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<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Power-flow study</span></span>

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<p>In <a href="Power_engineering" title="Power engineering">power engineering</a>, a <b>power-flow study</b> (also known as <b>power-flow analysis</b> or <b>load-flow study</b>) is a <a href="Numerical_analysis" title="Numerical analysis">numerical analysis</a> of the flow of <a href="Electric_power" title="Electric power">electric power</a> in an interconnected system. A power-flow study usually uses simplified notations such as a <a href="One-line_diagram" class="mw-redirect" title="One-line diagram">one-line diagram</a> and <a href="Per-unit_system" title="Per-unit system">per-unit system</a>, and focuses on various aspects of <a href="AC_power" title="AC power">AC power</a> parameters, such as <a href="Voltage" title="Voltage">voltage</a>, voltage angles, real power and reactive power. It analyzes the power systems in normal steady-state operation.

</p><p>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 <a href="Busbar" title="Busbar">bus</a>, and the real and reactive power flowing in each line.

</p><p>Commercial power systems are usually too complex to allow for hand solution of the power flow. Special-purpose <a href="Network_analyzer_(AC_power)" title="Network analyzer (AC power)">network analyzers</a> 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.

</p><p>In addition to a power-flow study, computer programs perform related calculations such as <a href="Short-circuit" class="mw-redirect" title="Short-circuit">short-circuit</a> fault analysis, stability studies (transient and steady-state), <a href="Unit_commitment" class="mw-redirect" title="Unit commitment">unit commitment</a> and <a href="Economic_dispatch" class="mw-redirect" title="Economic dispatch">economic dispatch</a>.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> In particular, some programs use <a href="Linear_programming" title="Linear programming">linear programming</a> to find the <i>optimal power flow</i>, the conditions which give the lowest cost per <a href="Kilowatt_hour" class="mw-redirect" title="Kilowatt hour">kilowatt hour</a> delivered.

</p><p>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 <a href="Unit_commitment_problem_in_electrical_power_production" title="Unit commitment problem in electrical power production">optimal operations of groups of generating units</a>.

</p><p>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.<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup>

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<div class="mw-heading mw-heading2"><h2 id="Model">Model</h2></div>

<p>An <i>alternating current power-flow model</i> is a model used in electrical engineering to analyze <a href="Power_grids" class="mw-redirect" title="Power grids">power grids</a>. It provides a <a href="Nonlinear_systems" class="mw-redirect" title="Nonlinear systems">nonlinear system</a> 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.

</p><p>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 <a href="Phasor" title="Phasor">phasor</a> analysis, another simplification. A further simplification is to use the <a href="Per-unit_system" title="Per-unit system">per-unit system</a> to represent all voltages, power flows, and impedances, scaling the actual target system values to some convenient base. A system <a href="One-line_diagram" class="mw-redirect" title="One-line diagram">one-line diagram</a> 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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<div class="mw-heading mw-heading2"><h2 id="Power-flow_problem_formulation">Power-flow problem formulation</h2></div>

<p>The goal of a power-flow study is to obtain complete voltage angles and magnitude information for each bus in a power system for specified load and generator real power and voltage conditions.<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> Once this information is known, real and reactive power flow on each branch as well as generator reactive power output can be analytically determined. Due to the nonlinear nature of this problem, numerical methods are employed to obtain a solution that is within an acceptable tolerance.

</p><p>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 <a href="Slack_bus" title="Slack bus">slack bus</a>.

</p><p>In the power-flow problem, it is assumed that the real power <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{D}}">

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</math></span><img src="./74d3b0f8a885007cddd6edc91d77b8bad9de7033.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.085ex; height:2.509ex;" alt="{\displaystyle P_{D}}" loading="lazy"></span> and reactive power <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q_{D}}">

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</math></span><img src="./9c0f49158427f64a4e824ea713df9170c2118194.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.431ex; height:2.509ex;" alt="{\displaystyle Q_{D}}" loading="lazy"></span> at each Load Bus are known. For this reason, Load Buses are also known as PQ Buses. For Generator Buses, it is assumed that the real power generated <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{G}}">

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<annotation encoding="application/x-tex">{\displaystyle P_{G}}</annotation>

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</math></span><img src="./fef28d13e351b70775fee01277b3f4dd4f37a5f2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.016ex; height:2.509ex;" alt="{\displaystyle P_{G}}" loading="lazy"></span> and the voltage magnitude <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |V|}">

