Text

theses

The increased demand on energy resources worldwide, along with the expectations of future depletion of fossil fuels: coal, oil, and gas, have encouraged considering the renewable energy as an alternative to traditional resources. Renewable resources including wind, solar, hydro, biomass, and geothermal are naturally abundant and can be harnessed to meet the energy demand without exacerbating the environmental contamination.

During the last few decades, wind energy has become one of the fastest growing and most promising renewable energy resources, due to their availability and minimal impact on the environment. As variable-speed wind turbine generators (WTGs) became advent, they gained increasing popularity due to their ability to work efficiently over wide ranges of wind speeds. The double fed induction generators (DFIGs) are widely used in variable-speed wind energy systems, since they consume less reactive power, inflict less mechanical stress on turbines, and allow decoupled control of the active and reactive power. This dissertation focuses on model reduction and optimal control of variable-speed wind turbines with DFIG systems.

To reduce the complexity of the power system model when a large number of WTGs is integrated to the power grid, and to obtain a simplified model that adequately simulates such systems, the methods of balancing transformation and singular perturbations were utilized to reduce the order of a DFIG-based wind turbine model. We show that the order of the considered wind turbine model can be reduced from eight to six via the balancing transformation. Further reduction via the aforementioned method results in a significant increase in the error bound. In contrast, the method of singular perturbations shows that the order of the model can be further reduced to four, or even to two, and still provide very good approximations to the system model, in terms of its transient step response. Moreover, we show that the reduction in model order achieved via singular perturbations is superior from the optimal performance point of view to that achieved via balancing when the linear-quadratic near-optimal controllers are considered, and when the wind turbulence and a large-signal disturbance are applied to the system.

The state-space model of wind farms of different sizes, under different wind speed conditions, was also studied in this thesis. Model order reduction methods: balanced truncation, balanced residualization, cross Gramians, and singular perturbation were applied to the one-mass model to obtain simplified equivalents to wind farms of different sizes. This helps in reducing the computational complexity when controlling such systems. Examining the controllability and observability of the system, in both the vector and diagonal forms of the input control matrix, showed a considerable loss of controllability and observability in the case of the latter form.

The main obstacles that are associated with wind energy conversion systems are their intermittent behavior and dependence on the geographical location and weather conditions. Such randomness and uncertainty introduce nonlinearities when modeling the system dynamics. Therefore, designing optimal and robust controllers are crucial to deal with all the nonlinearities and uncertainties associated with wind energy systems. Based on time scale decomposition, an optimal controller for a DFIG-based wind turbine was designed by decomposing the algebraic Riccati equation (ARE) of the singularly perturbed wind turbine system into two reduced-order AREs that correspond to the slow and fast time scales. In addition, we derive a mathematical expression to obtain the optimal regulator gains with respect to the optimal pure-slow and pure-fast, reduced-order Kalman filters and linear quadratic Gaussian (LQG) controllers. Using this method allows the design of the linear controllers for the slow and fast subsystems independently, thus, achieving complete separation and parallelism in the design process. This solves the corresponding numerical ill-conditioning problem, and reduces the complexity that arises when the number of wind turbines integrated to the power system increases. The reduced-order systems were compared to the original full-order system to validate the performance of the proposed method when a wind turbulence and a large-signal disturbance are applied to the system. In addition, we showed that the similarity transformation does not preserve the performance index value in the case of Kalman filter and the corresponding LQG controller.

Studying the wind turbine with DFIG system as a high order singularly perturbed system led to the introduction of a new recursive algorithm for solving the algebraic Sylvester equation that defines the cross Gramian of singularly perturbed linear systems. The cross Gramian matrix provides aggregate information about the controllability and observability of a linear system. The solution was obtained in terms of reduced-order algebraic Sylvester equations that correspond to the slow and fast subsystems of the singularly perturbed system. The rate of convergence of the proposed algorithm is O(epsilon), where epsilon is a small singular perturbation parameter that indicates the separation of slow and fast state variables. Several real physical system examples were solved to demonstrate the efficiency of the proposed algorithm.

ETD_11351

Ph.D.

Includes bibliographical references

External ETD doctoral

doi:10.7282/t3-98ne-4q70

The author owns the copyright to this work.