# Power Flow Computations

The typical goal of PowerModels is to build a JuMP model that is used to solve power network optimization problems. The JuMP model abstraction enables PowerModels to have state-of-the-art performance on a wide range of problem formulations including those with discrete variables and complex non-linear constraints, such as semi-definite cones. That said, for the specific case of power flow computations, in some specific applications performance gains can be had by avoiding the JuMP model abstraction and solving the problem more directly. To that end, PowerModels includes Julia-native solvers for AC power flow in polar voltage coordinates and the DC power flow approximation. This section provides an overview of the different power flow options that are available in PowerModels and under what circumstances they may be beneficial.

## Generic Power Flow

The general purpose power flow solver in PowerModels is,

This function builds a JuMP model for a wide variety of the power flow formulations supported by PowerModels. For example it supports,

• ACPPowerModel - a non-convex nonlinear AC power flow using complex voltages in polar coordinates
• ACRPowerModel - a non-convex nonlinear AC power flow using complex voltages in rectangular coordinates
• SOCWRPowerModel - a convex quadratic relaxation of the power flow problem
• DCPPowerModel - a linear DC approximation of the power flow problem

The typical ACPPowerModel and DCPPowerModel formulations are available via the shorthand form solve_ac_pf and solve_dc_pf respectively.

The solve_pf solution method is both formulation and solver agnostic and can leverage the wide range of solvers that are available in the JuMP ecosystem. Many of these solvers are commercial-grade, which in turn makes solve_pf the most reliable power flow solution method in PowerModels.

Note

Use of solve_pf is highly recommended over the other solution methods for increased robustness. Applications that benefit from the Julia native solution methods are an exception to this general rule.

### Warm Starting

In some applications an initial guess of the power flow solution may be available. In such a case, this information may be able to decrease a solver's time to convergence, especial when solving systems of nonlinear equations. The _start postfix can be used in the network data to initialize the solver's variables and provide a suitable solution guess. The most common values are as follows,

For each generator,

• pg_start - active power injection starting point
• qg_start - reactive power injection starting point

For each bus,

• vm_start - voltage magnitude starting point for the ACPPowerModel model
• va_start - voltage angle starting point for the ACPPowerModel model
• vr_start - real voltage starting point for the ACRPowerModel model
• vi_start - imaginary voltage starting point for the ACRPowerModel model

The following helper function can be used to use the solution point in the network data as the starting point for solve_ac_pf.

Warning

Warm starting a solver is a very delicate task and can easily result in degraded performance. Using PowerModels' default flat-start values is recommended before experimenting with warm starting a solver.

## Native AC Power Flow

The AC Power Flow problem is ubiquitous in power system analysis. The problem requires solving a system of nonlinear equations, usually via a Newton-Raphson type of algorithm. In PowerModels, the package NLSolve is used for solving this system of nonlinear equations. NLsolve provides a variety of established solution methods. The following function is used to solve AC Power Flow problem with voltages in polar coordinates with NLsolve.

PowerModels.compute_ac_pfFunction

Computes a nonlinear AC power flow in polar coordinates based on the admittance matrix of the network data using the NLsolve package. See the NLsolve documentation for solver configuration parameters.

Returns a solution data structure in PowerModels Dict format

source

compute_ac_pf will typically provide an identical result to solve_ac_pf. However, the existence of solution degeneracy around generator injection assignments and multiple power flow solutions can yield different results. The primary advantage of compute_ac_pf over solve_ac_pf is that it does not require building a JuMP model. If the initial point for the AC Power Flow solution is near-feasible then compute_ac_pf can result in a significant runtime saving by converging quickly and reducing data-wrangling and memory allocation overheads. This initial guess is provided using the standard _start values. The set_ac_pf_start_values! function provides a convenient way of setting a suitable starting point.

Tip

If compute_ac_pf fails to converge try solve_ac_pf instead.

## Native DC Power Flow

At its core the DC Power Flow problem simply requires solving a system of linear equations. This can be done natively in Julia via the \ operator. The following function can be used to solve a DC Power Flow using Julia's built-in linear systems solvers.

PowerModels.compute_dc_pfFunction

computes a linear DC power flow based on the susceptance matrix of the network data using Julia's native linear equation solvers.

returns a solution data structure in PowerModels Dict format

source

The compute_dc_pf method should provide identical results to solve_dc_pf. The primary advantage of compute_dc_pf over solve_dc_pf is that it does not require building a JuMP model. This results in significant memory saving and marginal performance saving due to reduced data-wrangling overhead. The primary use-case of this model is to compute the voltage angle values from a collection of bus injections when working with a formulation that does not explicitly model these values, such as a PTDF or LODF formulation. The solve_opf_ptdf_branch_power_cuts utility function provides an example of how compute_dc_pf is typically used.

This solver does not support warm starting.

## Branch Flow Values

By default none of the Power Flow solvers produce branch flow values. If needed, these can be computed with the network data functions,

Both of these methods require a complete network data with a valid voltage solution for computing the branch flows. For example, one common work flow to recover branch flow values is,

result = solve_ac_pf(network, ...)
# check that the solver converged
update_data!(network, result["solution"])
flows = calc_branch_flow_ac(network)
update_data!(network, flows)

Internally compute_ac_pf and compute_dc_pf utilize an admittance matrix representation of the network data, which may be useful in other contexts. The foundational type for both representations is AdmittanceMatrix{T}.

PowerModels.AdmittanceMatrixType

Stores data related to an Admittance Matrix. Work with both complex (i.e. Y) and real-valued (e.g. B) valued admittance matrices. Only supports sparse matrices.

• idx_to_bus - a mapping from 1-to-n bus idx values to data model bus ids
• bus_to_idx - a mapping from data model bus ids to 1-to-n bus idx values
• matrix - the sparse admittance matrix values
source

In the case of an full admittance matrix and simplified susceptance the type is AdmittanceMatrix{Complex{Float64}} and AdmittanceMatrix{Float64}, respectively.

The following functions can be used to compute the admittance matrix and susceptance matrix from PowerModels network data.