by R. Austria

## Divergence Continued

In Part 1, we discussed divergence of the power flow solution. Below is our previous list of reasons for divergence, and additional comments.

1. The power system is going to “blow-up.” — the term “blow-up” just applies to the numerical solution.
2. The power system is in voltage collapse. — yes, if … the conditions for collapse are present in the model.
3. The power system is unstable. — possibly, if one were to associate instability with collapse.
4. The initial conditions defined were bad or poor. — yes, this is true in the sense that the numerical solution is dependent on a good initial voltage estimate.
5. There is something wrong with the software. — now who can tell?
6. There is something wrong with the solution algorithm. — not the formulation but perhaps the implementation into computer code.
7. There is something wrong with the data. — yes, of course. The old concept – garbage in, garbage out.
8. There is something wrong with the analyst. — quite possibly.
9. There is something wrong with my computer. — do you have a 386?
10. The solution hit a bad iteration. — like hitting a crack in the pavement that breaks your differential and off goes the wheel! Not a bad analogy except that its much easier to do the bad iteration than it is to break the car.
11. The solution was unable to resolve the inversion of matrices with very large numbers. — in this day and age of 16 significant numbers, still possible.
12. We hit a bifurcation. — yes, after we exclude all the other simple reasons.
13. We hit a saddle-node bifurcation. — even better.
14. We hit the wall. — like, “this software is driving me crazy, I want to hit my head against the wall?”

Divergence of the power flow solution has traditionally been associated with the singularity of the Jacobian matrix. Since the Newton method requires an inverse of the Jacobian as part of its solution algorithm, singularity of the Jacobian means division by zero. Early researchers have characterized a singular Jacobian as voltage collapse. Today, this is qualified as a steady-state voltage collapse to distinguish from voltage collapse that accounts for dynamic responses of the power system.

Divergence may come from two causes:

• Physical – the initial conditions for the power flow define a voltage collapse condition. There is insufficient reactive power to supply load and losses in some portion of the modeled grid. The Jacobian is singular and the algorithm is attempting a zero divide.
• Numerical – the solution algorithm leaves feasible space and is unable to return. This may come about from a number of causes including a bad initial voltage vector estimate, extreme range of impedance values (known as diversity) or poor formation of or approximations in the Jacobian.

Another factor to consider in this discussion is that many commercial power flow software include modeling of local controls including tap-changing or phase-shifting transformers, switched shunt and series compensation, inter-area and HVDC controls and load shedding. Often, these local controls are implemented within the iteration algorithm with embedded heuristics which can lead to either divergence or an oscillatory solution. The term non-convergence is applied to the general case of numerical divergence or oscillating iterations.

In summary, convergence of the power flow is challenged by the various phenomena, both physical and numerical, that may lead to alternate solutions characterized by divergence, blow-up, voltage collapse or oscillatory iterations.

## Next in this series: Mitigation Techniques

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