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</math></span><img src="./9ddcffc28643ac01a14dd0fb32c3157859e365a7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.081ex; height:2.843ex;" alt="{\displaystyle |V|}" loading="lazy"></span> and voltage phase <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta }">

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</math></span><img src="./6e5ab2664b422d53eb0c7df3b87e1360d75ad9af.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:2.176ex;" alt="{\displaystyle \theta }" loading="lazy"></span> are known. Therefore, for each Load Bus, both the voltage magnitude and angle are unknown and must be solved for; for each Generator Bus, the voltage angle must be solved for; there are no variables that must be solved for the Slack Bus. In a system with <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N}">

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</math></span><img src="./f5e3890c981ae85503089652feb48b191b57aae3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle N}" loading="lazy"></span> buses and <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}">

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</math></span><img src="./4b0bfb3769bf24d80e15374dc37b0441e2616e33.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}" loading="lazy"></span> generators, there are then <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2(N-1)-(R-1)}">

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</math></span><img src="./c80e583b3addac2363d0793f12d57adce4aac985.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.455ex; height:2.843ex;" alt="{\displaystyle 2(N-1)-(R-1)}" loading="lazy"></span> unknowns.

</p><p>In order to solve for the <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2(N-1)-(R-1)}">

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</math></span><img src="./c80e583b3addac2363d0793f12d57adce4aac985.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.455ex; height:2.843ex;" alt="{\displaystyle 2(N-1)-(R-1)}" loading="lazy"></span> unknowns, there must be <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2(N-1)-(R-1)}">

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<annotation encoding="application/x-tex">{\displaystyle 2(N-1)-(R-1)}</annotation>

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</math></span><img src="./c80e583b3addac2363d0793f12d57adce4aac985.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.455ex; height:2.843ex;" alt="{\displaystyle 2(N-1)-(R-1)}" loading="lazy"></span> 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:

</p><p><span class="mwe-math-element mwe-math-element-block"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0=-P_{i}+\sum _{k=1}^{N}|V_{i}||V_{k}|(G_{ik}\cos \theta _{ik}+B_{ik}\sin \theta _{ik})}">

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</p><p>where <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{i}}">

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</math></span><img src="./3ba1396129f7be3c7f828a571b6649e6807d10d3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.292ex; height:2.509ex;" alt="{\displaystyle P_{i}}" loading="lazy"></span> is the net active power injected at bus <i>i</i>, <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G_{ik}}">

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</math></span><img src="./03a436fb23d456fca8331befe32ab7751eba8f18.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.483ex; height:2.509ex;" alt="{\displaystyle G_{ik}}" loading="lazy"></span> is the real part of the element in the <a href="Ybus_matrix" class="mw-redirect" title="Ybus matrix">bus admittance matrix</a> Y<sub>BUS</sub> corresponding to the <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i_{th}}">

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</math></span><img src="./fc3edcaf1baa0a1686e254b9ca3e073d9d727942.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.575ex; height:2.509ex;" alt="{\displaystyle i_{th}}" loading="lazy"></span> row and <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k_{th}}">

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</math></span><img src="./9cd0f63ee7f784ff38aa730e26c2b4605e41f267.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.984ex; height:2.509ex;" alt="{\displaystyle k_{th}}" loading="lazy"></span> column, <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B_{ik}}">

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</math></span><img src="./525c994819e3c7e5e50a61587f8091caaf0266ef.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.42ex; height:2.509ex;" alt="{\displaystyle B_{ik}}" loading="lazy"></span> is the imaginary part of the element in the Y<sub>BUS</sub> corresponding to the <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i_{th}}">

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</math></span><img src="./fc3edcaf1baa0a1686e254b9ca3e073d9d727942.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.575ex; height:2.509ex;" alt="{\displaystyle i_{th}}" loading="lazy"></span> row and <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k_{th}}">

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</math></span><img src="./9cd0f63ee7f784ff38aa730e26c2b4605e41f267.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.984ex; height:2.509ex;" alt="{\displaystyle k_{th}}" loading="lazy"></span> column and <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta _{ik}}">

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</math></span><img src="./2967afb20f6b8fc3502bc7477223d286bd257d3f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.747ex; height:2.509ex;" alt="{\displaystyle \theta _{ik}}" loading="lazy"></span> is the difference in voltage angle between the <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i_{th}}">

