116:, then the voltage and current are 90 degrees out of phase. For two quarters of each cycle, the product of voltage and current is positive, but for the other two quarters, the product is negative, indicating that on average, exactly as much energy flows into the load as flows back out. There is no net energy flow over each half cycle. In this case, only reactive power flows: There is no net transfer of energy to the load; however, electrical power does flow along the wires and returns by flowing in reverse along the same wires. The current required for this reactive power flow dissipates energy in the line resistance, even if the ideal load device consumes no energy itself. Practical loads have resistance as well as inductance, or capacitance, so both active and reactive powers will flow to normal loads.
1323:
inductors and capacitors. Energy flows in one direction from the source to the load. In AC power, the voltage and current both vary approximately sinusoidally. When there is inductance or capacitance in the circuit, the voltage and current waveforms do not line up perfectly. The power flow has two components – one component flows from source to load and can perform work at the load; the other portion, known as "reactive power", is due to the delay between voltage and current, known as phase angle, and cannot do useful work at the load. It can be thought of as current that is arriving at the wrong time (too late or too early). To distinguish reactive power from active power, it is measured in units of "
1249:. It is zero when the current leads or lags the voltage by 90 degrees. When the voltage and current are 180 degrees out of phase, the power factor is negative one, and the load is feeding energy into the source (an example would be a home with solar cells on the roof that feed power into the power grid when the sun is shining). Power factors are usually stated as "leading" or "lagging" to show the sign of the phase angle of current with respect to voltage. Voltage is designated as the base to which current angle is compared, meaning that current is thought of as either "leading" or "lagging" voltage. Where the waveforms are purely sinusoidal, the power factor is the cosine of the phase angle (
140:
mechanism for controlling the power factor in electric power transmission; capacitors (or inductors) are inserted in a circuit to partially compensate for reactive power 'consumed' ('generated') by the load. Purely capacitive circuits supply reactive power with the current waveform leading the voltage waveform by 90 degrees, while purely inductive circuits absorb reactive power with the current waveform lagging the voltage waveform by 90 degrees. The result of this is that capacitive and inductive circuit elements tend to cancel each other out.
442:
1662:" is an important source of reactive power in the above power balance equation, which is generated by the capacitative nature of the transmission network itself. By making decisive switching actions in the early morning before the demand increases, the system gain can be maximized early on, helping to secure the system for the whole day. To balance the equation some pre-fault reactive generator use will be required. Other sources of reactive power that will also be used include shunt capacitors, shunt reactors,
1245:. For two systems transmitting the same amount of active power, the system with the lower power factor will have higher circulating currents due to energy that returns to the source from energy storage in the load. These higher currents produce higher losses and reduce overall transmission efficiency. A lower power factor circuit will have a higher apparent power and higher losses for the same amount of active power. The power factor is 1.0 when the voltage and current are in
4566:
38:
5145:
4077:
1352:, or large coils of wire to store energy in the form of a magnetic field. When a voltage is initially placed across the coil, the inductor strongly resists this change in a current and magnetic field, which causes a time delay for the current to reach its maximum value. This causes the current to lag behind the voltage in phase. Inductors are said to "sink" reactive power, and thus to cause a lagging power factor.
144:
1336:
by the capacitor and not have to be transferred over the transmission lines. This practice saves energy because it reduces the amount of energy that is required to be produced by the utility to do the same amount of work. Additionally, it allows for more efficient transmission line designs using smaller conductors or fewer bundled conductors and optimizing the design of transmission towers.
1345:
causes an opposing voltage to develop across the capacitor. This voltage increases until some maximum dictated by the capacitor structure. In an AC network, the voltage across a capacitor is constantly changing. The capacitor opposes this change, causing the current to lead the voltage in phase. Capacitors are said to "source" reactive power, and thus to cause a leading power factor.
4034:
2372:
3011:
3287:
In general, engineers are interested in the active power averaged over a period of time, whether it is a low frequency line cycle or a high frequency power converter switching period. The simplest way to get that result is to take the integral of the instantaneous calculation over the desired period:
1335:
is used to reduce apparent power flow to a load by reducing reactive power supplied from transmission lines and providing it locally. For example, to compensate an inductive load, a shunt capacitor is installed close to the load itself. This allows all reactive power needed by the load to be supplied
123:
values of voltage and current. Apparent power is taken into account when designing and operating power systems, because although the current associated with reactive power does no work at the load, it still must be supplied by the power source. Conductors, transformers and generators must be sized to
4060:
are a bad thing. They will increase the RMS current (since there will be non-zero terms added) and therefore apparent power, but they will have no effect on the active power transferred. Hence, harmonic currents will reduce the power factor. Harmonic currents can be reduced by a filter placed at the
1674:
While active power and reactive power are well defined in any system, the definition of apparent power for unbalanced polyphase systems is considered to be one of the most controversial topics in power engineering. Originally, apparent power arose merely as a figure of merit. Major delineations of
1356:
can source or sink reactive power, and provide a measure of control to system operators over reactive power flow and thus voltage. Because these devices have opposite effects on the phase angle between voltage and current, they can be used to "cancel out" each other's effects. This usually takes the
1313:
For instance, a power factor of 0.68 means that only 68 percent of the total current supplied (in magnitude) is actually doing work; the remaining current does no work at the load. Power Factor is very important in Power sector substations. Form the national grid the sub sectors are required to have
4055:
is nonzero. Therefore, the only product terms that have a nonzero average are those where the frequency of voltage and current match. In other words, it is possible to calculate active (average) power by simply treating each frequency separately and adding up the answers. Furthermore, if voltage of
2142:
The 1920 committee found no consensus and the topic continued to dominate discussions. In 1930, another committee formed and once again failed to resolve the question. The transcripts of their discussions are the lengthiest and most controversial ever published by the AIEE. Further resolution of
1322:
In a direct current circuit, the power flowing to the load is proportional to the product of the current through the load and the potential drop across the load. The power that happens because of a capacitor or inductor is called reactive power. It happens because of the AC nature of elements like
139:
Conventionally, capacitors are treated as if they generate reactive power, and inductors are treated as if they consume it. If a capacitor and an inductor are placed in parallel, then the currents flowing through the capacitor and the inductor tend to cancel rather than add. This is the fundamental
3521:
Since an RMS value can be calculated for any waveform, apparent power can be calculated from this. For active power it would at first appear that it would be necessary to calculate many product terms and average all of them. However, looking at one of these product terms in more detail produces a
1344:
Stored energy in the magnetic or electric field of a load device, such as a motor or capacitor, causes an offset between the current and the voltage waveforms. A capacitor is a device that stores energy in the form of an electric field. As current is driven through the capacitor, charge build-up
107:, the two quantities reverse their polarity at the same time. Hence, the instantaneous power, given by the product of voltage and current, is always positive, such that the direction of energy flow does not reverse and always is toward the resistor. In this case, only active power is transferred.
