Optimization Methods in Energy and Power Systems Lecture

Optimization Methods in Energy and Power Systems Lecture 11: Stochastic Programming Mahdi Pourakbari Kasmaei, 2019 Thursday, 16 May 2019

Stochastic Programming ü Introduction of Stochastic Programming ü Stochastic Process ü Stochastic Programming • Two- and Multi-Stage Problems ü Quality Metrics M. Pourakbari Kasmaei, 2019 2

Stochastic Programming (Quality Metrics) ü Quality Metrics are used to reveal the interest of applying stochastic programming. • Deterministic problems are obtained out of stochastic programming by fixing the random variable to an expected or forecaste values ü Two main metrics are: • Expected Value of Perfect Information (EVPI) • Value of the Stochastic Solution (VSS) M. Pourakbari Kasmaei, 2019 3

Stochastic Programming (Quality Metrics: EVPI) ü Expected Value of Perfect Information (EVPI) demonstrates the price that a decision maker is willing to pay for obtaining perfect information about the future. ü Let us assume a two-sage stochastic programming Model. The EVPI for Maximization and Minimization Problems are calculated as follows. PI and S stand for Perfect information-, and Stochasticbased models, respectively. M. Pourakbari Kasmaei, 2019 4

Stochastic Programming (Quality Metrics: EVPI) ü For EVPI calculations, the scenario-variable model is used. ü The calculations are done as follows. • Relax the non-anticipativity condition. • Solve the relaxed Problem M. Pourakbari Kasmaei, 2019 5

Stochastic Programming (Quality Metrics: EVPI) Example 11. 1: Obtain the EVPI for the two-stage problem presented in Lecture 9, Ex. 9. 5. Let us solve it together! M. Pourakbari Kasmaei, 2019 6

Stochastic Programming (Quality Metrics: EVPI) Example 11. 2: Obtain the EVPI for the three-stage problem presented in Lecture 10, Ex. 10. 2. Let us solve it together! M. Pourakbari Kasmaei, 2019 7

Stochastic Programming (Quality Metrics: VSS) ü Value of Stochastic Solution (VSS) is used to quantify the advantage of using a stochastic programming over the deterministic one. ü Let us assume a two-sage stochastic programming Model. The VSS for Maximization and Minimization Problems are calculated as follows. D and S stand for Deterministic, and Stochastic models, respectively. M. Pourakbari Kasmaei, 2019 8

Stochastic Programming (Quality Metrics: VSS) ü For EVPI calculations, the following steps are considered. • Random variables of the stochastic process are replaced by their respective expected values. • The optimal values for the first-stage variables are obtained. • The original Stochastic problem can then be solved by fixing the values of the first-stage variables (to those provided by the deterministic one). M. Pourakbari Kasmaei, 2019 9

Stochastic Programming (Quality Metrics: VSS) Example 11. 3: Obtain the VSS for the two-stage problem presented in Lecture 9, Ex. 9. 4. Let us solve it together! M. Pourakbari Kasmaei, 2019 10

Stochastic Programming (Quality Metrics: VSS) Example 11. 4: Obtain the VSS for the three-stage problem presented in Lecture 10, Ex. 10. 1. Let us solve it together! M. Pourakbari Kasmaei, 2019 11

Thanks! M. Pourakbari Kasmaei, 2019 12