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{{Short description|Numerical analysis of electric power flow}}
In [[power engineering]],
Power-flow or load-flow studies are important for planning future expansion of power systems as well as in determining the best operation of existing systems. The principal information obtained from the power-flow study is the magnitude and phase angle of the voltage at each [[busbar|bus]], and the real and reactive power flowing in each line.
Commercial power systems are usually too complex to allow for hand solution of the power flow. Special
In addition to a power-flow study, computer programs perform related calculations such as [[short-circuit]] fault analysis, stability studies (transient and steady-state), [[unit commitment]] and [[economic dispatch]].<ref>{{Cite book | last1 = Low | first1 = S. H. | chapter = Convex relaxation of optimal power flow: A tutorial | doi = 10.1109/IREP.2013.6629391 | title = 2013 IREP Symposium Bulk Power System Dynamics and Control - IX Optimization, Security and Control of the Emerging Power Grid | pages = 1–06 | year = 2013 | isbn = 978-1-4799-0199-9 | s2cid = 14195805 }}</ref> In particular, some programs use [[linear programming]] to find the ''optimal power flow'', the conditions which give the lowest cost per [[kilowatt hour]] delivered.
A load flow study is especially valuable for a system with multiple load centers, such as a refinery complex. The power
In term of its approach to uncertainties, load
==Model==
An
Usually analysis of a three-phase power system is simplified by assuming balanced loading of all three phases.
==Power-flow problem formulation==
The goal of a power-flow study is to obtain complete
The solution to the power-flow problem begins with identifying the known and unknown variables in the system. The known and unknown variables are dependent on the type of bus. A bus without any generators connected to it is called a Load Bus. With one exception, a bus with at least one generator connected to it is called a Generator Bus. The exception is one arbitrarily-selected bus that has a generator. This bus is referred to as the [[slack bus]].
In the power-flow problem, it is assumed that the real power
In order to solve for the <math>2(N-1) - (R-1)</math> unknowns, there must be <math>2(N-1) - (R-1)</math> equations that do not introduce any new unknown variables. The possible equations to use are power balance equations, which can be written for real and reactive power for each bus.
The real power balance equation is:
where <math>P_{i}</math> is the net active power injected at bus ''i'', <math>G_{ik}</math> is the real part of the element in the [[Ybus matrix|bus admittance matrix]] Y<sub>BUS</sub> corresponding to the <math>i_{th}</math> row and <math>k_{th}</math> column, <math>B_{ik}</math> is the imaginary part of the element in the Y<sub>BUS</sub> corresponding to the <math>i_{th}</math> row and <math>k_{th}</math> column and <math>\theta_{ik}</math> is the difference in voltage angle between the <math>i_{th}</math> and <math>k_{th}</math> buses (<math>\theta_{ik}=\theta_i-\theta_k</math>). The reactive power balance equation is:
where <math>Q_i</math> is the net reactive power injected at bus ''i''.
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==Newton–Raphson solution method==
There are several different methods of solving the resulting nonlinear system of equations. The most popular{{according to whom|date=November 2023}} is
where <math>\Delta P</math> and <math>\Delta Q</math> are called the mismatch equations:
<math display=block>\Delta Q_{i} = -Q_{i} + \sum_{k=1}^N |V_i||V_k|(G_{ik}\sin\theta_{ik}-B_{ik}\cos\theta_{ik})</math>▼
▲<math>\Delta Q_{i} = -Q_{i} + \sum_{k=1}^N |V_i||V_k|(G_{ik}\sin\theta_{ik}-B_{ik}\cos\theta_{ik})</math>
and <math>J</math> is a matrix of partial derivatives known as a [[Jacobian matrix and determinant|Jacobian]]:
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The linearized system of equations is solved to determine the next guess (''m'' + 1) of voltage magnitude and angles based on:
▲: <math>|V|^{m+1} = |V|^m + \Delta |V|\,</math>
The process continues until a stopping condition is met. A common stopping condition is to terminate if the [[Matrix norm|norm]] of the mismatch equations is below a specified tolerance.
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==Other power-flow methods==
* [[Gauss–Seidel method]]: This is the earliest devised method. It shows slower rates of convergence compared to other iterative methods, but it uses very little memory and does not need to solve a matrix system.
* [[Fast-decoupled-load-flow method]] is a variation on
* [[Holomorphic embedding load flow method]]: A recently developed method based on advanced techniques of complex analysis. It is direct and guarantees the calculation of the correct (operative) branch, out of the multiple solutions present in the power
* [[Backward-Forward Sweep (BFS) method]]: A method developed to take advantage of the radial structure of most modern distribution grids. It involves choosing an
* [[Laurent Power Flow (LPF) method]]: Power flow formulation that provides guarantee of uniqueness of solution and independence on initial conditions for electrical distribution systems. The LPF is based on the current injection method (CIM) and applies the Laurent series expansion. The main characteristics of this formulation are its proven numerical convergence and stability, and its computational advantages, showing to be at least ten times faster than the BFS method both in balanced and unbalanced networks.<ref>Giraldo, J. S., Montoya, O. D., Vergara, P. P., & Milano, F. (2022). A fixed-point current injection power flow for electric distribution systems using Laurent series. Electric Power Systems Research, 211, 108326. https://doi.org/10.1016/j.epsr.2022.108326</ref> Since it is based on the system's admittance matrix, the formulation is able to consider radial and meshed network topologies without additional modifications (contrary to the compensation-based BFS<ref>Shirmohammadi, D., Hong, H. W., Semlyen, A., & Luo, G. X. (1988). A compensation-based power flow method for weakly meshed distribution and transmission networks. IEEE Transactions on power systems, 3(2), 753-762. https://doi.org/10.1109/59.192932</ref>). The simplicity and computational efficiency of the LPF method make it an attractive option for recursive power flow problems, such as those encountered in time-series analyses, metaheuristics, probabilistic analysis, reinforcement learning applied to power systems, and other related applications.
==DC power
{{Expand section|date=August 2025}}
==References==
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