Diagnosing Load Flow Convergence Issues

Load flow studies form the basis for the steady-state analysis of power systems and are performed for a wide range of applications from planning to operations. The studies are typically conducted using computer software packages such as PSSE, DIgSILENT PowerFactory, ETAP, PowerWorld and SKM Power*Tools.

The load flow (or power flow) calculation solves for four steady-state electrical quantities (at each bus and branch element):

  • Voltage V at each bus
  • Phase angle Φ at each bus
  • Active power flow P through each branch element
  • Reactive power flow Q through each branch element

We can use these four basic electrical quantities to calculate other quantities of interest, for example:

  • Apparent power S through each branch element
  • Current I flowing through each branch element
  • Loading of branch elements (e.g. transformers, cables, etc) based on continuous and short-term equipment ratings
  • Voltage drop between buses
  • Total system load
  • Total amount of generation dispatched
  • Network losses

These calculated quantities are of interest to electrical network operators and design engineers in order to ensure that the power system operates efficiently and within limits. The rest of this article will talk about the generic structure of a commercial power flow solver / algorithm and how to debug convergence problems.

Algorithm Structure

Power flow solution algorithms are typically comprised of inner and outer loops:

  • The inner loop calculates the main iterations of the solver (e.g. Newton-Raphson, Gauss-Seidel, etc) in order to minimise the power (or current) mismatch equations.
  • The outer loop is used to check and satisfy control conditions (e.g. auto tap changer / shunt control) and limiting behaviour (e.g. generator reactive power limits).

Note that automatic tap or shunt control is sometimes included in the inner loop, but only when continuous tap/shunt steps are used. When automatic tap and shunt control is incorporated in the outer loop, the power flow algorithm will attempt to switch the positions of the taps and shunts at the end of every inner loop iteration to satisfy the control conditions (e.g. bus voltage or branch power flow setpoints).

Diagnosing Convergence Issues

When a load flow fails to converge, the first step to diagnosing the problem is to identify whether the algorithm has failed in the inner or outer loop.

An inner loop problem signifies that the Newton-Raphson equations could not be solved, which could be due to ill-conditioning, voltage stability issues, etc. This usually results in a program error message stating that the algorithm has either diverged or stagnated.

Inner loop problem (from DIgSILENT PowerFactory)

On the other hand, when the problem is in the outer loop, the Newton-Raphson equations can be solved, but control conditions can’t be met (e.g. voltage setpoint could not be reached with transformer taps at maximum / minimum settings or are hunting between two tap steps). Typically, a maximum number of outer loop iterations must be reached before an error is reported by the program.

Outer loop problem (from DIgSILENT PowerFactory)

Tackling Inner Loop Problems

As mentioned earlier, inner loop problems suggest that the system is ill-conditioned and the Newton-Raphson algorithm either diverges or is stuck in a local minimum (i.e. stagnated). Mismatches in generation, poor distribution of generator dispatch and local voltage stability issues are among the common causes for an inner loop problem:

  • Mismatches in generation and load: when there is a large mismatch in generation and load (i.e. over- or under-generation), the discrepancy in active power has to be supplied or absorbed by the slack element. Depending on the location of the slack element, this can lead to large power flows from one part of the network to another, which can cause imbalances in the voltage profile and lead to convergence problems.
  • Distribution of generator dispatch: similar to mismatches in generation, poor distribution of dispatched generation can lead to large unwanted power flows from one part of the network to another.
  • Local voltage stability: relates to busbars reaching the nose point of the PV curve (voltage collapse). This could be very localised due to large loads placed at weak radial parts of the network without sufficient reactive compensation, or could affect larger areas where multiple buses have depressed voltages.

