# Power Flow Solution Techniques

(This Blog is an introductory discussion of the AC power flow at a beginner level. Other Blogs on this site discuss more advanced aspects of the power flow, including convergence and alternative solution methods.)

The Power Flow is a steady-state representation of a meshed three-phase electrical network. It is sometimes characterized as a “snapshot” of electrical operating conditions given a set of assumed electrical customers (loads) and supplies (generators) linked together through a transmission system (grid). A single-phase equivalent of the positive sequence network is used since balanced three-phase conditions are assumed.

# Converging the Power Flow 3: Mitigation

by R.  Austria A power flow that doesn’t converge is annoying, to say the least. For one, any information you try to use from a non-convergent solution is moot and questionable (recall that a power flow is a solution of a set of equations representing Kirchhoff’s Laws for electric circuits) since the condition it represents may not be physically possible. So what then to do about it?

# Converging the Power Flow 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.

# On Using Linear Approximation and Distribution Factors

In the accelerated environments of today’s electric energy markets, fast analyses of power flows are a must. Emerging real-time and day-ahead markets require that analysis of infrastructure capacity be performed in a compressed timeframe. Whereas the electric demand of consumers and industry may retain its well-known cyclical nature, varying by time of day, by season and by local weather and social patterns, the supply side of the equation has drastically changed. Competition has engendered even the traditional suppliers of energy to be more flexible and anticipatory to pricing and demand signals, affecting operating and bidding strategy in timeframes that range from the next operating hour to when the next new generation facility can be interconnected. In addition, newer energy sources such as wind and solar power introduce new dependencies which vary hour-to-hour to the supply mix.

Power flows are the tools by which system analysts determine the impact of coincident electric demand and supply bids on the infrastructure of the AC interconnected transmission grid. The power flow tool used most prevalently, known as the AC power flow, requires specialized knowledge to use, is subject to convergence issues and modeling constraints and requires continuous monitoring for data integrity. Databases used for transmission analysis model, for example, the whole Western or Eastern U.S. Interconnection, and parts of Canada and Mexico, maintained through cooperative interchange among many operating entities.

In compressed timeframes, approximate but fast and consistent power flow solutions tend to be preferred over accurate but unwieldy ones. Hence, we find the increasing popularity of linear approximations to the power flow. The reason for widespread use may have to do with the fact that linear methods are much easier to apply, are not subject to convergence issues, and require less data.

### Assumptions

A linear approximation, sometimes also referred to as a DC power flow, applies assumptions to the AC power flow equations that simplify the calculation and makes it non-iterative, and thus not subject to convergence issues that plague the AC power flow. Some of the key assumptions include:

• The resistance component of transmission lines is much smaller than the reactive component.
• Reactive power flows are negligible thus accurate representation of generator reactive capability, reactive demand and power conditioning devices or static var devices is not required.
• Voltage magnitudes are fixed at either nominal or some assumed initial vector.
• Line losses are negligible, constant or can be approximated with a heuristic method that does not require solving the AC power flow. Various methods for handling losses lead to various forms of DC power flows as evidenced by a variety of commercial software.

### Distribution Factors

The DC power flow maintains energy balance by ensuring that the total MW supplied equals the total MW consumed. Any changes to the initial energy balance condition may be approximated by extending the linearization paradigm through the use of distribution factors.

Distribution factors, or dfax, are a measure of the impact of injections and network changes on the grid applied over the initial energy balance condition or base case power flow. They are a function of network topology (noting that the assumption that resistance is a small component relative to the reactance of transmission lines is still applicable here) alone. There are a number of forms of dfax, including:

• Power injection dfax – measure the change in MW flow on transmission branches as a function of an injection of unit MW at a certain node. (See Figure 1) (Note: To maintain energy balance, a positive MW injection is offset by a negative injection, or absorption, at another location. The balancing location is specified either explicitly as a pair to the injection, or explicitly via a “swing bus” defined in the power flow data.) Figure 1:  Illustration of Power Injection DFAX.

• Line outage dfax – measure the change in flow on branches as a function of the outage of a transmission branch. (see Figure 2)  Figure 2:  Illustration of Line Outage DFAX.

• Power transfer dfax – measure the total change in MW flow on a transmission branches for a set of injections and absorptions at various buses.

• Line outage transfer dfax – are a form of power transfer dfax with specific line outages or contingencies. Hence, this form of dfax apply only to the post-contingency state.

