Siemens Energy, Inc.
Power Technology
Issue 111
Dealing with Power Flow Solution Difficulties
Feng Dong
Staff Software Engineer
feng.dong@siemens.com
Ted Kostyniak
Principal Software Engineer
ted.kostyniak@siemens.com
Baldwin Lam
Senior Manager Consulting
baldwin.lam@siemens.com
Power Flow Study and Solution Method
A power flow solution is the most frequently performed type of study of electric power systems that is
applied in day-to-day operation as well as in short-term and long-term planning. It can be described by
the following statement:
“Given the power consumptions at all nodes (buses) in the electric power system and the power
productions at all the generating facilities, find the power flowing on each branch (line and transformer) of
the interconnecting network.”
PSS®E has several steady state power flow solution methods. Each of them iteratively adjusts the
voltages at all buses in the system in order to satisfy Kirchhoff’s current law and the system’s load
demand. In other words, the net amount of real and reactive power flowing into and out of each bus
should be practically zero (or within a small tolerance) and the total power supply and demand in the
system should be balanced. There are two families of solution methods available in PSS®E:
 Gauss-Seidel methods include regular and modified Gauss-Seidel. Generally, these solution methods
are very fast per iteration, but the overall time to reach a converged solution may be slower than
when using the Newton-Raphson based methods.
 Newton-Raphson methods include the Full Newton-Raphson, Decoupled Newton-Raphson and
Fixed-slope Decoupled Newton-Raphson. These methods use a first order expansion of the power
flow equations in the solution, which is repeated until convergence is reached. The overall solution
times are generally fast and the solutions are able to converge to within reasonably small tolerances
(such as 0.1 MW and Mvar).
However, any of these five solution methods can fail to reach convergence, resulting in either a diverged
or un-converged condition. It is often a challenge for the PSS®E user to figure out the cause of the
solution failure.
Causes of Solution Failures
First, more than one iterative method may be required. A failure in convergence may come from the
failure of the power flow solution method, when the solution leaves the feasible space and is unable to
return. It is rare, however, to find a situation that has a solution, but the bus voltage vector cannot be
found by the application of one or more of the five methods. There are many problems that are difficult or
impossible to solve using a single iterative method, but can be solved readily by the successive
application of more than one method.
Second, divergence may be due to an infeasible operating point. The power flow may represent a voltage
collapse condition, where there is insufficient reactive power to supply load and losses in some portion of
the system, or where the load demand exceeds the transfer capability of the network. The voltage
Power Technology
March 2012
collapse condition is often associated with a singular Jacobian matrix (the solution matrix used in the
Newton-Raphson method).
Third, non-convergence of the power flow can be caused by too many control adjustments in the network
models, such as tap-changing or phase-shifting transformers, switchable shunt capacitors or reactors,
area interchange control, HVDC lines and FACTS device controls. The controls can operate in a discrete
or continuous adjustment mode, each with its local control objective. The power flow solution
automatically adjusts the control settings to meet the desired bus voltages or branch flows of the
respective controlling equipment. Incompatible or poorly coordinated control adjustments can result in
non-convergence.
Addressing Solution Failures
In power flow solutions, the network models should represent the physical power system accurately, in
terms of the network topology and parameters of all components in the system. PSS®E provides several
functions to facilitate data checking, such as BRCH for checking abnormal branch parameters and TPCH
for checking irregular transformer tap control settings. It is often difficult to identify the data errors, due to
the large number of parameters in a power system and the wide range in magnitude of each data
category. Some typical solution convergence problems related to power flow data or modeling are listed
in the following:
 Bad initial values of the bus voltage vector. The Newton-Raphson solution methods, particularly the
Full Newton-Raphson algorithm, are sensitive to the initial values, sometimes causing a solution to
blow up abruptly. Adjusting the initial voltage magnitudes and phase angles at the problematic buses
(e.g., by not leaving them at the default values of one per unit voltage and zero degree angle) may
aid the solution process.
