An Affine Arithmetic Approach to the Power Flow Problem Using MCP Formulation
Electric power systems continue to increase in complexity because of the deployment of market mechanisms, the integration of renewable generation and distributed energy resources (e.g., wind and solar), and the introduction of dynamic demand from sources such as electric vehicles and price sensitive loads. These revolutionary changes and their added uncertainty and dynamicity call for significant modifications in power system operations models including unit commitment (UC), economic load dispatch (ELD) and optimal power flow (OPF). Planning and operation of these "smart" electric grids are expected to be impacted significantly, because of the intermittent nature of various supply and demand resources that have penetrated into the system with the recent advances.
The main focus of this thesis is on the application of the Affine Arithmetic (AA) method to power system operational problems. The AA method is a very efficient and accurate tool to incorporate uncertainties, as it considers all the information between dependent variables, by considering their correlations, and hence provides less conservative bounds compared to the Interval Arithmetic (IA) method. Moreover, the AA method does not require assumptions to approximate the probability distribution function (pdf) of random variables.
In order to take advantage of the AA method in power flow analysis problems, first a novel formulation of the power flow problem within an optimization framework that includes complementarity constraints is proposed. The power flow problem is formulated as a mixed complementarity problem (MCP), which can take advantage of robust and efficient state-of-the-art nonlinear programming and complementarity problems solvers. Based on the proposed MCP formulation, it is formally demonstrated that the Newton Raphson (NR) solution of the power flow problem is essentially a step of the traditional General Reduced Gradient (GRG) algorithm. The solution of the proposed MCP model is compared with the commonly used NR method using a variety of small-, medium-, and large-sized systems, such as IEEE 14-bus, 30-bus, 57-bus, 118-bus and 300-bus test systems and real 1211-bus and 2975-bus systems in order to examine the flexibility and robustness of this approach.
The MCP-based approach is then used in a power flow problem under uncertainties, in order to obtain the operational ranges for the variables based on the AA method considering active and reactive power demand uncertainties. The proposed approach does not rely on the pdf of the uncertain variables and is therefore shown to be more efficient than the traditional solution methodologies, such as Monte Carlo Simulation (MCS). Also, because of the characteristics of the MCP-based method, the provided bounds take into consideration the limits of real and reactive power generation.
The thesis also proposes a novel AA-based method to solve the OPF problem with uncertain generation sources and hence determine the operating margins of the thermal generators in systems under these conditions. In the AA-based OPF problem, all the state and control variables are treated in affine form, comprising a center value and the corresponding noise magnitudes, to represent forecast, model error, and other sources of uncertainty without the need to assume a pdf. The AA-based approach is benchmarked against the MCS-based intervals, and is shown to obtain bounds close to the ones obtained using the MCS method, although they are slightly more conservative. Furthermore, the proposed algorithm to solve the AA-based OPF problem is shown to be efficient as it does not need the pdf approximations of the random variables and does not rely on many iterations to converge to a solution. The applicability of the suggested approach is also tested on a large real European system.