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<annotation encoding="application/x-tex">{\displaystyle i_{th}}</annotation>

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</math></span><img src="./fc3edcaf1baa0a1686e254b9ca3e073d9d727942.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.575ex; height:2.509ex;" alt="{\displaystyle i_{th}}" loading="lazy"></span> and <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k_{th}}">

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</math></span><img src="./9cd0f63ee7f784ff38aa730e26c2b4605e41f267.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.984ex; height:2.509ex;" alt="{\displaystyle k_{th}}" loading="lazy"></span> buses (<span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta _{ik}=\theta _{i}-\theta _{k}}">

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<annotation encoding="application/x-tex">{\displaystyle \theta _{ik}=\theta _{i}-\theta _{k}}</annotation>

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</math></span><img src="./9f25a241de7b76905fe18ac77b0c0507133c60ab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.755ex; height:2.509ex;" alt="{\displaystyle \theta _{ik}=\theta _{i}-\theta _{k}}" loading="lazy"></span>). The reactive power balance equation is:

</p><p><span class="mwe-math-element mwe-math-element-block"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0=-Q_{i}+\sum _{k=1}^{N}|V_{i}||V_{k}|(G_{ik}\sin \theta _{ik}-B_{ik}\cos \theta _{ik})}">

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<annotation encoding="application/x-tex">{\displaystyle 0=-Q_{i}+\sum _{k=1}^{N}|V_{i}||V_{k}|(G_{ik}\sin \theta _{ik}-B_{ik}\cos \theta _{ik})}</annotation>

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</math></span></span>

</p><p>where <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q_{i}}">

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<annotation encoding="application/x-tex">{\displaystyle Q_{i}}</annotation>

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</math></span><img src="./b9f7193081d440425e522698e80817b5d558df03.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.638ex; height:2.509ex;" alt="{\displaystyle Q_{i}}" loading="lazy"></span> is the net reactive power injected at bus <i>i</i>.

</p><p>Equations included are the real and reactive power balance equations for each Load Bus and the real power balance equation for each Generator Bus. Only the real power balance equation is written for a Generator Bus because the net reactive power injected is assumed to be unknown and therefore including the reactive power balance equation would result in an additional unknown variable. For similar reasons, there are no equations written for the Slack Bus.

</p><p>In many transmission systems, the impedance of the power network lines is primarily inductive, i.e. the phase angles of the power lines impedance are usually relatively large and very close to 90 degrees. There is thus a strong coupling between real power and voltage angle, and between reactive power and voltage magnitude, while the coupling between real power and voltage magnitude, as well as reactive power and voltage angle, is weak. As a result, real power is usually transmitted from the bus with higher voltage angle to the bus with lower voltage angle, and reactive power is usually transmitted from the bus with higher voltage magnitude to the bus with lower voltage magnitude. However, this approximation does not hold when the phase angle of the power line impedance is relatively small.<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup>

</p>

<div class="mw-heading mw-heading2"><h2 id="Newton–Raphson_solution_method">Newton–Raphson solution method</h2></div>

<p>There are several different methods of solving the resulting nonlinear system of equations. The most popular is a variation of the <a href="Newton%E2%80%93Raphson_method" class="mw-redirect" title="Newton–Raphson method">Newton–Raphson method</a>. The Newton-Raphson method is an <a href="Iterative_method" title="Iterative method">iterative method</a> which begins with initial guesses of all unknown variables (voltage magnitude and angles at Load Buses and voltage angles at Generator Buses). Next, a <a href="Taylor_Series" class="mw-redirect" title="Taylor Series">Taylor Series</a> 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:

</p><p><span class="mwe-math-element mwe-math-element-block"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{bmatrix}\Delta \theta \\\Delta |V|\end{bmatrix}}=-J^{-1}{\begin{bmatrix}\Delta P\\\Delta Q\end{bmatrix}}}">

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<annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}\Delta \theta \\\Delta |V|\end{bmatrix}}=-J^{-1}{\begin{bmatrix}\Delta P\\\Delta Q\end{bmatrix}}}</annotation>

</semantics>

</math></span></span>

</p><p>where <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta P}">

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<annotation encoding="application/x-tex">{\displaystyle \Delta P}</annotation>

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</math></span><img src="./f6b96a223eb5eec6d7bee974542df2effaeff123.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.681ex; height:2.176ex;" alt="{\displaystyle \Delta P}" loading="lazy"></span> and <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta Q}">

