Instability when considering investment in Markov Chain model

[AngeBlanchard]

Hi Oscar,

I'm trying to model an hourly electric dispatch. I consider a full year of 365 nodes, one node per day with 24 hours within. Renewable production being uncertain, I model their production through stochastic variables that take their values at the beginning of each day. Moreover, I model disruption that can occur on gas import routes via a Markov chain, i.e. either we can import gas to burn it into the turbines, either we cannot.

With this being said, I try to find the optimal gas storage capacity to invest in, in order to hedge against the risk of gas supply disruption. The idea being that a storage asset can help to handle a crisis situation when gas imports are no longer available.

This is how I tried to handle this problem (I only write the code section of interest) :

# The first node is deterministic and here to decide the investment level in UHS
        if t == 1
            @constraint(sp, CAPA_STORAGE.out >= CAPA_STORAGE.in) # we allow the investment in gas storage
            for i in 1:24
                JuMP.fix(ξ_wind[i], Ω[1][1][(t-1)*24 + i]; force = true)
                JuMP.fix(ξ_pv[i], Ω[1][2][(t-1)*24 + i]; force = true)
                JuMP.fix(ξ_load[i], Ω[1][3][(t-1)*24 + i]; force = true)
            end # here the electric generation from renewables is forced deterministically for the 24 hours of the first node
            @stageobjective(sp, 
            sum( all generating costs at hour i, for i in 1:24) # pseudo code to keep it short
            + CAPA_STORAGE.out * INV_COST_STOCK_H2 # investment cost
            ) 
    # Otherwise, the investment is prohibited
        else
            @constraint(sp, CAPA_STORAGE.out == CAPA_STORAGE.in) # storage capacity can't change anymore
            SDDP.parameterize(sp, Ω) do ω 
                for i in 1:24
                    JuMP.fix(ξ_wind[i], ω[1][(t-1)*24 + i]; force = true)
                    JuMP.fix(ξ_pv[i], ω[2][(t-1)*24 + i]; force = true)
                    JuMP.fix(ξ_load[i], ω[3][(t-1)*24 + i]; force = true)
                end
                @stageobjective(sp, 
                sum( all generating costs at hour i, for i in 1:24)
                ) 
            end 

My problem is: even though I eventually get policies of rather good quality (2% discrepancy with the lower bound after 100 iterations), the level of investment is not stable at all and when I train the model with a low investment cost in gas storage (sensitivity analysis) I sometimes get policies that invest less than in cases whereby costs are higher ! Which is nonsense obviously.

Do you have any idea of what's happening ? Is it just that I need more iterations ? Or is there a better alternative to model investment in this case ?

Thanks a lot,
Ange

PS: how do you manage to get the code in markdown with all the colors ? Even with ``` it does not seem to work for me ?

[odow]

Do you have any idea of what's happening ? Is it just that I need more iterations ?

Yes. This is a pretty common issue in the literature that is often overlooked: just because the costs are converging, does not mean that the primal policy has converged 😄.

See https://odow.github.io/SDDP.jl/stable/explanation/theory_intro/#Termination-criteria

With 365 nodes, I'd guess you'd need thousands of cuts before the policy is "nice." Certainly many, many, more than 100.

See also: #178

how do you manage to get the code in markdown with all the colors ? Even with ``` it does not seem to work for me ?

You need ```julia. I've edited your post, so take a look.

[AngeBlanchard]

Well, then I guess I'll have to start the training phase and go to chill outside for hours haha.
Thanks for your help and the provided links ! I close.
Ange