6. STOCHASTIC ANALYSIS, MODELLING AND SIMULATION OF
7.4 Multiple reservoir IDP model configuration with CSUDP
7.4.2 CSUDP model parameter sensitivity analysis
For purposes of testing the performance of the configured CSUDP model, it is necessary to establish the sensitivity of the model to various parameters such as choice of initial trajectory,
and corridor width. It is also important to establish the best combination of parameters that ensure that a global optimum can be reached in a given model run. In the application of the customised CSUDP model to the case study, sensitivity analyses to study the performance of the optimisation model was initiated by providing as input: net basin supply data for the year 1900 and a randomly chosen initial trajectory for each lake. The installed capacity of the hydro power plants was also randomly varied from 980 MW to 1530 MW. The value of delx, for the discretisation interval of lake levels was fixed at the permitted minimum value of 0.02 m for Lakes Victoria and Kyoga and 0.025 m for Lake Albert. The splicing option where an initially defined coarse interval of delx is tightened gradually to the desired value during subsequent iterations was not activated in this trial due to the fact that although it can significantly help to reduce execution time, there is a danger of not attaining the global optimum (Labadie, 2003).
During the search for the global optimum in the CSUDP model, a tie breaking option is also available for selection from the user interface for use in resolving situations where non unique optima exist. The program either saves the first optimal solution it encounters as it optimises over the discretized range of lake levels or saves the last non unique solution (Labadie, 2003).
To select the first optimal solution encountered, the user must select the option “first tie value taken” from the user interface or “last tie taken” to return the last non unique solution found.
Although the program is not capable of identifying all possible non unique solutions, by repeating the program execution for each tie breaking option, the user can determine whether a unique solution has been found (Labadie, 2003). For purposes of gaining experience with application of the model the option “first tie taken” was selected in this initial trial. The results in Table 7.3 illustrates the effect of varying the installed capacity on model output in the water year 1900 based on the selected CSUDP user interface parameters.
Table 7.3 Effect of varying installed capacity with the year 1900 Installed Capacity of
Hydrower Plants (MW)
No. of iterations required for converging to optimal solution
Maximum Energy Generation (MWh)
980 262 90396144
1530 263 115000128
The results presented in Table 7.3 required approximately 4 days of computing time on an IBM Lenovo ThinkPad (T60p) Laptop fitted with an Intel Core Duo processor with clock speed of
2.33 GHz and 3GB of RAM. The model was then run with all the hydropower plants online i.e. with a total installed capacity of 1,530 MW and with the adopted CSUDP user inter face settings for delx, splice and tie breaking options left unchanged for the water year 1900, but with three different sets of initial trajectories specified for each lake. Model runs with trial initial trajectories denoted 1,2 and 3 are presented in Table 7.4.
Table 7.4 Effect of varying the initial trajectory with the year 1900 Trial trajectory
number
No. of iterations required for converging
to optimal solution
Optimum Annual Energy Generation
(MWh)
1 263 115000128
2 219 113259605
3 243 113282162
In Table 7.4, Trajectories 2 and 3 were arbitrarily selected while Trajectory 1 corresponds to the median values of the historical net basin supply series in each month for each lake.
Although models run with trajectory 1 required a slightly higher number of iterations to converge, it produced the highest annual energy generation.
Trial Trajectory 1, and all previous CSUDP user inter face settings for delx and splice for the water year 1900 were retained for the next sensitivity test where the objective was to investigate the effect of varying the tie break option. The results in Table 7.5 illustrates the effect of selecting either the “first tie taken” or “last tie taken” in the CSUDP user interface.
Table 7.5 Effect of varying the tie breaking option with the year 1900 Tie option
No. of iterations for converging to optimal
solution
Optimum Annual Energy Generation
(MWh)
First tie value taken 263 115000128
Last tie value taken 266 114957596
The effect of repeating program execution for each tie breaking option in Table 7.5 seems to indicate that non unique optima exist. However, the first optimal solution encountered (first tie taken) is only marginally higher than the last non unique solution (last tie taken) whose location is likely to be a “saddle point”.
To study the effect of initial corridor width upon the convergence behaviour of the CSUDP model with the IDP process as the optimisation algorithm, the model was run for five different corridor width options, with all settings that yielded maximum annual generation in Table 7.5.
Total energy production during the year 1900 and the number of iterations of the IDP model required to converge to the optimal result for the different delx settings are shown in Table 7.6.
As indicated previously the delx setting can be related to corridor width by the relationship delx
= 0.5 x corridor width.
Table 7.6 Effect of varying the delx setting on optimal energy generated for the year 1900
Corridor width option
delx settings (m) No. of iterations
required for converging
to optimal solution
Optimum Annual Energy Generation
(MWh)
Lake Victoria Lake Kyoga Lake Albert
1 0.10 0.10 0.10 49 -
2 0.05 0.05 0.10 85 -
3 0.02 0.20 0.20 79 114657552
4 0.02 0.10 0.10 101 114877152
5 0.02 0.02 0.025 263 115000128
In Table 7.6, corridor width Options 1 and 2 produced infeasible results since the discretisation intervals violate the provisions of Equation 7.9, hence no objective function results were returned. This experiment confirmed corridor width Option 5 as the one that yields the maximum annual energy generation for the case study. However it is also associated with a large number of iterations and therefore requires a lot of computing effort.
An attempt was made to run the IDP model with the settings that yield the optimal operation policies for a period of at least 80 years at a monthly time scale so as define monthly operating rules that relate release from any lake in a given month to the initial storage and net basin supply. At a later stage this approach would involve the determination of generalized lake- reservoir operating rules by multiple regression analysis. Unfortunately, insurmountable difficulties were encountered due to the excessive amount of computer time required for the IDP model to converge to the optimal solution. The length of computing time involved was in the order of several weeks. Due occasional interruption of power supply during these extended model runs, none of the trials was successfully concluded.
In order to reduce the computation times to reasonable time frames by reducing the dimensionality of the IDP model, subsequent analysis was confined to illustration of the prescribed optimal operating policies by the IDP model to a typical water year and evaluation of the defined operation rules of over an annual time step for period 1899 – 2008.