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<annotation encoding="application/x-tex">{\displaystyle \Delta Q}</annotation>

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</math></span><img src="./04f1b35d7435ced944ad7f40f992aae97c602695.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.774ex; height:2.509ex;" alt="{\displaystyle \Delta Q}" loading="lazy"></span> are called the mismatch equations:

</p><p><span class="mwe-math-element mwe-math-element-block"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta P_{i}=-P_{i}+\sum _{k=1}^{N}|V_{i}||V_{k}|(G_{ik}\cos \theta _{ik}+B_{ik}\sin \theta _{ik})}">

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<annotation encoding="application/x-tex">{\displaystyle \Delta P_{i}=-P_{i}+\sum _{k=1}^{N}|V_{i}||V_{k}|(G_{ik}\cos \theta _{ik}+B_{ik}\sin \theta _{ik})}</annotation>

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<annotation encoding="application/x-tex">{\displaystyle \Delta Q_{i}=-Q_{i}+\sum _{k=1}^{N}|V_{i}||V_{k}|(G_{ik}\sin \theta _{ik}-B_{ik}\cos \theta _{ik})}</annotation>

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</math></span></span>

</p><p>and <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle J}">

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<annotation encoding="application/x-tex">{\displaystyle J}</annotation>

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</math></span><img src="./359e4f407b49910e02c27c2f52e87a36cd74c053.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.471ex; height:2.176ex;" alt="{\displaystyle J}" loading="lazy"></span> is a matrix of partial derivatives known as a <a href="Jacobian_matrix_and_determinant" title="Jacobian matrix and determinant">Jacobian</a>:

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<annotation encoding="application/x-tex">{\displaystyle J={\begin{bmatrix}{\dfrac {\partial \Delta P}{\partial \theta }}&{\dfrac {\partial \Delta P}{\partial |V|}}\\{\dfrac {\partial \Delta Q}{\partial \theta }}&{\dfrac {\partial \Delta Q}{\partial |V|}}\end{bmatrix}}}</annotation>

</semantics>

</math></span><img src="./039b91de9f2762353eef1b7b2e540594fb053525.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.534ex; margin-bottom: -0.304ex; width:22.601ex; height:12.843ex;" alt="{\displaystyle J={\begin{bmatrix}{\dfrac {\partial \Delta P}{\partial \theta }}&{\dfrac {\partial \Delta P}{\partial |V|}}\\{\dfrac {\partial \Delta Q}{\partial \theta }}&{\dfrac {\partial \Delta Q}{\partial |V|}}\end{bmatrix}}}" loading="lazy"></span>.

</p><p>The linearized system of equations is solved to determine the next guess (<i>m</i> + 1) of voltage magnitude and angles based on:

</p><p><span class="mwe-math-element mwe-math-element-block"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta _{m+1}=\theta _{m}+\Delta \theta \,}">

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<annotation encoding="application/x-tex">{\displaystyle \theta _{m+1}=\theta _{m}+\Delta \theta \,}</annotation>

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<annotation encoding="application/x-tex">{\displaystyle |V|_{m+1}=|V|_{m}+\Delta |V|\,}</annotation>

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</p><p>The process continues until a stopping condition is met. A common stopping condition is to terminate if the <a href="Matrix_norm" title="Matrix norm">norm</a> of the mismatch equations is below a specified tolerance.

</p><p>A rough outline of solution of the power-flow problem is:

</p>

<ol><li>Make an initial guess of all unknown voltage magnitudes and angles. It is common to use a "flat start" in which all voltage angles are set to zero and all voltage magnitudes are set to 1.0 p.u.</li>

<li>Solve the power balance equations using the most recent voltage angle and magnitude values.</li>

<li>Linearize the system around the most recent voltage angle and magnitude values</li>

<li>Solve for the change in voltage angle and magnitude</li>

<li>Update the voltage magnitude and angles</li>

<li>Check the stopping conditions, if met then terminate, else go to step 2.</li></ol>

<div class="mw-heading mw-heading2"><h2 id="Other_power-flow_methods">Other power-flow methods</h2></div>

<ul><li><a href="Gauss%E2%80%93Seidel_method" title="Gauss–Seidel method">Gauss–Seidel method</a>: 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.</li>

<li>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 <a href="Jacobian_matrix_and_determinant" title="Jacobian matrix and determinant">Jacobian</a> 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.<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup></li>

