By Debora Mahlke

Encouraged through functional 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 resolution, the writer proposes a unique decomposition strategy which is dependent upon the idea that of splitting the underlying situation tree into subtrees. in response to the formulated types from strength construction, the set of rules is computationally investigated and the numerical effects are mentioned.

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**Extra info for A Scenario Tree-Based Decomposition for Solving Multistage Stochastic Programs: With Application in Energy Production (Stochastic Programming)**

**Sample text**

4 Let Γ be a scenario tree with the node set N = {1, . . , N }. 12) provide a complete linear description of PΓ,L,l . Proof. 12)}. 50 Chapter 4.

Up down ∈ {0, 1} and zit ∈ {0, 1} Finally, we introduce the binary variables zit modeling the switching process of the plant, respectively. In particular, up = 1 if and only if the plant is switched on in time period t and was not zit down indicates if the plant is shut operating in period t − 1. Analogously, zit down in time t. 20 Chapter 3. 3: Variables pit produced power of plant i ∈ I [MW] sjt storage level of storage j ∈ J [MWh] sin kt sout lt charging power of unit k ∈ Kj and j ∈ J [MW] discharging power of unit l ∈ Lj and j ∈ J [MW] xt imported power [MW] cpow it cstor jt cimp t costs of plant i ∈ I [e] costs of storage j ∈ J [e] import costs [e] zit state variable for the production of plant i ∈ I [1] up zit start-up variable of plant i ∈ I [1] down zit in zkt out zlt in,up zkt out,up zlt in yjt out yjt shut-down variable of plant i ∈ I [1] state variable of charging unit k ∈ Kj and j ∈ J [1] state variable of discharging unit l ∈ Lj and j ∈ J [1] start-up variable of charging unit k ∈ Kj and j ∈ J [1] start-up variable of discharging unit l ∈ Lj and j ∈ J [1] state variable for charging the storage j ∈ J [1] state variable for discharging the storage j ∈ J [1] For each energy storage j ∈ J , the variable sjt ∈ R+ represents the current storage level.

More precisely, we construct the point by considering those nodes k ∈ N \Ns , for which their predecessor p(k) is in Ns . If xp(k) = 1, then we set xl to one for all nodes l ∈ desc(k). Analogously, if xp(k) = 0, we set xl to zero for all nodes l ∈ desc(k). Finally, we choose N − T points cs,L n for n ∈ N \Ns . Here, the power plant is switched oﬀ in node n. If t(n) − 1 ≤ L the power plant is operating on the path(p(n)). 11) by equality, the power plant is operating on the path(vt(n)−1 ), too. This means that the scenario path Ns is aﬀected and xdown vt(n) = 1.