A power flow that doesn’t converge is annoying, to say the least. For one, any information you try to use from a non-convergent solution is moot and questionable (recall that a power flow is a solution of a set of equations representing Kirchhoff’s Laws for electric circuits) since the condition it represents may not be physically possible. So what then to do about it?
There are two basic approaches: (1) to tweak the modeling so that convergence is more likely, or (2) tweak the algorithm to reduce or eliminate the inclination to diverge.
Over years and years of use of (and frustration with) power flow solutions, a number of modeling tweaks have provided some relief from non-convergence. If you only have access to commercial software and do not have influence on the developers to try your ideas for improving the solution, then data input tweaks are all you have. Following is a short, and a not by far comprehensive, list:
- Change the swing bus. The swing bus for the power flow is primarily required by the numerical solution rather than by the physical nature of the power system. Although I have heard planners talk about the swing bus as load-following generation, the power flow solution does not need to have load-following capability. The power flow needs to have at least one swing bus, and you can choose any one you want for numerical reasons. Changing the choice of swing bus may sometimes help convergence where it is otherwise difficult. Usually this works when you are applying a contingency, such as a line outage, to a previously converged power flow.
- Check for local controls. Commercialized power flow software may include features that permit automatic switching of bus shunts, transformer taps, DC controls and phase shifters, or remote control of bus voltages that are in fact modifications of the Newton power flow equations. But rather than add these local controls into the power flow equations, these are applied on a heuristic basis with the typical heuristic being something like — “if the change in /V/ is less than a parameter x, then change the tap, and reformulate the power flow solution”. In essence, the power flow is solved multiple times, once before the local controls are applied, and repeatedly after each heuristic condition for a local control is met. This then brings up two causes for nonconvergence:
- Oscillatory iteration. The solution switches back and forth from one set of heuristics to another without converging. This can occur often with conflicting local objectives where for example a switched shunt and a transformer tap are controlling voltage at the same bus but for different setpoints. To address this, manually check data for settings of transformer taps, phase shifters, switched shunts for oscillatory iteration. Since these controls are local, identifying interacting controls may be a matter of observing the convergence monitor of the power flow and identifying any repetitive adjustments of local controls.
- Bad matrix formulation. This occurs when local controls are applied to change the Jacobian directly and the new Jacobian is near singular. This occurs most frequently when using a generator to control voltage at a remote bus. Removing the remote bus control and reverting to the standard model of having the generator control its own bus eliminates this source of non-convergence.
- Equivalence remote portions of the power flow model to reduce “spread” or diversity. Spread here refers to the range in values for branch impedances on the model base MVA. Low voltage branches tend to have high per unit impedances. When combined with high voltage impedance in the admittance matrix, the large values tend to swamp the low values leading to difficult convergence or non-convergence.
- Reduce the Newton acceleration factor to slow down any potential divergence and remain within feasible space.
- Increase the iteration limit in cases where convergence is slow.
- Increase mismatch tolerance in cases with high diversity, slow convergence or near voltage collapse.
- Open reactive limits, especially at generators. This effectively increases the feasible space.
If you’re thinking of trying any one or several of the above tweaks, be forewarned that these are hit-or-miss affairs. If you’ve run out of ideas, options or patience, it may be time to consider another commercial package with a new formulation or write your own power flow solver.
One of the aspects of the Newton power flow is that it uses complex variables that account for real and reactive power flow. A neat approximation is to assume that only the real power flow matters. Add to this further approximations on the R/X ratio of transmission lines, and the voltage drop in conductors and WALA! you have the DC or linear power flow. The DC power flow is a marvel of simplication. Just MW generation to match MW load and losses, and the resulting equations can be solved directly, not iteratively. No more annoying divergence. The DC power flow allows for a multitude of analyses that can be done quickly without ever even thinking about non-convergence. That’s just analyst heaven.
But the DC does come with a heavy penalty. By ignoring voltage magnitudes and reactive power flow, it incurs an arror. For a more detailed treatment of the issues relating to DC power flows, please see the Pterra Technical article “On Distribution Factors“.
Another interesting tweak is the non-divergent power flow. Years ago, my revered colleague N. Dag Reppen experimented with a technique to slow down divergence in the Newton power flow solution. The basic idea was to monitor the solution progress. If certain parameters indicate that divergence is imminent, an adjustment to the incremental change in the voltage vector is applied to slow it down. If for example, one full step in an iteration was 1.0, then the algorithm reutrned to the previous iteration and reduced the step to 0.5. If that still didn’t slow down divergence, then it reduced the step further to 0.25, and so on. If the step had been reduced to a very small number, then the solution was terminated, just before divergence. It’s important to note here that non-divergent does not mean convergent; it simply means that the solution is stopped. In fact a solution that is stopped by the non-divergent algorithm is always a non-converged case. So what’s the advantage if you still don’t have a converged case? A couple of things:
- The algorithm may allow the solution to slow down just enough to avoid divergence and then continue on to convergence. One way to think of this is that the solution vector just skirts the infeasible space but is still able to reach convergence. In applications to ng>contingency analysis, the non-divergent algorithm is able to reduce the number of non-convergent cases compared to a Newton solution that does not use the algorithm. For some this is major benefit since it reduces the number of power flow cases that you have test manually to figure out why they are not converging.
- Even if the solution is stopped by the non-divergent algorithm, it may still contain useful information. For instance, the typical stopped solution will have mismatches that are high at certain nodes while still being within tolerance at others. the nodes where mismatches are high may be the locus of a reactive deficiency.
The non-divergent algorithm has proved useful enough that it has been adopted by at least one commercial power flow product. However, it still leaves a hole in which non-convergent cases may fall with no hope of recovery.
This leads us to the concept of the Interior Point Newton Method
But that’s all our time for now and we’ll take this interesting topic up at the next installment of this series on Converging the Power Flow. In order not to miss the next blog, please subscribe.