<li><a href="Holomorphic_embedding_load_flow_method" class="mw-redirect" title="Holomorphic embedding load flow method">Holomorphic embedding load flow method</a>: 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.</li>

<li>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).<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup></li>

<li>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.<sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> 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<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup>). 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.</li></ul>

<div class="mw-heading mw-heading2"><h2 id="DC_power_flow">DC power flow</h2></div>

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<p>DC power flow (also known as direct current load flow (DCLF)) gives estimations of lines power flows on AC power systems. Despite the name, DC power flow is not an analysis on <a href="Direct_current" title="Direct current">direct current</a>, but rather on alternating current; the name comes from the linearity of the analysis, which resembles analysis on direct current. DC power flow looks only at <a href="Active_power" class="mw-redirect" title="Active power">active power</a> flows and neglects <a href="Reactive_power" class="mw-redirect" title="Reactive power">reactive power</a> flows. This method is non-iterative and absolutely convergent but less accurate than AC Load Flow solutions. DC power flow is used wherever repetitive and fast load flow estimations are required.<sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup>

</p>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

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.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("./mw/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("./mw/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("./mw/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("./mw/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}

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</style><cite id="CITEREFLow2013" class="citation book cs1">Low, S. H. (2013). "Convex relaxation of optimal power flow: A tutorial". <i>2013 IREP Symposium Bulk Power System Dynamics and Control - IX Optimization, Security and Control of the Emerging Power Grid</i>. pp. <span class="nowrap">1–</span>06. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1109%2FIREP.2013.6629391">10.1109/IREP.2013.6629391</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-1-4799-0199-9</bdi>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:14195805">14195805</a>.</cite></span>

</li>

<li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><cite id="CITEREFAienHajebrahimiFotuhi-Firuzabad2016" class="citation journal cs1">Aien, Morteza; Hajebrahimi, Ali; Fotuhi-Firuzabad, Mahmud (2016). "A comprehensive review on uncertainty modeling techniques in power system studies". <i>Renewable and Sustainable Energy Reviews</i>. <b>57</b>: <span class="nowrap">1077–</span>1089. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2016RSERv..57.1077A">2016RSERv..57.1077A</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2Fj.rser.2015.12.070">10.1016/j.rser.2015.12.070</a>.</cite></span>

</li>

<li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"><cite id="CITEREFGraingerStevenson1994" class="citation book cs1">Grainger, J.; Stevenson, W. (1994). <i>Power System Analysis</i>. New York: McGraw–Hill. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-07-061293-5</bdi>.</cite></span>

</li>

<li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text"><a rel="nofollow" class="external text" href="http://www.eeh.ee.ethz.ch/uploads/tx_ethstudies/modelling_hs08_script_02.pdf">Andersson, G: Lectures on Modelling and Analysis of Electric Power Systems</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20170215042633/http://www.eeh.ee.ethz.ch/uploads/tx_ethstudies/modelling_hs08_script_02.pdf">Archived</a> 2017-02-15 at the <a href="Wayback_Machine" title="Wayback Machine">Wayback Machine</a></span>

</li>

<li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text"><cite id="CITEREFStottAlsac1974" class="citation journal cs1 cs1-prop-long-vol">Stott, B.; Alsac, O. (May 1974). "Fast Decoupled Load Flow". <i>IEEE Transactions on Power Apparatus and Systems</i>. PAS-93 (3): <span class="nowrap">859–</span>869. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1974ITPAS..93..859S">1974ITPAS..93..859S</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1109%2Ftpas.1974.293985">10.1109/tpas.1974.293985</a>. <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0018-9510">0018-9510</a>.</cite></span>

</li>

<li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</a></b></span> <span class="reference-text">Petridis, S.; Blanas, O.; Rakopoulos, D.; Stergiopoulos, F.; Nikolopoulos, N.; Voutetakis, S. An Efficient Backward/Forward Sweep Algorithm for Power Flow Analysis through a Novel Tree-Like Structure for Unbalanced Distribution Networks. <i>Energies</i> 2021, <i>14</i>, 897. <a rel="nofollow" class="external free" href="https://doi.org/10.3390/en14040897">https://doi.org/10.3390/en14040897</a>, <a rel="nofollow" class="external free" href="https://www.mdpi.com/1996-1073/14/4/897">https://www.mdpi.com/1996-1073/14/4/897</a></span>

