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Numerical analysis of electric power flow
In power engineering, the power flow study, or load flow study, is a numerical analysis of the flow of electric power in an interconnected system. A power flow study usually uses simplified notations such as a one line diagram and per unit system, and focuses on various aspects of AC power parameters, such as voltages, voltage angles, real power and reactive power. It analyzes the power systems in normal steady state operation. 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 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 purpose network analyzers were built between 1929 and the early 1960s to provide laboratory scale physical models of power systems. Large scale digital computers replaced the analog methods with numerical solutions. 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. 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 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 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.

An alternating current power flow model is a model used in electrical engineering to analyze power grids. It provides a nonlinear system of equations which describes the energy flow through each transmission line. The problem is non linear because the power flow into load impedances is a function of the square of the applied voltages. Due to nonlinearity, in many cases the analysis of large network via AC power flow model is not feasible, and a linear (but less accurate) DC power flow model is used instead. 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.

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 Newton–Raphson that exploits the approximate decoupling of active and reactive flows in well behaved power networks, and additionally fixes the value of the Jacobian during the iteration in order to avoid costly matrix decompositions. Also referred to as "fixed slope, decoupled NR". Within the algorithm, the Jacobian matrix gets inverted only once, and there are three assumptions. Firstly, the conductance between the buses is zero. Secondly, the magnitude of the bus voltage is one per unit. Thirdly, the sine of phases between buses is zero. Fast decoupled load flow can return the answer within seconds whereas the Newton Raphson method takes much longer. This is useful for real time management of power grids. 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 flow equations. Backward Forward Sweep (BFS) method: A method developed to take advantage of the radial structure of most modern distribution grids. It involves choosing an initial voltage profile and separating the original system of equations of grid components into two separate systems and solving one, using the last results of the other, until convergence is achieved. Solving for the currents with the voltages given is called the backward sweep (BS) and solving for the voltages with the currents given is called the forward sweep (FS). 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. 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). 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.

Direct current load flow gives estimations of lines power flows on AC power systems. Direct current load flow looks only at active power flows and neglects reactive power flows. This method is non iterative and absolutely convergent but less accurate than AC Load Flow solutions. Direct current load flow is used wherever repetitive and fast load flow estimations are required.