Since dfax are linear, when dfax are superposed, the resulting impact is the algebraic sum of the individual impacts. Hence, line outage transfer dfax may be determined from the PIdfax and LOdfax of Figures 1 and 2.

The attractiveness of dfax for real-time and transaction analysis applications may be attributed to its conceptual simplicity and uncomplicated solutions. Software that rely primarily on dfax methods are widely used, not only at control centers but in planning environments.

### Exactly how approximate?

Dfax are approximations. Errors may vary from small to significant. Some factors that impact the magnitude of the approximation error from dfax are:

• Voltage profile of the starting power flow model. Approximation errors increase the larger the deviations from nominal voltage.

• Voltage level of transmission lines. Error is greater for lower voltage lines than higher voltage lines.

• Utilization level of the grid. Error increases with a more severely loaded grid such as may be experienced during the peak load hour of the year.

• Large injections tend to result in larger errors. In combination with the previous bullet, the higher the ratio of the amount of injection with respect to the loading level or demand, the higher the error.

• The presence of underground or submarine cables or series compensated lines also contribute to approximation error.

Sometimes, there is a mitigation effect to the errors in that errors that overestimate may compensate for errors that under estimate, resulting in less net error than expected. This is more a coincidental than an intrinsic characteristic of the method.

In some applications, dfax that are calculated without consideration of losses may be supplemented with loss factors, or lfax, which estimate the change in transmission losses as a function of power injections or line outages. Loss factors are themselves an approximation of the losses that would be determined if a full AC power flow solution were applied.

### Case Studies

Both case studies used for illustrative purposes are based on real-life cases in the Eastern Interconnection (North America).

For our first case study, we consider the system shown in Figure 3, wherein a new power plant A is proposed for interconnection. Substation A is an existing facility that serves as termination for four 161 kV transmission lines. Figure 3:  First case study.

The limiting contingency for power injections from Plant A is the outage of Line 4. Before any new generation, this contingency results in a thermal load on line 3 of 85% of rating. What is the maximum MW injection from Plant A that will not overload line 3?

Using dfax, the predicted limit is 95 MW. Checking with AC power flow solutions, there are three possible answers, as follows:

1. The voltage at Substation A is 102% of nominal before the addition of Plant A. If this voltage is maintained, the limit based on an AC power flow solution is 75 MW.

If Plant A is able to provide the reactive power to raise the voltage at Substation A to 105%, the limit on injected power is 100 MW.

2. If Plant A is a reactive power sink such as an induction generator and reduces the voltage at Substation A to 100% nominal, the maximum power at Plant A is 60 MW.

3. The answer predicted by dfax is close to no. 2 above and appears to assume that Plant A supplies significant reactive power to raise the voltage at Substation A.

For our second case study, we consider a 500 kV interface that is the path for transactions of power from west to east. The limiting constraint is the thermal rating of one of the EHV lines on loss of the other line as shown in Figure 4. For this contingency, the starting case shows an AC power flow loading on the remaining EHV line of 89% of its thermal rating. Figure 4:  Second case study.

A power transfer dfax calculation on the starting case predicts that an incremental transfer of 1869 MW can be supported before the EHV line hits its thermal limit.

Using AC power flows to gradually increment the west to east power transfer shows a resulting transfer limit of 1500 MW. The dfax and AC power flow results are plotted in Figure 5. As in the first case study, the dfax solution assumes voltages are either steady or slightly rising as transfers are increased. In reality, increasing reactive losses on the EHV line results in dropping voltages along the transfer path, as shown in Figure 6. The dfax method in this case thus overestimates the actual transfer limit. Figure 5:  Power Transfer Limits determined by dfax and AC power flow. Figure 6:  Underlying reactive power losses as transfers increase.

### Conclusions

Proponents of linear models have justified their use by the fact that market uncertainty already embeds an inherent approximation in the power flow calculations that is not overcome by slower but more accurate power flow solutions. However, there is the risk of the error from uncertainty adding on to the error from linear approximation moving analytical results further from the reality that it purports to represent.

However, the ease of use and relative simplicity of linear models make it attractive for real-time and short timeframe assessments of power transactions, market dispatch and loading relief. Where reactive losses and voltages are expected to be a factor, additional margins need to be applied to ensure that the linear methods do not give an overly optimistic estimate.

In general, some caution is needed when applying linear models. Where the application does not require fast turnaround or involves future conditions that are not day ahead but months or years ahead, using more accurate methods remains the safe and prudent approach.