High Performance Algorithms Enabling Real-Time Security Assessment of Sustainable Electric Power Systems

Power systems are transitioning towards low-carbon emission Renewable Energy Sources (RES) such as wind and photovoltaic power. This type of generation is highly dependent weather conditions, which can cause fluctuations in the system operating point. These fluctuations can happen rapidly as more and more generation is based on RES, making the traditionally timeconsuming offline approaches for stability and security assessment insufficient.

Therefore, there will be a need for real-time stability and security assessment methods. This PhD is part of the Security Assessment of Renewable Power Systems (SARP) project. The main goal of the project is to push forward the technology needed to ensure secure and stable system operation of future power systems with a high share of RES based production.

In the PhD the factorization method for Thévenin equivalent computations has been investigated, which led to the development of a fast and efficient method for computations of Thévenin equivalents. The voltage stability boundary monitoring method acounting for non-linearities in Thévenin voltages have been omtimized using a binary search with polynomial fitting.

Lastly an algorithm for fast refactorization using hierarchical matrices have been developed. Thévenin equivalent computations is used in a long range of methods recently developed for real-time stability and security assessment methods. It is therefore important that these can be computed fast and efficiently to ensure the methods can operate in real-time.

Thévenin equivalents can be computed using an LU factorization to optimally invert the bus admittance matrix. Both direct and incomplete factorization methods can be used for the computations. Clark Kent LU factorization (KLU) is almost twice as fast as the standard LU factorization with no cost of accuracy.

However, factorization time is seen to be a negligible part of the computations. The most inefficient part is to compute the impedance matrix for the current sources of the system. Incomplemte LU factorization (ILU) reduced the fill-in of the coefficient matrix for computing Thévenin voltages. ILU reduces the runtime for computing Thévenin voltages at the cost of accuracy.

As runtimes have a hard time competing with the direct methods for acceptable error levels the applicability of ILU is severely limited. The impedance matrix for the current sources is used to determine the coefficient matrix for computing Thévenin voltages, however this is inefficient to compute due to the density.

Therefore the factor-solve method, which avoids these computations, is developed by using KLU. KLU factorization brings a matrix on block triangular form, where each block consists of all nodes, that are connected. For a system of linear equations it uses block back substitution to find the solution.

The factor-solve method uses block back substitution to determine the Thévenin voltages and thereby avoids computing the coefficient matrix. This makes it possible to determine the Thévenin voltages in linear time compared to the reference implementation, which using the coefficient matrix had close to quadratic complexity.

Computations for Thévenin impedances is still quadratic, however these can easily be parallelized resulting in runtimes lower than 3 seconds for systems up to 30.000 buses, whereas Thévenin voltages can be determined in less than 6 milliseconds. The Thévenin impedances only depend on system topology and therefore doesn’t need to be recomputed as often.

The distance to the boundary of voltage instability can be determined more accurately by taking in to account non-linearity in Thévenin voltages as the load changes. The maximum power transfer to a non-controlled load is used to determine this distance, and repeated changes in load impedance is required to find this.

The maximum power transfer is found between the current load and the Thévenin impedance (which is the boundary normally determined by the impedance match criterion). Doing this naïvely by splitting the interval evenly results in poor runtime and accuracy. Therefore, a binary search is used instead. This determines the load impedances that the power transfer should be computed to find the maximum power transfer.

The binary search is further improved by combining the search with a polynomial fitting. Feasible points are fitted to a second order polynomial, and the maximum of the polynomial is found. By utilization of parallelization it is possible to determine the margin for 2.000 non-controlled loads in a 3.000 bus system in less than 6 seconds, while also determining the maximum power transfer accurately.

Even though runtimes are improved the complexity is still quadratic, therefore the scaling needs to be improved. The computations are dominated by a large block in the block triangular form and therefore the computations for this is sought to be optimized. The voltage stability method requires the matrix to be refactorized for each change in load impedance and therefore efficient refactorization would improve runtimes.

The hierarchical matrix structure constructs a hierarchical tree for a chosen level of discretization, where the leaf nodes and edges between these represent the matrix to be factorized. The factorization is then done by eliminating the leaf nodes and compressing fill-in generated to the parent level by approximating using a low-rank approximation such as Singular Value Decomposition.

To do fast refactorization the effect a change in the diagonal have on the computations is tracked in the initial factorization. This makes it possible to do refactorization, where only the nodes affected by the change is recomputed. This lowers runtime considerably, however work still remains for the factorization to work for all systems and levels of discretization.