Power Analysis with GPower repeated measures MANOVA within-between interaction

I am running a longitudinal experiment over 9 months. I have 2 dependent measures (water, electricity) and 5 groups (conditions, 4 experimental, 1 control). The conditions are the "between"factor, and time is the "within" factor with 9 levels (measurement points in time). I am struggling to calculate the required sample size to get 1-ß = 0.8 with alpha = 0.5. In GPower, I choose "Repeated measures MANOVA within-between interaction" as I have multiple IVs (my conditions) and I enter 5 as my number of groups. I then enter 9 as my number of measurements (as we will compare water and electricity consumption over 9 months). However, I don't know what to enter as an f(V) effect size, since I could not find any conventions for Pillais trace corrected f. Can someone help me out by clarifying what "small", "medium", and "large" conventions there are for f(V)? Previous literature reports small effect sizes, but in Cohen's ds and partial eta squares...so I am lost.

Thank you.

For a power calculation of this nature, I would recommend using the "benchmarks" with the common effect size most closely associated with the test statistic in question. In this case, the Pillai's trace is a statistic built on the eigenvalues of the observational matrix. The eigenvalues can most closely be associated with a variance partitioning. Thus, the comparable "common" effect size would be the coefficient of determination (or the $$\eta^2$$ or partial-$$\eta^2$$, using the terminology of ANOVA/ANCOVA). As such, if I were running a power calculation like this, I would use the benchmarks of 0.02/0.13/0.26 for small/moderate/large (https://imaging.mrc-cbu.cam.ac.uk/statswiki/FAQ/effectSize, the MR $$\eta^2$$ row).