</li>

<li id="cite_note-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-7">^</a></b></span> <span class="reference-text">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. <a rel="nofollow" class="external free" href="https://doi.org/10.1016/j.epsr.2022.108326">https://doi.org/10.1016/j.epsr.2022.108326</a></span>

</li>

<li id="cite_note-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-8">^</a></b></span> <span class="reference-text">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. <a rel="nofollow" class="external free" href="https://doi.org/10.1109/59.192932">https://doi.org/10.1109/59.192932</a></span>

</li>

<li id="cite_note-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-9">^</a></b></span> <span class="reference-text"><a rel="nofollow" class="external text" href="https://link.springer.com/content/pdf/bbm%3A978-3-642-17989-1%2F1.pdf">Seifi, H. &. (2011). Appendix A: DC Load Flow. In H. &. Seifi, Electric power system planning: issues, algorithms and solutions (pp. 245-249). Berlin: Springer</a></span>

</li>

</ol></div></div>

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</style><div id="Electricity_delivery729" style="font-size:114%;margin:0 4em"><a href="Electricity_delivery" title="Electricity delivery">Electricity delivery</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">Concepts</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Automatic_generation_control" title="Automatic generation control">Automatic generation control</a></li>

<li><a href="Backfeeding" title="Backfeeding">Backfeeding</a></li>

<li><a href="Demand_factor" title="Demand factor">Demand factor</a></li>

<li><a href="Droop_speed_control" title="Droop speed control">Droop speed control</a></li>

<li><a href="Electric_power" title="Electric power">Electric power</a></li>

<li><a href="Electric_power_quality" title="Electric power quality">Electric power quality</a></li>

<li><a href="Electrical_fault" title="Electrical fault">Electrical fault</a></li>

<li><a href="Energy_demand_management" title="Energy demand management">Energy demand management</a></li>

<li><a href="Energy_return_on_investment" title="Energy return on investment">Energy return on investment</a></li>

<li><a href="Grid_energy_storage" title="Grid energy storage">Grid energy storage</a></li>

<li><a href="Grid_strength" class="mw-redirect" title="Grid strength">Grid strength</a></li>

<li><a href="Home_energy_storage" title="Home energy storage">Home energy storage</a></li>

<li><a href="Load-following_power_plant" title="Load-following power plant">Load-following</a></li>

<li><a href="Merit_order" title="Merit order">Merit order</a></li>

<li><a href="Nameplate_capacity" title="Nameplate capacity">Nameplate capacity</a></li>

<li><a href="Peak_demand" title="Peak demand">Peak demand</a></li>

<li><a href="Power_factor" title="Power factor">Power factor</a></li>

<li><a href="Power_system_reliability" title="Power system reliability">Power system reliability</a></li>

<li><a href="Repowering" title="Repowering">Repowering</a></li>

<li><a href="Utility_frequency" title="Utility frequency">Utility frequency</a></li>

<li><a href="Variable_renewable_energy" title="Variable renewable energy">Variability</a></li>

<li><a href="Vehicle-to-grid" title="Vehicle-to-grid">Vehicle-to-grid</a></li></ul>

</div></td><td class="noviewer navbox-image" rowspan="8" style="width:1px;padding:0 0 0 2px"><div><span typeof="mw:File"></span></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Sources</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:7em"><a href="Non-renewable_resource" title="Non-renewable resource">Non-renewable</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Fossil_fuel_power_station" title="Fossil fuel power station">Fossil fuel power station</a>

<li><a href="Natural_gas" title="Natural gas">Natural gas</a></li>

<li><a href="Oil_shale" title="Oil shale">Oil shale</a></li>

<li><a href="Petroleum" title="Petroleum">Petroleum</a></li></ul></li>

<li><a href="Nuclear_power" title="Nuclear power">Nuclear</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:7em"><a href="Renewable_energy" title="Renewable energy">Renewable</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Biofuel" title="Biofuel">Biofuel</a></li>

<li><a href="Biogas" title="Biogas">Biogas</a></li>

<li><a href="Biomass" title="Biomass">Biomass</a></li>

<li><a href="Geothermal_power" title="Geothermal power">Geothermal</a></li>

<li><a href="Hydroelectricity" title="Hydroelectricity">Hydro</a></li>

<li><a href="Marine_energy" title="Marine energy">Marine</a>

<ul><li><a href="Marine_current_power" title="Marine current power">Current</a></li>

