6. STOCHASTIC ANALYSIS, MODELLING AND SIMULATION OF
7.3 Multiple reservoir IDP model formulation
η = overall generation efficiency, typically taken to be 0.85, Q = is the turbine discharge in (m3.s-1), and
H = is the gross head in metres i.e. 16 m as determined from the feasibility studies.
*
F
t = are optimal objective functions from period t to the end of the cycle T, given the current state of the reservoir storage levels (S
i t, 1 ), andr
t = is the immediate return (hydropower generated) during period t as defined by Equation 7.1.In Equation 7.3, optimization begins at some point in the future (Stage T) and proceeds backwards in time to the present. The rewards associated with stage T+1 are a boundary condition taken to be zero. Equation 7.3 is solved recursively, subject to upper and lower limits on discretized reservoir storage levels and releases. State transformation equations for the lake reservoirs, which are governed by the principle of continuity are presented in Equations 7.4 – 7.6.
For Lake Victoria:
1,t 1, 1t 1,t 1,t
S S
I R
(7.4) For Lake Kyoga:2,t 2, 1t 2,t 1,t 2,t
S S
I R R
(7.5)For Lake Albert:
3,t 3, 1t 3,t 2,t 3,t
S S
I R R
(7.6)where:
,
I
i t = net basin supply or net inflow to reservoir i during period t, (i = 1, 2, 3) (MCM).,
R
i t = release from reservoir i during period t, (i = 1, 2, 3) (MCM)., 1
S
i t ,S
i t, = beginning and ending period storage levels for reservoir i during period t, ( i = 1, 2, 3) (MCM).Upper and lower limits on storage levels and reservoir releases are imposed for purposes of safeguarding against damage to shoreline, river bank settlements, infrastructure and satisfaction of other riparian interests such as and navigation. The Incremental Dynamic Programming (IDP) model is an iterative method in which Equation 7.3 is used to search for
an improved trajectory among discrete states in the neighbourhood of an initial trajectory.
Figure 7.6 illustrates the procedure for a single reservoir (one dimension of the state space). It commences with selection of a trial trajectory in the state-stage domain chosen in such a way that the specified set of initial and final storage levels of the system yield a release that falls within the acceptable range of release constraints (
R R
min,
max) of the system and then evaluates the return function (power generated) in the neighbourhood, or so called "corridor," of this trajectory. At the end of each iteration step a locally improved trajectory is obtained and then used as the trial trajectory in the next step. The step is repeated until a near optimal trajectory that determines a feasible operating policy of the system is found.
Figure 7.6 Illustration of the state increment, initial trajectory and corridor width concepts in one dimension for the IDP model (Allen & Bridgeman, 1986).
Thus instead of searching for the optimal operating policy over the entire state-stage domain as in the case of traditional dynamic programming, the IDP process circumvents the “curse of dimensionality” by narrowing down successive searches for the optimal rule. When implemented for a three lake-reservoir system the IDP procedure constructs a symmetrical corridor around trial trajectories of initial lake levels at each stage and subsequently searches for the optimal trajectory and corresponding objective function value within the corridor.
Searching incrementally over the range of permissible storage levels, the backward moving recursive equation, finds the best initial storage volumes of the lake-reservoirs that maximizes the value of the hydropower generation over the planning period for feasible scenarios of the final storage volumes in each time period to complete a cycle. The computation steps for a
Figure 7.7 Incremental dynamic programming procedure (after Nandalal & Bogardi, 2007).
As the iterations are carried out, improvement in the objective function reduces. Since less superior trajectories are obtained as the iterations progresses for a given corridor width, a
No Start
Establish initial trajectory and its return
Begin the first cycle with the initial trial trajectory
Begin the first iteration of the cycle
Construct a corridor around the trial trajectory
Begin next iteration with current corridor width Begin next cycle with a
reduced corridor width
Find the optimal trajectory and its return within current corridor Set current optimal
trajectory as trial
Set current optimal trajectory as trial
Last cycle?
Stop Current optimal
trajectory represents the optimal solution of the
problem
No
Convergence of algorithm
fulfilled?
Yes
Yes
Current optimal trajectory represents a near optimal solution of
the problem Last cycle?
Last iteration?
Yes
Yes No No
convergence criteria is set and computed upon completion of each iteration within the corridor.
The reference to less superior trajectories here is to indicate that the improvement in trajectories reduces but the trajectories themselves improve. A maximum number iterations is also predefined which must not be exceeded within each corridor search. The maximum number of iterations is usually a multiple value of the number of time periods (months) for a given cycle (year). The convergence criterion that needs to be satisfied before a corridor width search can be changed is defined by Nandalal & Borgardi (2007) as:
* *
1
* *
1 0
i i
i
OBF OBF OBF OBF
; i = 1,2,… … …, I (7.7) where
*
OBFi = return from the optimal trajectory for the i th iteration of a given cycle (I = 0,1,2,…), and
I = maximum number of iterations per cycle.
In a situation where successive values of
i, do not indicate a significant improvement on the value of the return function, the iterative process is terminated and the next cycle is begun with a reduced corridor width around the optimal trajectory of the completed cycle. Upon completion of the final iteration in each cycle, another convergence criterion is set to determine the convergence of the algorithm towards the optimal solution. Nandalal and Borgardi (2007), define this final convergence criteria as:* *
1
* 1
j j
j
OBF OBF
OBF
; (7.8)
where
*
OBFj = return from the optimal trajectory for the j th cycle (j = 1, 2, 3, ……) λ = arbitrary value of the convergence criteria that terminates the IDP algorithm.
The final trajectory, i.e. the beginning of month reservoir storage levels that yield the optimum return, is identified as the solution of the optimization problem. The backward looking version of the dynamic programming evaluates many reservoir storage levels that are not attainable
from the given initial storage levels, resulting in a large computational burden and slow convergence rates.