1371:
Transmission connected generators are generally required to support reactive power flow. For example, on the United
Kingdom transmission system, generators are required by the Grid Code Requirements to supply their rated power between the limits of 0.85 power factor lagging and 0.90 power factor
1330:
Energy stored in capacitive or inductive elements of the network gives rise to reactive power flow. Reactive power flow strongly influences the voltage levels across the network. Voltage levels and reactive power flow must be carefully controlled to allow a power system to be operated within
3528:
601:
itself. This is done because otherwise using the product V I to define S would result in a quantity that depends on the reference angle chosen for V or I, but defining S as V I* results in a quantity that doesn't depend on the reference angle and allows to relate S to P and Q.
2039:
2816:
1653:
2156:
80:. The portion of instantaneous power that results in no net transfer of energy but instead oscillates between the source and load in each cycle due to stored energy is known as instantaneous reactive power, and its amplitude is the absolute value of
3414:
2133:
1896:
2828:
1042:
838:
1802:
3153:
4029:{\displaystyle {\begin{aligned}&A\cos(\omega _{1}t+k_{1})\cos(\omega _{2}t+k_{2})\\={}&{\frac {A}{2}}\cos \left+{\frac {A}{2}}\cos \left\\={}&{\frac {A}{2}}\cos \left+{\frac {A}{2}}\cos \left\end{aligned}}}
2590:
414:. Understanding the relationship among these three quantities lies at the heart of understanding power engineering. The mathematical relationship among them can be represented by vectors or expressed using
3510:
4206:
To distinguish between linear and nonlinear loads, we may say that linear time-invariant loads are characterized so that an application of a sinusoidal voltage results in a sinusoidal flow of current.
1908:
3533:
2833:
699:
3424:
of the waveform. In practical applications, this would be done in the digital domain, where the calculation becomes trivial when compared to the use of rms and phase to determine active power:
1372:
leading at the designated terminals. The system operator will perform switching actions to maintain a secure and economical voltage profile while maintaining a reactive power balance equation:
132:). Another consequence is that adding the apparent power for two loads will not accurately give the total power unless they have the same phase difference between current and voltage (the same
41:
The blinking of non-incandescent city lights is shown in this motion-blurred long exposure. The AC nature of the mains power is revealed by the dashed appearance of the traces of moving lights.
3284:
This definition is useful because it applies to all waveforms, whether they are sinusoidal or not. This is particularly useful in power electronics, where non-sinusoidal waveforms are common.
2692:
2367:{\displaystyle S={\sqrt {\left(|V_{\mathrm {a} }^{2}|+|V_{\mathrm {b} }^{2}|+|V_{\mathrm {c} }^{2}|\right)\left(|I_{\mathrm {a} }^{2}|+|I_{\mathrm {b} }^{2}|+|I_{\mathrm {c} }^{2}|\right)}}}
1378:
2418:
517:
1221:
347:
4061:
input of the device. Typically this will consist of either just a capacitor (relying on parasitic resistance and inductance in the supply) or a capacitor-inductor network. An active
1101:
897:
3217:
595:
3294:
1293:
2045:
1808:
4222:
3006:{\displaystyle {\begin{aligned}P&=0\\Q&=|S|=V_{\mathrm {RMS} }I_{\mathrm {RMS} }=I_{\mathrm {RMS} }^{2}|X|={\frac {V_{\mathrm {RMS} }^{2}}{|X|}}\end{aligned}}}
1267:
213:
1154:
2646:
2619:
2449:
568:
3279:
3250:
4282:
2680:
3064:
3036:
4360:
Emanuel, Alexander (July 1993). "On The
Definition of Power Factor and Apparent Power in Unbalanced Polyphase Circuits with Sinusoidal Voltage and Currents".
1314:
minimum amount of power factor. Otherwise there are many loss. Mainly the required vary around 0.90 to 0.96 or more. Better the power factor less the loss.
1699:. In 1920, a "Special Joint Committee of the AIEE and the National Electric Light Association" met to resolve the issue. They considered two definitions.
4247:
915:
711:
3421:
406:. Since reactive power transfers no net energy to the load, it is sometimes called "wattless" power. It does, however, serve an important function in
1705:
3076:
2457:
218:
Engineers use the following terms to describe energy flow in a system (and assign each of them a different unit to differentiate between them):
4417:
2822:
For a perfect capacitor or inductor, there is no net power transfer; so all power is reactive. Therefore, for a perfect capacitor or inductor:
4134:
IEEE Standard
Definitions for the Measurement of Electric Power Quantities Under Sinusoidal, Nonsinusoidal, Balanced, or Unbalanced Conditions
1366:
125:
5090:
3430:
1269:) between the current and voltage sinusoidal waveforms. Equipment data sheets and nameplates will often abbreviate power factor as "
1695:
power distribution, it became clear that the definition of apparent power and the power factor could not be applied to unbalanced
4332:
2034:{\displaystyle S_{V}=|P_{\mathrm {a} }+P_{\mathrm {b} }+P_{\mathrm {c} }+j(Q_{\mathrm {a} }+Q_{\mathrm {b} }+Q_{\mathrm {c} })|}
72:, results in net transfer of energy in one direction is known as instantaneous active power, and its time average is known as
4199:
4174:
4149:
2150:
theory was proposed in 1993 by
Alexander Emanuel for unbalanced linear load supplied with asymmetrical sinusoidal voltages:
4218:
5148:
4565:
153:
The complex power is the vector sum of active and reactive power. The apparent power is the magnitude of the complex power.
4987:
4065:
circuit at the input would generally reduce the harmonic currents further and maintain the power factor closer to unity.
619:
4293:
2811:{\displaystyle P=S=V_{\mathrm {RMS} }I_{\mathrm {RMS} }=I_{\mathrm {RMS} }^{2}R={\frac {V_{\mathrm {RMS} }^{2}}{R}}\,\!}
1648:{\displaystyle \mathrm {Generator\ MVARs+System\ gain+Shunt\ capacitors=MVAR\ Demand+Reactive\ losses+Shunt\ reactors} }
5095:
4410:
1327:", or var. These units can simplify to watts but are left as var to denote that they represent no actual work output.
4715:
4269:
1348:
Induction machines are some of the most common types of loads in the electric power system today. These machines use
2656:
A perfect resistor stores no energy; so current and voltage are in phase. Therefore, there is no reactive power and
2378:
4878:
4803:
4674:
1332:
464:
4243:
1166:
296:
5050:
4982:
4972:
4848:
4748:
4403:
2424:
that is, the root of squared sums of line voltages multiplied by the root of squared sums of line currents.
