mahlke

ISBN-10: 3834814091

ISBN-13: 9783834814098

Optimization difficulties related to doubtful facts come up in lots of parts of commercial and fiscal functions. Stochastic programming offers an invaluable framework for modeling and fixing optimization difficulties for which a likelihood distribution of the unknown parameters is offered. stimulated by way of useful optimization difficulties taking place in power platforms with regenerative strength provide, Debora Mahlke formulates and analyzes multistage stochastic mixed-integer types. for his or her answer, the writer proposes a singular decomposition process which is dependent upon the concept that of splitting the underlying state of affairs tree into subtrees. in accordance with the formulated versions from power construction, the set of rules is computationally investigated and the numerical effects are mentioned.

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**Additional info for A Scenario Tree-Based Decomposition for Solving Multistage Stochastic Programs**

**Example text**

In order to describe the amount of charged power of a charging unit k ∈ Kj of a storage j ∈ J , we use the variable sin kt ∈ R+ . Likewise, the discharged power of discharging unit l ∈ Lj is described by sout lt ∈ R+ . For the description of the operational state of a unit k ∈ Kj , we introduce in ∈ {0, 1} and accordingly for a discharging unit the decision variable zkt in,up out the variable z ∈ {0, 1}. The start-up variables zkt ∈ {0, 1} and l ∈ Lj lt out,up zlt ∈ {0, 1} indicate whether unit k ∈ Kj or l ∈ Lj is switched on in time period t, respectively.

In this chapter we describe the mathematical modeling of the problem, taking technical as well as economical aspects into account. The problem considered here is to optimize the energy supply of a city, where energy is generated by conventional power plants, purchased on the spot market or obtained from wind energy. Due to the growing proportion of energy produced from renewable energy sources, not only ﬂuctuating demand has to be taken into account but also ﬂuctuations of the power supply. In this system, energy storages are used to decouple supply and demand, achieving a better capacity utilization and a higher eﬃciency of the power plants.

S}. Additionally, we denote the set of nodes corresponding to a path from the root node to n by path(n). By πn , we refer to the probability of a scenario to pass a node n ∈ N . , π1 = 1. 7. t. W1 x1 Tn xp(n) + Wn xn xn = = b1 bn for all n ∈ N \ {1} ∈ Xn for all n ∈ N where xn denotes the decision variables of node n and Tn and Wn are matrices of corresponding size as well as the vector cn . The set Xn represents the restrictions requiring some or all of the variables of node n to be integer.

### A Scenario Tree-Based Decomposition for Solving Multistage Stochastic Programs by Debora Mahlke

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