Power flow, or load flow, is widely used in power system operation and planning. The power flow model of a power system is built using the relevant network, load, and generation data. Power engineers are required to plan, design, and maintain the power system to operate reliably and within safe limits. Numerous power flow studies are required to ensure that power is adequately delivered at all time despite normal load fluctuations and undesirable events such as contingencies. Daily fluctuations in the power system operations cause power flow mismatches at busbars. As a result, the busbars voltage magnitude and angle adjust instantly until an equilibrium is reached between the load and the transmitted power. This new equilibrium point can also be obtained from simulation using power flow methods.
Different Techniques to Solve Power Flow Equations
Outputs of the power flow model include voltages at different buses, line flows in the network, and system losses. These outputs are obtained by solving nodal power balance equations. Since these equations are nonlinear, the following iterative techniques are commonly used to solve this problem.
Power flow studies consist in determining the voltage magnitude and angle at each busbar until there is an equilibrium in active and reactive power at which point power mismatches are insignificant. Calculating the final values of the busbars voltage is not always straightforward especially for heavily loaded and complex systems.
The starting point of solving power flow problems is to identify the known and unknown variables in the system. Based on these variables, buses are classified into three types: slack, generation, and load buses as shown in the following table:
|Types of Bus||Variables|
|Real Power, P||Reactive Power, Q||Voltage Magnitude, |V|||Voltage Angle, δ|
|Generator Bus (PV)||Known||Unknown||known||Unknown|
|Load Bus (PQ)||known||known||Unknown||Unknown|
Newton Raphson method is a numerical technique for solving non-linear equations. It is often classified as iterative root finding scheme. The reason it is called root finding is, it is geared towards solving equations like f(x)=0 (or f(x)=0). The solution to such an equation, call it x* (or x*), is clearly a root of the function f(x) (or f(x)). The first order Newton-Raphson (NR) method is considered as the state of the art for power flow calculations. This method has been widely used in industry applications.
It is iterative because it requires a series of successive approximations to the solutions. The procedure is generally as follows. First, guess a solution. Unless we are very fortunate, the guess will be, of course, wrong. So, we determine an update to the "old" solution that moves to a "new" solution with the intention that the "new" solution is closer to the correct solution than was the "old" solution.
A key aspect to this type of procedure is the way we obtain the update. If we can guarantee that the update is always improving the solution, such that the "new" solution is in fact always closer to the correct solution than the "old" solution, then such a procedure can be guaranteed to work if only we are willing to compute enough updates, i.e., if only we are willing to iterate enough times.
In Gauss Seidel method, the computations appear to be serial. Further, each component of new iteration depends upon all previously computed components. Updates cannot be done simultaneously. In addition to this, new iteration depends on the order in which equations are examined. If this ordering is changed, the components of new iteration (and not just their order) also change. These limitations persuade engineers and researchers to go for Newton Raphson method.
In high voltage transmission systems, the voltage angles between adjacent buses are relatively small. In addition, X/R ratio is high. These two properties result in a strong coupling between real power and voltage angle and between reactive power and voltage magnitude. In contrary, the coupling between real power and voltage magnitude, as well as reactive power and voltage angle, is weak. Considering adjacent buses, real power flows from the bus with a higher voltage angle to the bus with a lower voltage angle. Similarly, reactive power flows from the bus with a higher voltage magnitude to the bus with a lower voltage magnitude.
Fast-decoupled power flow technique includes two steps:
As the size of matrix becomes very large for a big bus system so for faster and less memory allocation, we prefer decoupled load flow where we take P independent of V and Q Independent of δ, and thus those Jacobian elements are taken as zero.
Comparison of the Load Flow Methods
| Load Flow Method|| Speed|| Accuracy|| Solution Robustness|
| Gauss-Seidel|| Very Slow|| Approximate|| Robust|
| Newton Raphson|| Slow|| Accurate|| Robust|
| Fast Decoupled|| Fast|| Accurate|| Less Robust|
Industry Application of Power Flow Methods
Commercial power systems are complicated. It is not possible to analyze power flow through hand calculations. Physical models of power systems were analyzed through network analyzers in laboratories between 1929 and early 1960. Afterwards, invention of digital computers replaced the analog methods with numerical methods. Initially, linear methods were proposed to analyze power flow analysis. Among these, Cramer's method, Gauss elimination and LU factorization are notable. However, these methods cannot handle complex, nonlinear and big power systems. Therefore, iterative techniques i.e., Gauss Seidel method, Newton Raphson method are developed to solve complex power systems.
There are a number of very high-quality commercial power flow programs on the market today, some of which include those developed by the Electric Power Research Institute (EPRI), Power Technologies Incorporated (PTI), Operation Technology, Inc., and EDSA. Most of today's commercial software packages are menu-driven from a Windows environment. Few of the commonly used software are discussed briefly as below:
 Power flow methods for improving convergence. P. J. Lagace. 24 December 2012. IECON 2012 - 38th Annual Conference on IEEE Industrial Electronics Society.
 Design and Power Flow Analysis of Electrical System Using Electrical Transient and Program Software. Raheel Muzzammel, Ibrahim Khail, Muhammad Huzaifa Tariq, Abubakar Muhammad Asghar, Ali Hassan. Energy and Power Engineering, 2019, 11, 186-199
 Development of Newton-Raphson Power-Flow Method Based on Second Order Multiplier. Mohamed Abokrisha ; Ahmed Diaa ; Ali Selim ; Salah Kamel. 2017 Nineteenth International Middle East Power Systems Conference (MEPCON).
 Analysis and Comparison of Load Flow Methods for Distribution Networks Considering Distributed Generation. M. Sedghi, M. Aliakbar-Golkar. International Journal of Smart Electrical Engineering, Vol.1, No.1, Winter 2012.
 Fast Decoupled Load Flow for Security Assessment. G. Z. Ben-Yaacov, G. Mijne and K. Struwig. Power System Security Assessment. Electricity Supply Commission, Johannesburg, South Africa.
 William D. Stevenson, Jr. 1982. Elements of Power Systems Analysis. New York: McGraw-Hill.