1054:
850:
4903:
4863:
4440:
4057:
5130:
5125:
4843:
4818:
4808:
4784:
4779:
4485:
4095:
5045:
4763:
4733:
4510:
411:
17:
1357:
form of capacitor banks being used to counteract the lagging power factor caused by induction motors.
5100:
4589:
4550:
3409:{\displaystyle P_{\text{avg}}={\frac {1}{t_{2}-t_{1}}}\int _{t_{1}}^{t_{2}}v(t)\,i(t)\,\mathrm {d} t}
3165:
1684:
705:
Consequentially, with reference to the power triangle, real power (units in watts, W) is derived as:
573:
350:
5075:
4883:
4823:
4480:
2128:{\displaystyle \mathrm {pf} _{V}={P_{\mathrm {a} }+P_{\mathrm {b} }+P_{\mathrm {c} } \over S_{V}}}
1891:{\displaystyle \mathrm {pf} _{A}={P_{\mathrm {a} }+P_{\mathrm {b} }+P_{\mathrm {c} } \over S_{A}}}
5174:
5014:
5004:
4994:
4062:
2683:
1310:. The concept of power dissipation in AC circuit is explained and illustrated with the example.
1272:
4935:
4798:
4581:
4470:
2147:
1663:
1252:
198:
5070:
4838:
4833:
4813:
1324:
96:
30:
This article is about power in AC systems. For information on utility-supplied AC power, see
4056:
the mains supply is assumed to be a single frequency (which it usually is), this shows that
1124:
4664:
4426:
3040:
2624:
2597:
2427:
1112:
610:
546:
403:
248:
111:
87:
3255:
3226:
8:
5035:
4868:
4768:
4743:
4696:
4505:
4495:
4460:
4090:
2659:
1676:
1353:
445:
Instantaneous power in AC systems when the current lags behind the voltage by 50 degrees.
50:
46:
376:
is the apparent power. Reactive power does not do any work, so it is represented as the
4909:
4520:
3049:
3021:
276:
128:
on an electrical grid and, under certain operating conditions, collapse the network (a
4336:
5169:
5060:
4940:
4545:
4265:
4195:
4170:
4145:
4105:
1696:
536:
441:
61:
may result in periodic reversals of the direction of energy flow. Its SI unit is the
31:
3066:
is defined as being positive for an inductor and negative for a capacitor, then the
5009:
4950:
4654:
4649:
4626:
4535:
4475:
4369:
4137:
535:
denotes current in phasor form, with the amplitude as RMS. Also by convention, the
528:
399:
392:
120:
3420:
This method of calculating the average power gives the active power regardless of
1037:{\displaystyle Q=|S|\sin {\varphi }=|I|^{2}X={\frac {|V|^{2}}{|Z|^{2}}}\times {X}}
833:{\displaystyle P=|S|\cos {\varphi }=|I|^{2}R={\frac {|V|^{2}}{|Z|^{2}}}\times {R}}
5040:
4999:
4977:
4858:
4828:
4793:
4753:
4555:
4251:
4141:
1659:
407:
5065:
5055:
4853:
4465:
4110:
3067:
435:
415:
93:
68:
The portion of instantaneous power that, averaged over a complete cycle of the
1797:{\displaystyle S_{A}=|S_{\mathrm {a} }|+|S_{\mathrm {b} }|+|S_{\mathrm {c} }|}
1302:
and the phase angle between voltage and current is 45.6°. The power factor is
5163:
5085:
4873:
4758:
4738:
4669:
4659:
4616:
4500:
4455:
4263:
IEEE 100 : the authoritative dictionary of IEEE standards terms.-7th ed.
4082:
3148:{\displaystyle Q=I_{\mathrm {RMS} }^{2}X={\frac {V_{\mathrm {RMS} }^{2}}{X}}}
1246:
5105:
5080:
4945:
4914:
4728:
4530:
4390:
4244:
Definition of Active Power in the
International Electrotechnical Vocabulary
2585:{\displaystyle P^{+}=3|V^{+}||I^{+}|\cos {(\arg {(V^{+})}-\arg {(I^{+})})}}
1242:
1236:
133:
129:
380:
of the vector diagram. Active power does do work, so it is the real axis.
92:
In a simple alternating current (AC) circuit consisting of a source and a
4930:
4898:
4691:
4679:
4599:
4525:
4515:
4445:
4100:
1692:
1118:
Combining, the complex power (units in volt-amps, VA) is back-derived as
388:
357:
These are all denoted in the adjacent diagram (called a power triangle).
262:
69:
49:
is the time rate of flow of energy past a given point of the circuit. In
449:
88:
Active, reactive, apparent, and complex power in sinusoidal steady-state
37:
4893:
4888:
4701:
4684:
4540:
4395:
1241:
The ratio of active power to apparent power in a circuit is called the
605:
Other forms of complex power (units in volt-amps, VA) are derived from
4373:
4609:
4604:
4490:
4450:
1349:
124:
carry the total current, not just the current that does useful work.
104:
100:
58:
293:: the angle of difference (in degrees) between current and voltage;
54:
4165:
Thomas, Roland E.; Rosa, Albert J.; Toussaint, Gregory J. (2016).
909:
Reactive power (units in volts-amps-reactive, var) is derived as:
4644:
4634:
4076:
1048:
For a purely reactive load, reactive power can be simplified to:
395:
4639:
1902:
that is, the arithmetic sum of the phase apparent powers; and
455:
3505:{\displaystyle P_{\text{avg}}={\frac {1}{n}}\sum _{k=1}^{n}VI}
844:
For a purely resistive load, real power can be simplified to:
1227:
These are simplified diagrammatically by the power triangle.
143:
4594:
4289:
2139:
that is, the magnitude of total three-phase complex power.
410:
and its lack has been cited as a significant factor in the
384:
234:
62:
1669:
353:
I vector), current leading voltage (quadrant IV vector).
4158:
402:. The unit for reactive power is var, which stands for
527:
denotes voltage in phasor form, with the amplitude as
126:
Insufficient reactive power can depress voltage levels
3531:
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2159:
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1911:
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467:
450:
Calculations and equations in sinusoidal steady-state
299:
201:
4072:
4039:
However, the time average of a function of the form
3281:
are the time-varying voltage and current waveforms.
4164:
2621:denotes the positive sequence voltage phasor, and
1160:and the apparent power (units in volt-amps, VA) as
906:denotes resistance (units in ohms, Ω) of the load.
