Modelling And Optimal Control Of The Cascaded Kainji And Jebba Hydroelectric Power System In Nigeria

The cascaded hydroelectric power stations at Kainji and Jebba on the River Niger play a vital role in the Nigerian electricity supply that has been in a state of crisis for over four decades. Sustainable solutions to this problem require strategic solutions like the Roadmap for Power Sector Reform of 2010 and tactical components that address knotty issues of operations and management that have so far been either outsourced or completely omitted. The latter devolves into addressing the nearly intractable problem of determining control strategies that optimize suitable performance indices constrained by the dynamical equations describing the reservoirs. The aim of this research is to develop dynamical models and control strategies for optimal power generation at Kainji and Jebba hydroelectric power stations. The objectives were to: (i) develop a mathematical model (incorporating reservoir characteristics, inputs such as inflows with outflows due to power generation, flood-control discharges and evaporation); (ii) numerically solve the resulting equations and determine the parameters using measured data; and (iii) compute control laws for optimal operation of the system.rnA nonlinear dynamical model derived using energy conservation and flow continuity conditions was solved numerically employing an Adams-Moulton-Bashforth predictor-corrector integrator with a Runge-Kutta starter (AMBPCRKS). Reservoir parameters were then estimated using information obtained from a database created using National Control Centre (NCC) data and observations of evaporation and power generation during the study period. The optimal control problem, formulated as the minimisation of a quadratic cost functional subject to the system’s dynamical equations, was then solved by direct and indirect methods. The indirect method, which is based on Pontryagin’s Minimum Principle (PMP), resulted in a two-point boundary value problem (TPBVP) which was subsequently solved by an innovated multiple shooting technique referred to as the Progressive Domain Extension Method (PDEM) and a conjugate gradient algorithm. rnThe findings from the study were that:rni. integrating model equations using the AMBPCRKS in an EXCEL-VBA® environment achieved an accuracy of the fifth order;rnii. the calibrated model using reservoir parameters with actual inflows and outflows were within 2% of observed values, thus making it a dependable tool for planning, operations and training;rniii. approximate finite-dimensional optimal controls computed by the steepest descent and conjugate gradient approaches converged to nearly identical values although the latter technique took fewer iterations; andrniv. PMP-based controls extracted through the TPBVP can be solved using PDEM and conjugate gradient approaches with a stop criterion of 10-6, with the results being almost identical and practically realisable.rnIn conclusion, the research successfully addressed an extant frustrating difficulty confronting the operators of the cascaded power stations by solving an optimal control problem using available information and innovative computational methods realized on the ubiquitous EXCEL-VBA® platform. It is recommended that the ownership of the stations and government be encouraged to adopt the results for planning, operations, training of engineers and management as well as to support further work to complete the control process.

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