Simplifying River Chains for Dynamic Programming

Introduction

If too many reservoirs are modelled in the dynamic programming phase of water value calculation, a performance hit may occur. This is the well known ‘problem of dimensionality’ in dynamic programming. It might be necessary to simplify the problem to avoid slow DP (Dynamic Programming) solutions. The following is a method for doing this on a chain of stations.

River Chain Simplification

Figure 1

Figure 1 represents a chain of reservoirs and stations, with the nodes represented by circles.

S_i = the storage at node i I_i = the inflows to node i Max_i = the storage capacity at node i f_i = the flow from node i to node i + 1 f_IN = the flow into node 0 M_i = the generating potential from node i to node i + 1 (MW/cumec)

Instead of modelling each and every reservoir separately on this chain we will apply a constraint which requires all reservoirs to be filled to the same proportion of their capacity. So at any time:

1. S_i = α Max_i, for some alpha.

The behaviour of this chain is then determined by the inflows, f_IN and the change in α. Change in storage ΔS_i is given by change in α.

2. ΔS_i = Max_i , = change in α

Change in storage is also related to flows and inflows:

3. ΔS₀ = I₀ + f_IN – f₀

ΔS_i = I_i + f_i – f_{i +1} , i > 0

Combining 2. and 3. Gives:

4.

And Generation is given by

5.

Defining

6.

Equation 5 becomes:

7.

Figure 2

We now have the dynamics of the constrained sub-system of stations. The next step is to represent these dynamics with a simpler sub-system so the DP process can model it. The simpler system is shown in figure 2.

The first node has no storage, so there is only the storage at the second node to model. In order for this sub-system to behave in the same way as the constrained sub-system it is necessary for the generation and outflow to be equal at any time.

In the simplified sub-system flows are given by:

8. f_A = f_IN + I_A

f_B = f_A + I_B - Max

= I_A + I_B + f_IN - Max

Since f_B must equal f_n for all inflow, f_IN and values, comparing Equations 8 and 4 gives:

9. ( from the inflows components )

10. ( from the component )

Generation is given by:

11. G* = M_A f_A + M_B f_B

= M_A I_A + M_B (I_A + I_B) + (M_A + M_B) f_IN - M_B Max

Since G* must equal G for all inflow, f_IN and values comparing Equations 11 and 7 gives:

12. M_A + M_B = σ₀ ( from the f_IN component )

13. ( from the component )

14. ( from the inflows components )

Combining 12 and 13 gives:

15.

Combining 14 and 9 gives:

16.

17.

Equations 10,13,15 enable a specification of the simplified sub-system to be calculated. Equations 16 and 17 find the inflow figures to use in the system.

Once the results of a simulation with the simplified model are known they can be converted back to results for the original sub-system by calculating and α:

= ( f_A + I_A - f_B ) / Max

α = S / Max

Equations 1 and 4 then give the storages and flows in the original sub-system.

Figure 3

If M_A = 0 for the system, then Equations 16 and 17 will not solve. This will occur only if there is zero storage in all but the last reservoirs. It is inappropriate to try and simplify this configuration, unless there are no inflows below the first node, in which case the simplified sub-system is shown in figure 3.

Where

18. Max = Max₀

19. I = I₀

20. S = S₀

21. M = σ0

Allocation of Spills

The flow figures calculated above will match both outflow and generation, if all flows are used for generation. If however the flows are divided into generating and spill flow figures then the match will only be with potential generation, or the generation that would have occurred had all channels been generating with the given power conversion factors. It is necessary than to allocate the total f_A and f_B figures into f_i figures for both spill and generation. If

22. f_A = f ^g_A + f ^s_A

23. f_B = f ^g_B + f ^s_B

where f ^g is the flow used for generation and f ^s is the spill. The potential generation is:

24. G* ^P = M_A f_A + M_B f_B

The real generation is given by

25. G* ^R = M_A f ^g _A + M_B f ^g _B

If the resulting flows for the original sub-system are also divided into spill and generation flows:

26. f_i = f ^g_i + f ^s_i

G* ^P will be equal to the potential generation of the f_i flows:

27. G* ^P = G ^P = Σ_k = _{0.. n} f_k M_k

What we need as well as this is for the real generation figures to be equal:

28. G* ^R = G ^R

If we force all spill to be the same proportion of the total flow in the solution. i.e.

29. c

Then real generation in the original sub-system will be

30. G^R = μ Σ_{k = 0..n} f_k M_k

Combining equations 27, 28 and 30 give:

29. μ = G* ^R / G* ^P

This value for μ will provide spill and release figures for all nodes in the original sub-system, which match both real and potential generation.