694:{\displaystyle S=|I|^{2}Z={\frac {|V|^{2}}{Z^{*}}}}
4028:
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3408:
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387:(symbol: W). Apparent power is often expressed in
341:
207:
2807:
1339:
5161:
4192:Power System Harmonics and Passive Filter Design
4355:
4353:
2143:this debate did not come until the late 1990s.
368:is the reactive power (in this case positive),
2648:denotes the positive sequence current phasor.
1681:Phenomena of Retardation in the Induction Coil
4411:
4283:"August 14, 2003 Outage – Sequence of Events"
3516:
2413:{\displaystyle \mathrm {pf} ={P^{+} \over S}}
1367:Voltage control and reactive power management
454:The formula for complex power (units: VA) in
103:at the same frequency. If the load is purely
4350:
4128:
4126:
512:{\displaystyle S=VI^{*}=|S|\angle \varphi }
4418:
4404:
4167:The Analysis and Design of Linear Circuits
3070:signs can be removed from S and X and get
1691:(1915). However, with the development of
53:circuits, energy storage elements such as
4219:"Importance of Reactive Power for System"
3397:
3384:
3196:
2806:
1360:
1216:{\displaystyle |S|={\sqrt {P^{2}+Q^{2}}}}
4425:
4123:
1331:acceptable limits. A technique known as
440:
342:{\displaystyle \varphi =\arg(V)-\arg(I)}
142:
36:
4183:
4169:(8 ed.). Wiley. pp. 812–813.
2651:
1670:Unbalanced sinusoidal polyphase systems
372:is the complex power and the length of
99:load, both the current and voltage are
14:
5162:
1096:{\displaystyle Q={\frac {|V|^{2}}{X}}}
892:{\displaystyle P={\frac {|V|^{2}}{R}}}
4399:
2686:). Therefore, for a perfect resistor
2451:denotes the positive sequence power:
119:Apparent power is the product of the
287:Phase of voltage relative to current
193:Phase of voltage relative to current
27:Power in alternating current systems
4362:IEEE Transactions on Power Delivery
4189:
3159:Instantaneous power is defined as:
1689:Theoretical Elements of Engineering
24:
5096:Renewable energy commercialization
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4194:. Wiley, IEEE Press. p. 2.
4075:
1115:(units in ohms, Ω) of the load.
391:(VA) since it is the product of
4319:The Analysis of Linear Circuits
4225:from the original on 2015-05-12
3212:{\displaystyle p(t)=v(t)\,i(t)}
1230:
590:{\displaystyle {\overline {I}}}
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3178:
3172:
3043:of the capacitor or inductor.
2992:
2984:
2950:
2942:
2872:
2864:
2578:
2574:
2561:
2547:
2534:
2524:
2513:
2498:
2493:
2478:
2353:
2331:
2323:
2301:
2293:
2271:
2256:
2234:
2226:
2204:
2196:
2174:
2027:
2023:
1978:
1926:
1790:
1773:
1765:
1748:
1740:
1723:
1675:the concept are attributed to
1666:and voltage control circuits.
1340:Capacitive vs. inductive loads
1306:. The apparent power is then:
1179:
1171:
1077:
1068:
1013:
1004:
991:
982:
962:
953:
934:
926:
873:
864:
809:
800:
787:
778:
758:
749:
730:
722:
668:
659:
639:
630:
499:
491:
336:
330:
318:
312:
13:
1:
5091:Renewable Energy Certificates
5051:Cost of electricity by source
4973:Arc-fault circuit interrupter
4849:High-voltage shore connection
4321:. pp. 398 (section 8.3).
4116:
1298:Example: The active power is
5106:Spark/Dark/Quark/Bark spread
4904:Transmission system operator
4864:Mains electricity by country
4441:Automatic generation control
4292:. 2003-09-12. Archived from
4142:10.1109/IEEESTD.2010.5439063
1308:700 W / cos(45.6°) = 1000 VA
582:
7:
5131:List of electricity sectors
5126:Electric energy consumption
4844:High-voltage direct current
4819:Electric power transmission
4809:Electric power distribution
4486:Energy return on investment
4096:Electric power transmission
4068:
349:. Current lagging voltage (
10:
5191:
5046:Carbon offsets and credits
4764:Three-phase electric power
3517:Multiple frequency systems
2146:A new definition based on
1364:
1288:{\displaystyle \cos \phi }
1234:
543:is used, which is denoted
412:Northeast blackout of 2003
383:The unit for power is the
29:
5139:
5114:
5101:Renewable Energy Payments
5024:
4961:
4923:
4777:
4714:
4625:
4590:Fossil fuel power station
4580:
4573:
4562:
4433:
3522:very interesting result.
4884:Single-wire earth return
4824:Electrical busbar system
4481:Energy demand management
1262:{\displaystyle \varphi }
208:{\displaystyle \varphi }
45:In an electric circuit,
5015:Residual-current device
5005:Power system protection
4995:Generator interlock kit
4250:April 23, 2015, at the
4063:power factor correction
2684:passive sign convention
1664:static VAR compensators
4799:Distributed generation
4471:Electric power quality
4391:"AC Power Java Applet"
4333:"Load differentiation"
4051:is zero provided that
4030:
3506:
3477:
3410:
3275:
3246:
3213:
3149:
3060:
3032:
3007:
2812:
2676:
2642:
2615:
2586:
2445:
2414:
2368:
2148:symmetrical components
2129:
2035:
1892:
1798:
1649:
1361:Reactive power control
1289:
1263:
1217:
1150:
1149:{\displaystyle S=P+jQ}
1097:
1038:
893:
834:
695:
591:
564:
513:
446:
343:
215:
209:
110:If the load is purely
42:
5071:Fossil fuel phase-out
4839:Electricity retailing
4834:Electrical substation
4814:Electric power system
4031:
3507:
3457:
3411:
3276:
3247:
3214:
3150:
3061:
3033:
3008:
2813:
2677:
2643:
2641:{\displaystyle I^{+}}
2616:
2614:{\displaystyle V^{+}}
2587:
2446:
2444:{\displaystyle P^{+}}
2415:
2369:
2130:
2036:
1893:
1799:
1650:
1333:reactive compensation
1325:volt-amperes reactive
1290:
1264:
1218:
1151:
1098:
1039:
894:
835:
696:
592:
565:
563:{\displaystyle I^{*}}
514:
444:
364:is the active power,
344:
210:
146:
40:
4427:Electricity delivery
3529:
3431:
3295:
3274:{\displaystyle i(t)}
3256:
3245:{\displaystyle v(t)}
3227:
3166:
3077:
3050:
3022:
2829:
2693:
2660:
2652:Real number formulas
2625:
2598:
2458:
2428:
2379:
2157:
2046:
1909:
1809:
1706:
1379:
1354:Induction generators
1273:
1253:
1167:
1125:
1055:
916:
851:
712:
620:
613:(units in ohms, Ω).