<li><a href="Osmotic_power" title="Osmotic power">Osmotic</a></li>

<li><a href="Ocean_thermal_energy_conversion" title="Ocean thermal energy conversion">Thermal</a></li>

<li><a href="Tidal_power" title="Tidal power">Tidal</a></li>

<li><a href="Solar_power" title="Solar power">Solar</a></li>

<li><a href="Sustainable_biofuel" title="Sustainable biofuel">Sustainable biofuel</a></li>

</div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Generation</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="AC_power" title="AC power">AC power</a></li>

<li><a href="Cogeneration" title="Cogeneration">Cogeneration</a></li>

<li><a href="Combined_cycle_power_plant" title="Combined cycle power plant">Combined cycle</a></li>

<li><a href="Cooling_tower" title="Cooling tower">Cooling tower</a></li>

<li><a href="Dispatchable_generation" title="Dispatchable generation">Dispatchable</a></li>

<li><a href="Energy_storage" title="Energy storage">Energy storage</a>

<ul><li><a href="Battery_energy_storage_system" title="Battery energy storage system">Battery</a></li></ul></li>

<li><a href="Induction_generator" title="Induction generator">Induction generator</a></li>

<li><a href="Inertial_response" title="Inertial response">Inertial response</a></li>

<li><a href="Inverter-based_resource" title="Inverter-based resource">Inverter-based resource</a></li>

<li><a href="Micro_combined_heat_and_power" title="Micro combined heat and power">Micro CHP</a></li>

<li><a href="Microgeneration" title="Microgeneration">Microgeneration</a></li>

<li><a href="Rankine_cycle" title="Rankine cycle">Rankine cycle</a></li>

<li><a href="Three-phase_electric_power" title="Three-phase electric power">Three-phase electric power</a></li>

<li><a href="Virtual_power_plant" title="Virtual power plant">Virtual power plant</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><div style="display: inline-block; line-height: 1.2em; padding: .1em 0;"><a href="Electric_power_transmission" title="Electric power transmission">Transmission</a><br>and <a href="Electric_power_distribution" title="Electric power distribution">distribution</a></div></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Ancillary_services" title="Ancillary services">Ancillary services</a></li>

<li><a href="Balancing_authority" title="Balancing authority">Balancing authority</a></li>

<li><a href="Contingency_(electrical_grid)" title="Contingency (electrical grid)">Contingency (electrical grid)</a></li>

<li><a href="Demand_response" title="Demand response">Demand response</a></li>

<li><a href="Distributed_generation" title="Distributed generation">Distributed generation</a></li>

<li><a href="Dynamic_demand_(electric_power)" title="Dynamic demand (electric power)">Dynamic demand</a></li>

<li><a href="Electric_power_distribution" title="Electric power distribution">Electric power distribution</a></li>

<li><a href="Electric_power_system" title="Electric power system">Electric power system</a></li>

<li><a href="Electric_power_transmission" title="Electric power transmission">Electric power transmission</a></li>

<li><a href="Electrical_busbar_system" title="Electrical busbar system">Electrical busbar system</a></li>

<li><a href="Electrical_grid" title="Electrical grid">Electrical grid</a></li>

<li><a href="Electricity_retailing" title="Electricity retailing">Electricity retailing</a></li>

<li><a href="Grid_balancing" title="Grid balancing">Grid balancing</a></li>

<li><a href="High-voltage_direct_current" title="High-voltage direct current">High-voltage direct current</a></li>

<li><a href="High-voltage_shore_connection" title="High-voltage shore connection">High-voltage shore connection</a></li>

<li><a href="Interconnector" title="Interconnector">Interconnector</a></li>

<li><a href="Load_management" title="Load management">Load management</a></li>

<li><a href="Mains_electricity_by_country" title="Mains electricity by country">Mains electricity by country</a></li>

<li><a href="Overhead_power_line" title="Overhead power line">Overhead power line</a>

<ul><li><a href="Conductor_gallop" title="Conductor gallop">Conductor gallop</a></li></ul></li>

<li><a href="Power_station" title="Power station">Power station</a></li>

<li><a href="Pumped-storage_hydroelectricity" title="Pumped-storage hydroelectricity">Pumped hydro</a></li>

<li><a href="Single-wire_earth_return" title="Single-wire earth return">Single-wire earth return</a></li>

