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.