 Narrow control bands for transformers. A narrow voltage band for a voltage controlling transformer
could cause the tap adjustments to oscillate from iteration to iteration; a narrow flow control band for a
phase shifting transformer may cause a solution to abort when the Jacobian matrix becomes singular.
 Improperly scheduled flow of phase shifters that may lead to a singular Jacobian matrix.
 Branches with high R/X ratio. The Decoupled Newton-Raphson solution methods may have difficulty
converging in cases with branches that have high R/X ratio (transmission lines and transformers
typically have low R/X ratios).
Because the Newton-Raphson solution methods are used most often by PSS®E users, some practical
techniques to address the non-convergence issues encountered with these methods are discussed
below.
 Change the power flow solution parameters. Reduce the Newton-Raphson solution acceleration
factor to slow down any potential divergence and keep the solution within the feasible region; or,
increase the iteration limit in cases where convergence is slow; or, increase the mismatch tolerance
in cases with slow convergence or near voltage collapse; or, increase the automatic adjustment
threshold tolerance to activate switched shunt adjustments sooner in the iteration process to allow
better voltage control before voltage collapse starts to occur.
 Change the power flow control options. Disable local controls to avoid the effects of small
adjustments to the Jacobian that may lead to divergence; or, ignore the reactive power limits of the
generators to increase the feasible solution space; or, prevent the automatic adjustments of
switchable shunts and transformer taps after a certain number of iterations to avoid oscillation.
 Change the network model. The system swing is meant to absorb the difference between total
system generation and the sum of system loads and losses. From a mathematical standpoint, any
generator bus could be assigned as the system swing. Sometimes, changing the swing bus may help
convergence; or, change the remote voltage control to local voltage control for more effective control;
or, change the control mode of some switchable shunts from discrete to continuous temporarily, in
order to better understand the reactive power needs of the system.
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Power Technology
March 2012
 Apply the non-divergent power flow solution method. The non-divergent Newton-Raphson power flow
solution is designed to terminate the iterative process before the bus voltage vector is driven to a
state where large mismatches and unrealistic voltages are present. The resulting voltage vector
obtained using the non-divergent solution method, although not sufficiently accurate to represent a
converged power flow solution, can often provide a relatively good indication of the state of the
network, in terms of severe voltage depression and reactive power deficiency in some parts of the
power system.
 Use a voltage-dependent load model. A load model may consist of several components in the power
flow solution: constant power, constant current (load changing in direct proportion to voltage), as well
as constant impedance or admittance (load changing according to the square of voltage). By using a
voltage-dependent load model, a drop in voltage in the power system would result in a reduction in
load, thus reducing the real and reactive power needs of the system. This may allow the system
voltages to stabilize, instead of collapsing. PSS®E has a user-adjustable constant power load
characteristic threshold (PQBRK), which will automatically change the constant power load
representation to a voltage-dependent representation in the power flow equations if the voltage at a
load bus falls below the threshold. Of course, the load models should reflect the characteristics of the
actual system and should not be arbitrarily changed.
 Use the optimal power flow. A power flow case that fails to converge, or is near or in voltage collapse
can be represented as an optimal power flow problem. The PSS®E OPF solution automatically
changes control variables to achieve the best solution with respect to a stated quantitative
performance measure, i.e., an objective function and a set of variable constraints to satisfy. If the
voltage collapse is due to a deficiency in reactive power support, PSS®E OPF can be used to identify
the location and size of reactive compensation devices.
In addition, the power flow solution convergence monitor can provide useful information for identifying the
cause of non-convergence (e.g., whether the solution is oscillating because of conflicting controls,
solution convergence is too slow, or the initial bus voltage estimates are poor), so that appropriate
measures can be taken to assist solution convergence. One or more of the above techniques may be
applied, as needed.
The above are just some ideas for the PSS®E user to consider when they encounter power flow solution
difficulties. It should be stressed that the solutions obtained after changing the power flow data or model
may no longer truly represent the original power system. But, instead of just getting a message that the
power flow solution is un-converged or has diverged, the PSS®E user can now have a better grasp of the
problems in the power system.
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