<li><a href="Smart_grid" title="Smart grid">Smart grid</a></li>

<li><a href="Substation" title="Substation">Substation</a></li>

<li><a href="Super_grid" title="Super grid">Super grid</a></li>

<li><a href="Transformer" title="Transformer">Transformer</a></li>

<li><a href="Transmission_system_operator" title="Transmission system operator">Transmission system operator</a> (TSO)</li>

<li><a href="Transmission_tower" title="Transmission tower">Transmission tower</a></li>

<li><a href="Utility_pole" title="Utility pole">Utility pole</a></li>

<li><a href="Voltage_control_and_reactive_power_management" title="Voltage control and reactive power management">Voltage control and reactive power management</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Failure modes</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Black_start" title="Black start">Black start</a></li>

<li><a href="Brownout_(electricity)" title="Brownout (electricity)">Brownout</a></li>

<li><a href="Cascading_failure" title="Cascading failure">Cascading failure</a></li>

<li><a href="Islanding" title="Islanding">Islanding</a></li>

<li><a href="Power_outage" title="Power outage">Power outage</a>

<li><a href="Rolling_blackout" title="Rolling blackout">Rolling blackout</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><div style="display: inline-block; line-height: 1.2em; padding: .1em 0;">Protective<br>devices</div></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Arc-fault_circuit_interrupter" title="Arc-fault circuit interrupter">Arc-fault circuit interrupter</a></li>

<li><a href="Circuit_breaker" title="Circuit breaker">Circuit breaker</a>

<ul><li><a href="Earth-leakage_circuit_breaker" title="Earth-leakage circuit breaker">Earth-leakage</a></li>

<li><a href="Sulfur_hexafluoride_circuit_breaker" title="Sulfur hexafluoride circuit breaker">Sulfur hexafluoride</a></li></ul></li>

<li><a href="Generator_interlock_kit" title="Generator interlock kit">Generator interlock kit</a></li>

<li><a href="Numerical_relay" title="Numerical relay">Numerical relay</a></li>

<li><a href="Power_system_protection" title="Power system protection">Power system protection</a></li>

<li><a href="Protective_relay" title="Protective relay">Protective relay</a></li>

<li><a href="Residual-current_device" title="Residual-current device">Residual-current device</a> (GFI)</li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><div style="display: inline-block; line-height: 1.2em; padding: .1em 0;">Economics<br>and policies</div></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Availability_factor" title="Availability factor">Availability factor</a></li>

<li><a href="Capacity_factor" title="Capacity factor">Capacity factor</a></li>

<li><a href="Carbon_offsets_and_credits" title="Carbon offsets and credits">Carbon offsets and credits</a></li>

<li><a href="Cost_of_electricity_by_source" title="Cost of electricity by source">Cost of electricity by source</a></li>

<li><a href="Energy_subsidy" title="Energy subsidy">Energy subsidies</a></li>

<li><a href="Environmental_tax" title="Environmental tax">Environmental tax</a></li>

<li><a href="Feed-in_tariff" title="Feed-in tariff">Feed-in tariff</a></li>

<li><a href="Fossil_fuel_phase-out" title="Fossil fuel phase-out">Fossil fuel phase-out</a></li>

<li><a href="Load_factor_(electrical)" title="Load factor (electrical)">Load factor</a></li>

<li><a href="Net_metering" title="Net metering">Net metering</a></li>

<li><a href="Pigouvian_tax" title="Pigouvian tax">Pigouvian tax</a></li>

<li><a href="Renewable_Energy_Certificate_(United_States)" title="Renewable Energy Certificate (United States)">Renewable Energy Certificates</a></li>

<li><a href="Renewable_energy_commercialization" title="Renewable energy commercialization">Renewable energy commercialization</a></li>

<li><a href="Renewable_Energy_Payments" title="Renewable Energy Payments">Renewable Energy Payments</a></li>

<li><a href="Spark_spread" title="Spark spread">Spark/Dark/Quark/Bark spread</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><div style="display: inline-block; line-height: 1.2em; padding: .1em 0;">Statistics and<br>production</div></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Electric_energy_consumption" title="Electric energy consumption">Electric energy consumption</a></li>

<li><a href="List_of_electricity_sectors" title="List of electricity sectors">List of electricity sectors</a></li></ul>

</div></td></tr><tr><td class="navbox-abovebelow" colspan="3"><div>

<ul><li><span class="noviewer" typeof="mw:File"><span title="Category"></span></span> Category</li></ul>

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