574:
547:
465:
404:volt-ampere reactive
297:
249:volt-ampere reactive
199:
5036:Availability factor
4988:Sulfur hexafluoride
4869:Overhead power line
4769:Virtual power plant
4744:Induction generator
4697:Sustainable biofuel
4506:Home energy storage
4496:Grid energy storage
4461:Droop speed control
4190:Das, J. C. (2015).
4091:War of the currents
3371:
3139:
3108:
2981:
2940:
2800:
2769:
2675:{\displaystyle P=S}
2351:
2321:
2291:
2254:
2224:
2194:
1295:" for this reason.
283:: volt-ampere (VA);
51:alternating current
47:instantaneous power
4910:Transmission tower
4521:Nameplate capacity
4317:Close, Charles M.
4026:
4024:
3502:
3406:
3343:
3271:
3242:
3209:
3145:
3117:
3086:
3056:
3028:
3003:
3001:
2959:
2918:
2808:
2778:
2747:
2672:
2638:
2611:
2582:
2441:
2410:
2364:
2335:
2305:
2275:
2238:
2208:
2178:
2125:
2031:
1888:
1794:
1645:
1304:cos(45.6°) = 0.700
1285:
1259:
1213:
1146:
1093:
1034:
889:
830:
691:
587:
560:
509:
447:
339:
216:
205:
149:The Power Triangle
43:
5157:
5156:
5061:Environmental tax
4941:Cascading failure
4710:
4709:
4546:Utility frequency
4374:10.1109/61.252612
4221:. 21 March 2011.
4201:978-1-118-86162-2
4176:978-1-119-23538-5
4151:978-0-7381-6058-0
4106:Mains electricity
4058:harmonic currents
3942:
3851:
3739:
3635:
3455:
3441:
3341:
3305:
3143:
3059:{\displaystyle X}
3031:{\displaystyle X}
2997:
2804:
2408:
2362:
2123:
1886:
1697:polyphase systems
1619:
1580:
1532:
1484:
1451:
1412:
1211:
1091:
1024:
887:
820:
689:
585:
537:complex conjugate
279:of complex power
32:Mains electricity
16:(Redirected from
5182:
5147:
5146:
5056:Energy subsidies
5010:Protective relay
4951:Rolling blackout
4578:
4577:
4568:
4536:Power-flow study
4476:Electrical fault
4420:
4413:
4406:
4397:
4396:
4378:
4377:
4357:
4348:
4347:
4345:
4344:
4335:. Archived from
4329:
4323:
4322:
4314:
4308:
4307:
4305:
4304:
4298:
4287:
4279:
4273:
4260:
4254:
4240:
4234:
4233:
4231:
4230:
4215:
4209:
4208:
4187:
4181:
4180:
4162:
4156:
4155:
4130:
4085:
4080:
4079:
4050:
4035:
4033:
4032:
4027:
4025:
4021:
4017:
4016:
4015:
4003:
4002:
3987:
3983:
3982:
3981:
3969:
3968:
3943:
3935:
3930:
3926:
3925:
3924:
3912:
3911:
3896:
3892:
3891:
3890:
3878:
3877:
3852:
3844:
3840:
3831:
3827:
3826:
3822:
3821:
3820:
3805:
3804:
3787:
3783:
3782:
3781:
3766:
3765:
3740:
3732:
3727:
3723:
3722:
3718:
3717:
3716:
3701:
3700:
3683:
3679:
3678:
3677:
3662:
3661:
3636:
3628:
3624:
3612:
3611:
3596:
3595:
3574:
3573:
3558:
3557:
3535:
3511:
3509:
3508:
3503:
3476:
3471:
3456:
3448:
3443:
3442:
3439:
3422:harmonic content
3415:
3413:
3412:
3407:
3402:
3370:
3369:
3368:
3358:
3357:
3356:
3342:
3340:
3339:
3338:
3326:
3325:
3312:
3307:
3306:
3303:
3280:
3278:
3277:
3272:
3251:
3249:
3248:
3243:
3218:
3216:
3215:
3210:
3154:
3152:
3151:
3146:
3144:
3138:
3133:
3132:
3116:
3107:
3102:
3101:
3065:
3063:
3062:
3057:
3037:
3035:
3034:
3029:
3012:
3010:
3009:
3004:
3002:
2998:
2996:
2995:
2987:
2980:
2975:
2974:
2958:
2953:
2945:
2939:
2934:
2933:
2914:
2913:
2912:
2896:
2895:
2894:
2875:
2867:
2817:
2815:
2814:
2809:
2805:
2799:
2794:
2793:
2777:
2768:
2763:
2762:
2743:
2742:
2741:
2725:
2724:
2723:
2681:
2679:
2678:
2673:
2647:
2645:
2644:
2639:
2637:
2636:
2620:
2618:
2617:
2612:
2610:
2609:
2591:
2589:
2588:
2583:
2581:
2577:
2573:
2572:
2550:
2546:
2545:
2516:
2511:
2510:
2501:
2496:
2491:
2490:
2481:
2470:
2469:
2450:
2448:
2447:
2442:
2440:
2439:
2419:
2417:
2416:
2411:
2409:
2404:
2403:
2394:
2389:
2373:
2371:
2370:
2365:
2363:
2361:
2357:
2356:
2350:
2345:
2344:
2334:
2326:
2320:
2315:
2314:
2304:
2296:
2290:
2285:
2284:
2274:
2264:
2260:
2259:
2253:
2248:
2247:
2237:
2229:
2223:
2218:
2217:
2207:
2199:
2193:
2188:
2187:
2177:
2167:
2134:
2132:
2131:
2126:
2124:
2122:
2121:
2112:
2111:
2110:
2109:
2096:
2095:
2094:
2081:
2080:
2079:
2068:
2063:
2062:
2057:
2040:
2038:
2037:
2032:
2030:
2022:
2021:
2020:
2007:
2006:
2005:
1992:
1991:
1990:
1971:
1970:
1969:
1956:
1955:
1954:
1941:
1940:
1939:
1929:
1921:
1920:
1897:
1895:
1894:
1889:
1887:
1885:
1884:
1875:
1874:
1873:
1872:
1859:
1858:
1857:
1844:
1843:
1842:
1831:
1826:
1825:
1820:
1803:
1801:
1800:
1795:
1793:
1788:
1787:
1786:
1776:
1768:
1763:
1762:
1761:
1751:
1743:
1738:
1737:
1736:
1726:
1718:
1717:
1654:
1652:
1651:
1646:
1644:
1617:
1578:
1530:
1482:
1449:
1410:
1309:
1305:
1301:
1294:
1292:
1291:
1286:
1268:
1266:
1265:
1260:
1222:
1220:
1219:
1214:
1212:
1210:
1209:
1197:
1196:
1187:
1182:
1174:
1155:
1153:
1152:
1147:
1102:
1100:
1099:
1094:
1092:
1087:
1086:
1085:
1080:
1071:
1065:
1043:
1041:
1040:
1035:
1033:
1025:
1023:
1022:
1021:
1016:
1007:
1001:
1000:
999:
994:
985:
979:
971:
970:
965:
956:
948:
937:
929:
898:
896:
895:
890:
888:
883:
882:
881:
876:
867:
861:
839:
837:
836:
831:
829:
821:
819:
818:
817:
812:
803:
797:
796:
795:
790:
781:
775:
767:
766:
761:
752:
744:
733:
725:
700:
698:
697:
692:
690:
688:
687:
678:
677:
676:
671:
662:
656:
648:
647:
642:
633:
596:
594:
593:
588:
586:
578:
569:
567:
566:
561:
559:
558:
518:
516:
515:
510:
502:
494:
486:
485:
408:electrical grids
360:In the diagram,
348:
346:
345:
340:
214:
212:
211:
206:
21:
5190:
5189:
5185:
5184:
5183:
5181:
5180:
5179:
5160:
5159:
5158:
5153:
5135:
5119:
5117:
5110:
5041:Capacity factor
5029:
5027:
5020:
5000:Numerical relay
4978:Circuit breaker
4966:
4964:
4957:
4919:
4859:Load management
4829:Electrical grid
4794:Demand response
4787:
4782:
4773:
4754:Microgeneration
4706:
4621:
4569:
4560:
4556:Vehicle-to-grid
4429:
4424:
4387:
4382:
4381:
4358:
4351:
4342:
4340:
4331:
4330:
4326:
4315:
4311:
4302:
4300:
4296:
4285:
4281:
4280:
4276:
4261:
4257:
4252:Wayback Machine
4241:
4237:
4228:
4226:
4217:
4216:
4212:
4202:
4188:
4184:
4177:
4163:
4159:
4152:
4132:
4131:
4124:
4119:
4081:
4074:
4071:
4040:
4023:
4022:
4011:
4007:
3998:
3994:
3977:
3973:
3964:
3960:
3959:
3955:
3954:
3950:
3934:
3920:
3916:
3907:
3903:
3886:
3882:
3873:
3869:
3868:
3864:
3863:
3859:
3843:
3841:
3839:
3833:
3832:
3816:
3812:
3800:
3796:
3795:
3791:
3777:
3773:
3761:
3757:
3756:
3752:
3751:
3747:
3731:
3712:
3708:
3696:
3692:
3691:
3687:
3673:
3669:
3657:
3653:
3652:
3648:
3647:
3643:
3627:
3625:
3623:
3617:
3616:
3607:
3603:
3591:
3587:
3569:
3565:
3553:
3549:
3532:
3530:
3527:
3526:
3519:
3472:
3461:
3447:
3438:
3434:
3432:
3429:
3428:
3398:
3364:
3360:
3359:
3352:
3348:
3347:
3334:
3330:
3321:
3317:
3316:
3311:
3302:
3298:
3296:
3293:
3292:
3257:
3254:
3253:
3228:
3225:
3224:
3167:
3164:
3163:
3134:
3122:
3121:
3115:
3103:
3091:
3090:
3078:
3075:
3074:
3051:
3048:
3047:
3023:
3020:
3019:
3000:
2999:
2991:
2983:
2982:
2976:
2964:
2963:
2957:
2949:
2941:
2935:
2923:
2922:
2902:
2901:
2897:
2884:
2883:
2879:
2871:
2863:
2856:
2850:
2849:
2839:
2832:
2830:
2827:
2826:
2795:
2783:
2782:
2776:
2764:
2752:
2751:
2731:
2730:
2726:
2713:
2712:
2708:
2694:
2691:
2690:
2661:
2658:
2657:
2654:
2632:
2628:
2626:
2623:
2622:
2605:
2601:
2599:
2596:
2595:
2568:
2564:
2560:
2541:
2537:
2533:
2523:
2512:
2506:
2502:
2497:
2492:
2486:
2482:
2477:
2465:
2461:
2459:
2456:
2455:
2435:
2431:
2429:
2426:
2425:
2399:
2395:
2393:
2382:
2380:
2377:
2376:
2352:
2346:
2340:
2339:
2330:
2322:
2316:
2310:
2309:
2300:
2292:
2286:
2280:
2279:
2270:
2269:
2265:
2255:
2249:
2243:
2242:
2233:
2225:
2219:
2213:
2212:
2203:
2195:
2189:
2183:
2182:
2173:
2172:
2168:
2166:
2158:
2155:
2154:
2117:
2113:
2105:
2104:
2100:
2090:
2089:
2085:
2075:
2074:
2070:
2069:
2067:
2058:
2050:
2049:
2047:
2044:
2043:
2026:
2016:
2015:
2011:
2001:
2000:
1996:
1986:
1985:
1981:
1965:
1964:
1960:
1950:
1949:
1945:
1935:
1934:
1930:
1925:
1916:
1912:
1910:
1907:
1906:
1880:
1876:
1868:
1867:
1863:
1853:
1852:
1848:
1838:
1837:
1833:
1832:
1830:
1821:
1813:
1812:
1810:
1807:
1806:
1789:
1782:
1781:
1777:
1772:
1764:
1757:
1756:
1752:
1747:
1739:
1732:
1731:
1727:
1722:
1713:
1709:
1707:
1704:
1703:
1672:
1382:
1380:
1377:
1376:
1369:
1363:
1342:
1320:
1307:
1303:
1299:
1274:
1271:
1270:
1254:
1251:
1250:
1239:
1233:
1205:
1201:
1192:
1188:
1186:
1178:
1170:
1168:
1165:
1164:
1126:
1123:
1122:
1081:
1076:
1075:
1067:
1066:
1064:
1056:
1053:
1052:
1029:
1017:
1012:
1011:
1003:
1002:
995:
990:
989:
981:
980:
978:
966:
961:
960:
952:
944:
933:
925:
917:
914:
913:
877:
872:
871:
863:
862:
860:
852:
849:
848:
825:
813:
808:
807:
799:
798:
791:
786:
785:
777:
776:
774:
