# 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

**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

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

- The power system is going to “blow-up.”
- The power system is in voltage collapse.
- The power system is unstable.
- The initial conditions defined were bad or poor.
- There is something wrong with the software.
- There is something wrong with the solution algorithm.
- There is something wrong with the data.
- There is something wrong with the analyst.
- There is something wrong with my computer.
- The solution hit a bad iteration.
- The solution was unable to resolve the inversion of matrices with very large numbers.
- We hit a bifurcation.
- We hit a saddle-node bifurcation.
- 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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