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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.
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A load flow study is especially valuable for a system with multiple load centers, such as a refinery complex. The power-flow study is an analysis of the system’s capability to adequately supply the connected load. The total system losses, as well as individual line losses, also are tabulated. Transformer tap positions are selected to ensure the correct voltage at critical locations such as motor control centers. Performing a load-flow study on an existing system provides insight and recommendations as to the system operation and optimization of control settings to obtain maximum capacity while minimizing the operating costs. The results of such an analysis are in terms of active power, reactive power, voltage magnitude and phase angle. Furthermore, power-flow computations are crucial for [[Unit commitment problem in electrical power production|optimal operations of groups of generating units]].
In term of its approach to uncertainties, load-flow study can be divided to deterministic load flow and uncertainty-concerned load flow. Deterministic load-flow study does not take into account the uncertainties arising from both power generations and load behaviors. To take the uncertainties into consideration, there are several approaches that has been used such as probabilistic, possibilistic, information gap decision theory, robust optimization, and interval analysis.<ref>{{Cite journal|title=A comprehensive review on uncertainty modeling techniques in power system studies|journal=Renewable and Sustainable Energy Reviews|volume=57|pages=1077–1089|doi=10.1016/j.rser.2015.12.070|year=2016|last1=Aien|first1=Morteza|last2=Hajebrahimi|first2=Ali|last3=Fotuhi-Firuzabad|first3=Mahmud|bibcode=2016RSERv..57.1077A }}</ref>
==Model==
An
Usually analysis of a three-phase power system is simplified by assuming balanced loading of all three phases. Sinusoidal steady-state operation is assumed, with no transient changes in power flow or voltage due to load or generation changes, meaning all current and voltage waveforms are sinusoidal with no DC offset and have the same constant frequency. The previous assumption is the same as assuming the power system is linear time-invariant (even though the system of equations is nonlinear), driven by sinusoidal sources of same frequency, and operating in steady-state, which allows to use [[phasor]] analysis, another simplification. A further simplification is to use the [[per-unit system]] to represent all voltages, power flows, and impedances, scaling the actual target system values to some convenient base. A system [[one-line diagram]] is the basis to build a mathematical model of the generators, loads, buses, and transmission lines of the system, and their electrical impedances and ratings.
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There are several different methods of solving the resulting nonlinear system of equations. The most popular{{according to whom|date=November 2023}} is a variation of the [[Newton–Raphson method]]. The Newton-Raphson method is an [[iterative method]] which begins with initial guesses of all unknown variables (voltage magnitude and angles at Load Buses and voltage angles at Generator Buses). Next, a [[Taylor Series]] is written, with the higher order terms ignored, for each of the power balance equations included in the system of equations. The result is a linear system of equations that can be expressed as:
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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* [[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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