762:
757:
756:
748:
740:
729:
721:
713:
710:
709:
683:
679:
672:
667:
666:
658:
657:
655:
643:
638:
637:
629:
621:
618:
617:
597:), rather than
577:
575:
572:
571:
554:
550:
548:
545:
544:
498:
490:
481:
477:
466:
463:
462:
452:
416:complex numbers
298:
295:
294:
200:
197:
196:
190:
181:
172:
163:
154:
152:
90:
35:
28:
23:
22:
15:
12:
11:
5:
5188:
5178:
5177:
5175:Electric power
5172:
5155:
5154:
5152:
5151:
5140:
5137:
5136:
5134:
5133:
5128:
5122:
5120:
5116:Statistics and
5115:
5112:
5111:
5109:
5108:
5103:
5098:
5093:
5088:
5083:
5078:
5073:
5068:
5066:Feed-in tariff
5063:
5058:
5053:
5048:
5043:
5038:
5032:
5030:
5025:
5022:
5021:
5019:
5018:
5012:
5007:
5002:
4997:
4992:
4991:
4990:
4985:
4975:
4969:
4967:
4962:
4959:
4958:
4956:
4955:
4954:
4953:
4943:
4938:
4933:
4927:
4925:
4921:
4920:
4918:
4917:
4912:
4907:
4901:
4896:
4891:
4886:
4881:
4876:
4871:
4866:
4861:
4856:
4854:Interconnector
4851:
4846:
4841:
4836:
4831:
4826:
4821:
4816:
4811:
4806:
4804:Dynamic demand
4801:
4796:
4790:
4788:
4778:
4775:
4774:
4772:
4771:
4766:
4761:
4756:
4751:
4746:
4741:
4736:
4734:Combined cycle
4731:
4726:
4720:
4718:
4712:
4711:
4708:
4707:
4705:
4704:
4699:
4694:
4689:
4688:
4687:
4682:
4677:
4672:
4667:
4657:
4652:
4647:
4642:
4637:
4631:
4629:
4623:
4622:
4620:
4619:
4614:
4613:
4612:
4607:
4602:
4597:
4586:
4584:
4575:
4571:
4570:
4563:
4561:
4559:
4558:
4553:
4548:
4543:
4538:
4533:
4528:
4523:
4518:
4513:
4511:Load-following
4508:
4503:
4498:
4493:
4488:
4483:
4478:
4473:
4468:
4466:Electric power
4463:
4458:
4453:
4448:
4443:
4437:
4435:
4431:
4430:
4423:
4422:
4415:
4408:
4400:
4394:
4393:
4386:
4385:External links
4383:
4380:
4379:
4368:(3): 841–852.
4349:
4324:
4309:
4274:
4255:
4235:
4210:
4200:
4182:
4175:
4157:
4150:
4136:. IEEE. 2010.
4121:
4120:
4118:
4115:
4114:
4113:
4111:Deformed power
4108:
4103:
4098:
4093:
4087:
4086:
4070:
4067:
4037:
4036:
4020:
4014:
4010:
4006:
4001:
3997:
3993:
3990:
3986:
3980:
3976:
3972:
3967:
3963:
3958:
3953:
3949:
3946:
3941:
3938:
3933:
3929:
3923:
3919:
3915:
3910:
3906:
3902:
3899:
3895:
3889:
3885:
3881:
3876:
3872:
3867:
3862:
3858:
3855:
3850:
3847:
3842:
3838:
3835:
3834:
3830:
3825:
3819:
3815:
3811:
3808:
3803:
3799:
3794:
3790:
3786:
3780:
3776:
3772:
3769:
3764:
3760:
3755:
3750:
3746:
3743:
3738:
3735:
3730:
3726:
3721:
3715:
3711:
3707:
3704:
3699:
3695:
3690:
3686:
3682:
3676:
3672:
3668:
3665:
3660:
3656:
3651:
3646:
3642:
3639:
3634:
3631:
3626:
3622:
3619:
3618:
3615:
3610:
3606:
3602:
3599:
3594:
3590:
3586:
3583:
3580:
3577:
3572:
3568:
3564:
3561:
3556:
3552:
3548:
3545:
3542:
3539:
3536:
3534:
3518:
3515:
3514:
3513:
3501:
3498:
3495:
3492:
3489:
3486:
3483:
3480:
3475:
3470:
3467:
3464:
3460:
3454:
3451:
3446:
3437:
3418:
3417:
3405:
3401:
3396:
3393:
3390:
3387:
3383:
3380:
3377:
3374:
3367:
3363:
3355:
3351:
3346:
3337:
3333:
3329:
3324:
3320:
3315:
3310:
3301:
3270:
3267:
3264:
3261:
3241:
3238:
3235:
3232:
3221:
3220:
3208:
3205:
3202:
3199:
3195:
3192:
3189:
3186:
3183:
3180:
3177:
3174:
3171:
3157:
3156:
3142:
3137:
3131:
3128:
3125:
3120:
3114:
3111:
3106:
3100:
3097:
3094:
3089:
3085:
3082:
3055:
3027:
3015:
3014:
2994:
2990:
2986:
2979:
2973:
2970:
2967:
2962:
2956:
2952:
2948:
2944:
2938:
2932:
2929:
2926:
2921:
2917:
2911:
2908:
2905:
2900:
2893:
2890:
2887:
2882:
2878:
2874:
2870:
2866:
2862:
2859:
2857:
2855:
2852:
2851:
2848:
2845:
2842:
2840:
2838:
2835:
2834:
2820:
2819:
2803:
2798:
2792:
2789:
2786:
2781:
2775:
2772:
2767:
2761:
2758:
2755:
2750:
2746:
2740:
2737:
2734:
2729:
2722:
2719:
2716:
2711:
2707:
2704:
2701:
2698:
2671:
2668:
2665:
2653:
2650:
2635:
2631:
2608:
2604:
2593:
2592:
2580:
2576:
2571:
2567:
2563:
2559:
2556:
2553:
2549:
2544:
2540:
2536:
2532:
2529:
2526:
2522:
2519:
2515:
2509:
2505:
2500:
2495:
2489:
2485:
2480:
2476:
2473:
2468:
2464:
2438:
2434:
2422:
2421:
2407:
2402:
2398:
2392:
2388:
2385:
2374:
2360:
2355:
2349:
2343:
2338:
2333:
2329:
2325:
2319:
2313:
2308:
2303:
2299:
2295:
2289:
2283:
2278:
2273:
2268:
2263:
2258:
2252:
2246:
2241:
2236:
2232:
2228:
2222:
2216:
2211:
2206:
2202:
2198:
2192:
2186:
2181:
2176:
2171:
2165:
2162:
2137:
