Intensive Programme “Renewable Energy Sources”

This paper is organized as follows. Section 2 describes negative numerical properties of the Newton-Raphson algorithm in more detailed way. In Section 3, each possible stability approach is fully introduced. Testing of stability algorithms is provided in Section 4 comparing the flexibility of individual procedures, reliability of obtained results, total number of iterations, total CPU time, level of robustness and complexity to the original Newton-Raphson code. In Section 5, some concluding remarks close the paper with the best-evaluated stability algorithm obtained.

2. Numerical instability of the newton-raphson method

Convergence problems can appear when solving the networks, which are kept “on the edge” of their operating conditions. These networks, referred to as ill-conditioned power systems, are very sensitive to small changes of state variables (V,θ). Non-convergent behaviour is strongly connected to the reactive power problem of examined power systems. Excessively high loadings, moving final voltage values far away from the initial guess, can cause numerical divergence. Often used PQ loads, i.e. constant voltage-independent loads, have only approximate values, which are higher than real demands in the system. When reaching critical loading values, the network is close to the voltage collapse and cannot be solved due to singularity of the Jacobian matrix.

Sparse network topology (e.g. radial systems) and small number of interconnections between individual areas of the system can also cause rather poor convergence due to insufficient reactive power distribution in the network. Too narrowed operation limits, such as var constraints for PV buses and ranges of tap-changing transformers, along with incorrectly chosen slack bus, negative line reactances (i.e. series capacitors) and long lines are the most common reasons of the numerical instability. Especially, lightly loaded long lines may consume significant volume of transmitted reactive power.

To avoid these problems, it is recommended to primarily switch PQ-loads to Z-loads, replace PV-generators by PQ-generators (with relaxed var limits), disconnect long lines, add additional shunt compensators, release tap ratio limits of LTC transformers, change the slack bus position and divide the network into smaller areas which can be solved separately.

In the industry, several modifications of the network topology are performed during the day. Due to above mentioned factors, the Newton-Raphson method may not be able to perform the load flow analysis of such networks previously successfully solved. Therefore, robust stability algorithms are essential for simulations of majority of load flow cases by the Newton-Raphson method.

3. Stability algorithms for newton-raphson method

1. Start Point Estimation Methods

High sensitivity of the Newton-Raphson method on starting values has been already introduced in Section 2. In load flow analysis, usually chosen voltage start (flat) point is 1.0 for all PQ buses with zero angles ensuring the numerical stability for majority of well-conditioned load flow cases.

Probably the most powerful approaches for better estimations of initial values are One-Shot Fast-Decoupled and One-Shot Gauss-Seidel methods. These procedures, often located in pre-processing part of the Newton-Raphson method, improve the start points to better values which may be closer to the final solution. Unfortunately due to their linear simplifications, networks with high R/X ratios can significantly influence the convergence. Possible approach lies in the use of special technique presented in [9] to slightly amend high R/X rates to avoid divergence. Currently, the One-Iteration Fast-Decoupled algorithm is also used in commercial software package PowerWorld Simulator [7].