## OpenDSS Solution Techniques

The normal circuit solution technique in the EPRI OpenDSS program may be concisely written as a simple fixed-point iterative method:

Vn+1 = [Ysystem]-1 IPC(Vn) n = 0, 1, 2, … until converged

In words, after building Ysystem, the process starts with a guess at the system voltage vector, V0, and computes the compensation currents from each power conversion (PC) element to populate the IPC vector. Using a sparse matrix solver, the new estimate of Vn+1 is computed as shown. This process is repeated until a convergence criterion is met. For distribution systems, convergence is typically achieved in 4-10 iterations for the initial power flow solution and 2-3 iterations for subsequent solutions in a time series. The Ysystem matrix is not refactored until there is a major change in the system configuration. Thus, this method is very fast for a quasi-static time-series (QSTS) simulation.

Lines, transformers, etc., are called power delivery (PD) elements and are completely modeled by their primitive nodal admittance matrix. Loads are considered to be PC elements that are modeled by a Norton equivalent in which the constant, linear Norton admittance is included in Ysystem . The nonlinear characteristics of the load, if any, are represented by the compensation current sources in the Norton equivalent. This formulation can accommodate a wide variety of load models in which IPC can be expressed as a consistent function of Vn. This formulation was adapted from the program’s heritage as a harmonics solver in which all the loads were constant impedance.

As with most fixed-point iterations, the initial guess has to be fairly close to the final answer. This is easy to achieve by simply performing a direct solution of the nodal admittance matrix with IPC = 0 for loads and generators. This allows for initialization of even the most unusual circuit and transformer configurations.

The circuit solution formulation is essentially the same whether the program is performing a power flow, short circuit, harmonics, stray voltage, or dynamics solution. This technique works for any number of phases and is not very sensitive to circuit configuration.