Some approaches to resolving inner loop problems are as follows:

  • Turn off enforce reactive power limits option: most programs have an option to limit the reactive power output of PV buses (generators, SVCs, etc) so that they are within their reactive power capability curves. When a PV bus reaches limits, it is converted to a PQ bus in the outer loop with the reactive power set to its limits and the inner loop is run again, which can then fail due to voltage stability reasons (i.e. the PV bus is supporting the local voltage, but with limits enforced, voltages can go wild). Turning this option off can help diagnose which generators, SVCs, etc are supplying / absorbing large amounts of reactive power and their associated problem buses.
  • Change the location of the slack element and distribute generator dispatch: so that there are more balanced flows through the network. Some programs have a distributed slack bus option that automatically caters for mismatches in generation by distributing the slack across all in-service generators according to their nominal ratings.
  • Switch in or add shunt compensation: in order to support weak buses that may have voltage stability problems.

Tackling Outer Loop Problems

Outer loop problems are normally associated with voltage control issues from the following elements:

  • Generators – reaching their reactive power capability limits
  • SVCs – reaching their reactive power capability limits
  • Transformer automatic tap changers – reaching maximum tap settings without satisfying voltage setpoint tolerances
  • Shunt controllers – reaching maximum shunt steps without satisfying voltage setpoint tolerances

Some approaches to resolving outer loop problems are as follows:

  • Increase the maximum number of outer loop iterations: particularly for larger systems, the outer loop solution may take many iterations.
  • Change voltage control setpoints and tolerances: the bus voltage setpoints and tolerances can be adjusted to help the solution along.
  • Lock taps and/or shunts: automatic transformer tap and shunt adjustment options can be de-selected to lock the taps and shunts at their initial positions.
  • Reset taps and/or shunts: the initial tap and shunt positions may not be well configured for the particular scenario. Reset tap positions by setting all transformers to neutral tap and re-configure initial shunt positions to better control system voltages (e.g. if you know that a bus has a low voltage, put some capacitors in service).

Importance of Inertia in Island Power Systems

The accurate modelling of inertia in an islanded power system is crucial when attempting to integrate low or inertia-less generation into the network, particularly those from intermittent sources, e.g. solar PV, converter-fed wind turbines, etc.

In the context of synchronous generators, the term “inertia” generally refers to the kinetic energy in the rotating mass of a generator shaft. The inertia depends on the speed of rotation and mass of the shaft, i.e. the heavier the shaft and faster the rotational speed, the higher the inertia. Under normal operation, the instant a new load is applied to the system, rotational energy (inertia) is converted into electrical energy to supply the load. As energy is removed from the system to supply the load, the speed of rotation (or frequency) decreases. This is commonly referred to as the inertial response and occurs before primary frequency control actions (e.g. from governors) take place. As a result, inertia has a direct impact on transient frequency deviations resulting from sudden changes in generation and load.

Effects of varying inertia on system frequency

In large interconnected systems, these frequency deviations are minor since the instantaneous mismatches in generation and load are very small relative to the amount of synchronous generation dispatched. However, in small island systems, these frequency deviations are much more significant and can even lead to frequency collapse.

By virtue of their remoteness and lack of other resources, the small island power systems that are normally found in island nations and archipelagos typically have high penetration of diesel engine generation. As diesel engines generally have low inertia, the inertia constants selected during system modelling can have a large effect on frequency swings. For example, consider the load acceptance (25% load step event) of a 1.5MVA diesel generator with varying inertia constants:

The simulations show that maximum frequency swings can vary from 1.235Hz up to 3.548Hz depending on the inertia constant selected.

Small power systems are usually equipped with under-frequency load shedding (UFLS) systems to prevent network collapse during frequency swings and active power deficit events (e.g. trip of generator). Integrating inertia-less sources such as solar PV plants would displace synchronous generation and thus reduce the total inertia in the system. Grid simulation studies are performed to predict whether or not the UFLS system is at risk of operating during normal day-to-day fluctuations of the solar PV system, and this requires fairly accurate modelling of the system inertia.

Selecting appropriate inertia values

Generator inertia values are usually found in vendor / manufacturer data sheets and are often expressed as a moment of inertia quantity (e.g. in kg.m2 or slug.ft2). While some software packages can accept these quantities directly as inputs, other programs require that they are converted into per-unit inertia constants (H).

In some cases, particularly in older systems or networks with temporary generators, the inertia data is not available and must be estimated. Based on our database of actual equipment data, the following inertia values can be used as guidance: