# Power Flow

### The steady-state foundation of interconnection analysis

### Power flow and interconnection

Circuit analysis and modeling are fundamental to the design and operation of nearly all electrical systems. From electronic chips to the electric grid, circuits of all sizes and complexity are modeled to allow engineers to advance, simulate, and (hopefully) improve the devices and systems that influence modern society. In the world of electric power systems, steady-state circuit analysis via power flow simulation is a core analytical element by which nearly every technical and economic decision is made. This is particularly relevant for contemporary generator interconnection analysis, because these simulations most often determine and motivate the selection of system upgrades required to safely and reliably energize new renewable energy and storage projects on the grid. Power flow simulations allow engineers to perform “what-if?” checks to assess the impacts of these projects on grid reliability under a variety of scenarios.

In this article, we’ll delve into the details of power flow, focusing primarily on its role and influence in the generator interconnection context. In particular, we’ll discuss AC and DC power flow and the roles these analysis types play in interconnection studies. While the discussion will be more of a technical deep-dive than some other articles in this series, we will not get into mathematical formulations. For mathematical background on power flow, we recommend checking out the following books: Power System Analysis and Design (Glover, Overbye, Sarma), Power System Analysis (Grainger, Stevenson), and Power Generation, Operation, and Control (Wood, Wollenberg, Sheblé).

### AC power flow

At its core, a power flow simulation seeks to determine the “steady state” of a grid at a particular snapshot in time. That is, how the grid appears after any short-term transients or oscillations have settled based on the balance of power between generation and load. Think of the grid as being like a pond: if you throw a rock in a still pond, the splashing and wave patterns that result are the short-term transients, and once they die down, the pond is still again. The still pond is the steady state. To perform a power flow simulation, every interconnected component in a grid must be represented and accurately modeled according to its AC physics, a simulation process commonly referred to as AC power flow.

The state of the grid is defined by the AC voltage magnitudes and angles at every node (“bus”) in the network, and AC power flow seeks to determine this grid state via the solution of a large set of non-linear, physics-based network equations, often referred to as power balance equations. The power balance equations capture the supply, demand, and transfer of real and reactive power throughout a power system. They are solved iteratively using numerical solvers like Newton-Raphson, which is generally the most popular in AC power systems analysis (other solvers exist and have specific roles in analysis). The iterative solution is necessary because not all variables in the system are exactly known initially – an “initial guess” of a system’s conditions is attempted, and solution iterations are performed in an attempt to satisfy the power balance equations to within some small tolerance. For example, system thermal losses is an initially unknown variable. It should be noted here that solver iterations do not represent the evolution of a grid’s physical behavior over a period of time – on its own, power flow does not capture any time domain phenomena. Going back to the pond analogy: the iterations of a power flow simulation do not represent the splashes and waves caused by the rock. They are purely a set of steps taken by the numerical solver to go from the initial steady state of the still pond to the later steady state of the still pond after the rock is thrown in it. Note that the initial steady state and the final steady state are typically not the same – you’re modeling an event representing a change to the state of the grid. These changes can be minor or major. If a large meteorite crashed into the pond instead of a small rock, displacing a large amount of water, the final steady state of the pond would look drastically different from the initial steady state. Other types of simulations that capture aspects of the transient behavior, such as dynamics/stability or electromagnetic transient analysis, may be conducted in an interconnection study.

An AC power flow calculation allows engineers to determine both real and reactive power flows throughout the network, serving as the basis by which system reliability is evaluated and validating system design decisions (e.g., facility sizing and upgrade determination). An AC power flow simulation also considers device control settings when solving, including switched shunt adjustment, transformer tap adjustment, phase shifter adjustment, and HVDC tap adjustment. These controlled devices typically exist to regulate certain voltage magnitudes or power flows in the grid. Power flow simulation has been a major contributor to the development of modern electric grids, and the fundamental approach has been used and trusted for nearly a century.

Assuming model parameters are accurate, AC power flow solutions capture the physics of a grid at steady-state. However, it is important to understand that multiple AC power flow solutions can (and often do) exist based on the non-linear nature of the problem – the simulation only attempts to find a valid solution, and that solution need not be unique. The solution found depends on many factors, including the initial guess and the numerical algorithms and heuristics applied during a simulation. It is possible for a solver to find (“converge to”) a poor solution that, while mathematically valid, is not realistic. Further, it is possible that the solver does not converge to any solution at all. This non-convergence might mean there is no valid steady state solution to the problem being modeled (or in other words, the grid has collapsed and is in blackout conditions), or it might mean the solver was just not robust enough to find it starting from the initial conditions. It is sometimes difficult to tell the difference, but separating real collapse issues from spurious numerical issues is critical in interconnection studies as project developers do not want to be cost allocated to mitigate problems that aren’t really there. Proposed mitigations for non-convergence can include construction of new transmission lines, which can cost hundreds of millions of dollars to build. Such mitigations often make a project no longer viable. Pearl Street’s SUGAR™ software is designed to be as robust as possible, and is often used to find power flow solutions where other tools fail, and can eliminate time-consuming and manual investigation of root causes of non-convergence.

### DC power flow

In real grid analysis, AC power flow solutions are very often difficult and/or non-trivial to perform. Simulating AC power flows repeatedly (e.g., thousands or millions of times) and with potentially many significant system changes requires substantial time and computational resources. This is especially true for contingency analysis. These and other challenges forced power engineers to recognize a trade-off in modeling accuracy versus simulation speed, and they devised an alternative power flow approach to provide approximate solutions that are sufficient in some applications: DC power flow.

DC power flow serves as an approximation of AC power flow, and can provide approximate results in a fraction of the time it takes to perform an equivalent AC solution. Generally, this is done via an approximation and linearization of the network power balance equations according to a set of correlative assumptions about system state variables. Most importantly, it only considers real power flows in the system and ignores reactive power. (Decoupled power flow, another approximate solution approach, can be used when approximating reactive power.) DC power flow’s formulation also does not consider device control.

DC power flow is certainly not a replacement of AC power flow – it is simply a solution approach with its own set of valid applications – but it does have its advantages. Its primary advantage is its ability to perform simulations quickly, so many more analyses can be performed in a much shorter period of time compared to AC power flow. Additionally, by its definition, a DC power flow solution always converges. DC power flow is often used to screen for potential overloads in transmission lines and transformers, and anything flagged as a possible cause for concern is typically verified with AC power flow.

### AC + DC: Applications for interconnection

Most interconnection studies at some point utilize both AC and DC power flow solutions. Additionally, both methods are useful prior to a project entering an ISO or utility queue to help evaluate viability. Significant decisions are made based on the results of both AC and DC power flow simulations, and it is important to understand their functionality and the roles they play to make better-informed decisions about project development and when performing interconnection study-related analysis. With this in mind, we’ll walk through a few interconnection-related applications where AC and DC power flow approaches are used.

*Network upgrades*

Perhaps the foremost objective of utilizing power flow solutions in interconnection analysis is to determine the need for network upgrades to mitigate voltage and thermal violations and system collapse (manifested as valid non-convergence). AC power flow is most often employed to identify the location and extent (i.e., size) of a necessary upgrade, and an AC solution is necessary to guide non-convergence mitigation. These solutions are found as the system is analyzed subject to stresses known as contingencies. We’ll cover contingency analysis in more detail in a later article.

When a constraint is identified and mitigated, DC power flow is typically employed to determine how the upgrade is cost allocated to various projects and their owners/developers. A DC power flow can quickly and reliably determine linear sensitivity factors, commonly known as distribution factors. We will dive into this in future articles as well. The determination of network upgrades highlights the interdependency of AC and DC power flow to provide accurate and relevant outputs for renewable energy development.

*Deliverability*

Project deliverability is often determined via DC power flow solutions. This metric seeks to demonstrate the ability (or inability) of a project to deliver power to load locations within a specific operator or utility footprint subject to contingency events. Deliverability requirements and criteria are often defined regionally.

*Analysis inputs and model creation*

AC power flow is used to create the inputs necessary for analysis at other stages of an interconnection study or in a prospecting analysis. The first steps in any interconnection study involves a model building process. These are performed using AC power flow, and the models are used in all subsequent steady-state analysis, including contingency analysis and study components where DC power flow may be employed. AC power flow is also used to initialize models that serve as the basis for transient stability portions of interconnection studies.

*Site prospecting*

When looking for potential sites for a new project, developers will often use a combination of AC and DC power flow analyses to evaluate site viability and capacity. Prospecting exercises primarily leverage DC power flow, because accuracy can be less critical in the early stages of a project’s development, especially when compared to the time available to evaluate a large geographic area for a potential project. In these early stages, speed can be more critical than simulation precision, so developers will use DC power flow-based techniques to approximate site capacity and study results, including contingency analysis and the subsequent identification of possible mitigations and cost. Note that the more aggressive a developer wants to be with taking facilities to their limits, the more important the accuracy difference between AC and DC may become.

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