Models of a Planned Economy Are there any mathematical models of a planned economy like in the soviet union? Specifically id like to know how we would model 5 year plans.

Do any exist?
Jan Philipp Dapprich did something like this in his PhD-Thesis. For an introductory read, his blog might be a little more accessible, this article is a good start. His research, in short, is based on the assumption that the drastic increase in computational power during the last decades can be harnessed to fix some of the flaws of the planning performed in systems like the soviet union.
For a short description of the model I provide the following quote from the article:

An agent-based model that was developed for the purpose of the simulation is used to model the behaviour of consumers. In a real socialist economy this would not be necessary, as there would be actual consumers. In the simulation it is, however, necessary in order to determine the consumer demand for various product. Consumers are given a set amount of credits and will then semi-randomly decide how they will spend those credits. From this the overall demand for the various products is derived. What’s crucial about the consumer model is that demand is sensitive to credit prices. Consumers are less likely to buy a product if it has a higher price. Based on a comparison of demand and the supply of consumer products given by the optimized production plan “market clearing prices” can be approximated. This is done by successively adjusting the price of products depending on the difference between supply and demand until supply and demand match. The previous plan target is now adjusted through a comparison of prices and values (MDVs). The MDVs are scaled in order to make them comparable to prices. Should prices of a product exceed MDVs, this suggests that consumers are sufficiently interested in the product to justify further production. Therefore, the target entry for the product is increased. Should prices be below MDVs, this suggests that consumers are not interested enough in the product to justify the current emphasis on the product and the target entry is consequently lowered. By adjusting the plan target this way we make sure that in the next plan period the proportions at which products are produced more closely resemble what consumers actually need. This process also takes into account the cost of producing a product in the form of the MDVs. Should there be a significant limitation on the amount of greenhouse gas emissions that are allowed for, this would result in higher MDVs for products whose production involves higher amount of emissions. In this case the targets for those products will be lowered and a greater emphasis will be put on producing more environmentally friendly products. A central hypothesis that I wish to confirm using the simulation is that this shift towards environmentally friendly products distinguishes the effect of my model from the labour value model of Cockshott and Cottrell, as labour values don’t take emissions into account.

This excerpt is not sufficient to explain his whole methodology since it is quite complex. For the sake of readability I decided to just scratch the surface here and refer to the blog post series for deeper insight.