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I have to generate scenarios for a stochastic optimization program. I want to reduce this number of scenarios but the assignment of a probability to each scenario is my problem. How can I assign a probability to each scenario?


Background

I have historical data for a renewable energy profile such as wind and solar, and I want to formulate a market clearing operation problem using a two-stage stochastic program to minimize the operation cost in the day ahead. I want to have scenarios for each uncertain variable (wind, solar) using a neural network so I don't make fitting to a probability distribution. I then want to assign a probability for each scenario to sum over all scenarios and multiply by its probability to calculate the expected cost. Also I have some confusions about scenarios when I have a lot of uncertain variables like wind, solar and electric load.

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    $\begingroup$ You should provide more context and information. Scenario sampling is usually done for a given fixed number of scenarios (e.g., see the sample average approximation). As the number of scenarios in such methods is large, you might assume equal probability for all the scenarios. Scenario reduction is performed when you are given a large number of scenarios and that makes solving the resulting deterministic equivalent problem hard. In this case, the scenario reduction methods assign probabilities to the remaining scenarios based on the sum of probabilities of omitted scenarions they represent. $\endgroup$ – Ehsan Dec 7 '19 at 20:44
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    $\begingroup$ Hi Shady, welcome to OR.SE, you can also have a look at the Q/A in the following link: or.stackexchange.com/q/179/39 $\endgroup$ – Oguz Toragay Dec 7 '19 at 22:36
  • $\begingroup$ @Ehsan I have historical data for renewable energy profile such as wind,solar,I want to make market clearing operation problem using 2 stage stoch program to min the operation cost in the day ahead so I want to have scenarios for each uncertain variable (wind,solar) using nueral network so I don't make fitting to a probability distribution then I want to assign a probability for each scenario to sum over all scenarios and multiply by its probability to calculate the expected cost Also I have confusion about scenarios when I have a lot of uncertain variables like wind, solar and electric load $\endgroup$ – shady mamdouh Dec 8 '19 at 10:41
  • $\begingroup$ @OguzToragay Sure $\endgroup$ – shady mamdouh Dec 8 '19 at 10:42
  • $\begingroup$ @Ehsan Please may you suggest some papers about the scenario generation that assume equal probabilities and the ways for scenario reduction and assignment of probabilities $\endgroup$ – shady mamdouh Dec 11 '19 at 19:34
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  • If you have historical data, you might use them as scenario inputs to a scenario reduction algorithm. Some references are available from here and here.
  • Fitting a probability distribution does not prevent you from using a scenario approach to modeling uncertainty. In fact, the scenario approach is a popular and viable approach for handling uncertainty modeling as the numerical integration of the recourse function leads to complex and difficult problems (see section 3.1.c of here). If you decide to generate random scenarios from the fitted probability distributions, some useful references are available from here and here.
  • Regarding the presence of multiple uncertain parameters, the uncertainty modeling is done separately for each uncertain parameter as long as they are independent. If not, you have to know joint probability distribution functions for the uncertain parameters and sample from them. For examples of such sampling, see here and here.
  • I'm not sure why you are using a neural network to build scenarios. If what you are doing is correct and the number of scenarios is large enough, then you might assume equal probability for all of the scenarios.

Besides the above references, the book chapter available from here and the presentation available from here are good introductions to the topic.

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    $\begingroup$ Nice collection of references. You might consider adding some appropriate ones to or.stackexchange.com/questions/869/… . I think you can still edit the community wiki answer despite the question being stupidly closed. $\endgroup$ – Mark L. Stone Dec 12 '19 at 12:36
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    $\begingroup$ @MarkL.Stone: Thanks. I added some textbooks for IP, SP, and RO. $\endgroup$ – Ehsan Dec 12 '19 at 16:58
  • $\begingroup$ @Ehsan thanks for your appreciated answer but I want to clarify a final thing, if there are 5 scenarios for a certain variable with their probabilities and say 4 scenarios for another variable; may I assume that I now have 20 scenarios with probabilities equal the products of every scenario with 1st variable by the corresponding probabilities of the 2nd variable?? $\endgroup$ – shady mamdouh Dec 25 '19 at 10:49
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    $\begingroup$ @shadymamdouh: As long as you can assume that scenarios for different parameters are independent, you can calculate the probability of combined scenarios according to the method you described. $\endgroup$ – Ehsan Dec 25 '19 at 14:13

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