Converging the Power Flow

Bread and Butter

The power flow is the bread-and-butter tool of power system analysts of large and small-scale transmission systems. It is used in the day-to-day operations of the grid to determine potential congestion, transmission loading relief and need for generation re-scheduling, among others. It is likewise used in short-term and long-term planning to study the potential for thermal overloads, voltage violations and voltage collapse. So it would be disconcerting if a tool of this importance and widespread application should fail, which on occasion it does.

This blog discusses the issues that lead to failures of the power flow, the techniques and methods used to address or, at least, get around the failures, and a novel approach to ensuring convergence of the power flow.

Power Flow Formulation

The power flow formulation is based on the application of Kirchhoff’s laws to meshed electric networks. The basic concept is that the sum of all flows into each and every node should be equal to zero. Note that the flows are in complex form; i.e., they comprise of real and reactive components, or MWs and and MVARs. If there are n nodes, then there are n complex equations. The resulting system of equations is non-linear. Solution methods are primarily iterative with the objective of reducing the sum of flows in all nodes to some acceptably small value known as the mismatch tolerance.


Convergence is the state when all nodes have met the mismatch tolerance. The main power flow solution methods are:

  • Gauss-Siedel method – updates the voltage one node at a time until all nodes are within the mismatch tolerance. Generally a slower method than the others that follow.
  • Newton-Raphson method – uses a first order expansion of the power flow equations to approach convergence. Generally faster than the Gauss-Siedel method and able to converge to small tolerances. However, the method is prone to the phenomenon of divergence, when mismatches increase instead of decrease from iteration to iteration. This occurs when the solution vector exits the feasible solution space at any point during the algorithm. Once outside feasible space, the solution gradient tends to further increase mismatches leading to solutions that “blow-up” in the numerical sense, characterized by zero or very high voltages. Several variations on the Newton-Raphson are in use, including:
    • Fast Decoupled – separates the loosely linked real and reactive components of the power flow equations in order to speed up solution
    • Fixed Newton – does not update the first order approximation matrix (known as the Jacobian) every iteration to reduce computational burden
    • Non-divergent power flow – applies a reduction to the Jacobian multiplier whenever the solution appears to exit feasible space. In certain situations, this may prevent divergence, or at least stop it before blow-up.
  • Interior-Point Newton method – forces the solution inside feasible space to avoid divergence. The interior point method uses a second order expansion of the power flow equations as a basis for its algorithm. the method is more computationally intensive than either the Gauss Siedel or Newton-Raphson but is less susceptible to numerical divergence.


Divergence of the power flow has been taken to mean different things by system analysts, including:

  1. The power system is going to “blow-up.”
  2. The power system is in voltage collapse.
  3. The power system is unstable.
  4. The initial conditions defined were bad or poor.
  5. There is something wrong with the software.
  6. There is something wrong with the solution algorithm.
  7. There is something wrong with the data.
  8. There is something wrong with the analyst.
  9. There is something wrong with my computer.
  10. The solution hit a bad iteration.
  11. The solution was unable to resolve the inversion of matrices with very large numbers.
  12. We hit a bifurcation.
  13. We hit a saddle-node bifurcation.
  14. We hit the wall.

One important step towards understanding the divergence of a power flow solution is to be able to determine whether the divergence is numerical or physical in nature. We’ll take this up in the next blog. Please subscribe.

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