2136:
2120:
2116:
2108:
2103:
2099:
2093:
2088:
2084:
2078:
2073:
2066:
2061:
2056:
2053:
2041:
2029:
2025:
2019:
2014:
2010:
2004:
1999:
1995:
1989:
1984:
1980:
1977:
1974:
1968:
1963:
1959:
1953:
1948:
1944:
1938:
1933:
1928:
1924:
1919:
1915:
1900:
1899:
1883:
1879:
1871:
1866:
1862:
1856:
1851:
1847:
1841:
1836:
1829:
1824:
1819:
1816:
1804:
1792:
1785:
1780:
1775:
1771:
1767:
1760:
1755:
1750:
1746:
1742:
1735:
1730:
1725:
1721:
1716:
1712:
1671:
1668:
1656:
1655:
1643:
1640:
1637:
1634:
1631:
1628:
1625:
1622:
1616:
1613:
1610:
1607:
1604:
1601:
1598:
1595:
1592:
1589:
1586:
1583:
1577:
1574:
1571:
1568:
1565:
1562:
1559:
1556:
1553:
1550:
1547:
1544:
1541:
1538:
1535:
1529:
1526:
1523:
1520:
1517:
1514:
1511:
1508:
1505:
1502:
1499:
1496:
1493:
1490:
1487:
1481:
1478:
1475:
1472:
1469:
1466:
1463:
1460:
1457:
1454:
1448:
1445:
1442:
1439:
1436:
1433:
1430:
1427:
1424:
1421:
1418:
1415:
1409:
1406:
1403:
1400:
1397:
1394:
1391:
1388:
1385:
1365:Main article:
1362:
1359:
1341:
1338:
1319:
1318:Reactive power
1316:
1284:
1281:
1278:
1258:
1235:Main article:
1232:
1229:
1225:
1224:
1208:
1204:
1200:
1195:
1191:
1185:
1181:
1177:
1173:
1158:
1157:
1145:
1142:
1139:
1136:
1133:
1130:
1105:
1104:
1090:
1084:
1079:
1074:
1070:
1063:
1060:
1046:
1045:
1032:
1028:
1020:
1015:
1010:
1006:
998:
993:
988:
984:
977:
974:
969:
964:
959:
955:
951:
947:
943:
940:
936:
932:
928:
924:
921:
901:
900:
886:
880:
875:
870:
866:
859:
856:
842:
841:
828:
824:
816:
811:
806:
802:
794:
789:
784:
780:
773:
770:
765:
760:
755:
751:
747:
743:
739:
736:
732:
728:
724:
720:
717:
703:
702:
686:
682:
675:
670:
665:
661:
654:
651:
646:
641:
636:
632:
628:
625:
584:
581:
557:
553:
521:
520:
508:
505:
501:
497:
493:
489:
484:
480:
476:
473:
470:
451:
448:
436:imaginary unit
378:imaginary axis
355:
354:
338:
335:
332:
329:
326:
323:
320:
317:
314:
311:
308:
305:
302:
284:
269:Apparent power
266:
252:
241:Reactive power
238:
204:
184:Apparent power
166:Reactive power
97:time-invariant
89:
86:
82:reactive power
26:
9:
6:
4:
3:
2:
5187:
5176:
5173:
5171:
5168:
5167:
5165:
5150:
5142:
5141:
5138:
5132:
5129:
5127:
5124:
5123:
5121:
5113:
5107:
5104:
5102:
5099:
5097:
5094:
5092:
5089:
5087:
5086:Pigouvian tax
5084:
5082:
5079:
5077:
5074:
5072:
5069:
5067:
5064:
5062:
5059:
5057:
5054:
5052:
5049:
5047:
5044:
5042:
5039:
5037:
5034:
5033:
5031:
5023:
5016:
5013:
5011:
5008:
5006:
5003:
5001:
4998:
4996:
4993:
4989:
4986:
4984:
4983:Earth-leakage
4981:
4980:
4979:
4976:
4974:
4971:
4970:
4968:
4960:
4952:
4949:
4948:
4947:
4944:
4942:
4939:
4937:
4934:
4932:
4929:
4928:
4926:
4924:Failure modes
4922:
4916:
4913:
4911:
4908:
4905:
4902:
4900:
4897:
4895:
4892:
4890:
4887:
4885:
4882:
4880:
4877:
4875:
4874:Power station
4872:
4870:
4867:
4865:
4862:
4860:
4857:
4855:
4852:
4850:
4847:
4845:
4842:
4840:
4837:
4835:
4832:
4830:
4827:
4825:
4822:
4820:
4817:
4815:
4812:
4810:
4807:
4805:
4802:
4800:
4797:
4795:
4792:
4791:
4789:
4786:
4781:
4776:
4770:
4767:
4765:
4762:
4760:
4759:Rankine cycle
4757:
4755:
4752:
4750:
4747:
4745:
4742:
4740:
4739:Cooling tower
4737:
4735:
4732:
4730:
4727:
4725:
4722:
4721:
4719:
4717:
4713:
4703:
4700:
4698:
4695:
4693:
4690:
4686:
4683:
4681:
4678:
4676:
4673:
4671:
4668:
4666:
4663:
4662:
4661:
4658:
4656:
4653:
4651:
4648:
4646:
4643:
4641:
4638:
4636:
4633:
4632:
4630:
4628:
4624:
4618:
4615:
4611:
4608:
4606:
4603:
4601:
4598:
4596:
4593:
4592:
4591:
4588:
4587:
4585:
4583:
4582:Non-renewable
4579:
4576:
4572:
4567:
4557:
4554:
4552:
4549:
4547:
4544:
4542:
4539:
4537:
4534:
4532:
4529:
4527:
4524:
4522:
4519:
4517:
4514:
4512:
4509:
4507:
4504:
4502:
4501:Grid strength
4499:
4497:
4494:
4492:
4489:
4487:
4484:
4482:
4479:
4477:
4474:
4472:
4469:
4467:
4464:
4462:
4459:
4457:
4456:Demand factor
4454:
4452:
4449:
4447:
4444:
4442:
4439:
4438:
4436:
4432:
4428:
4421:
4416:
4414:
4409:
4407:
4402:
4401:
4398:
4392:
4389:
4388:
4375:
4371:
4367:
4363:
4356:
4354:
4339:on 2015-10-25
4338:
4334:
4328:
4320:
4313:
4299:on 2007-10-20
4295:
4291:
4284:
4278:
4271:
4270:0-7381-2601-2
4267:
4264:
4259:
4253:
4249:
4246:
4245:
4239:
4224:
4220:
4214:
4207:
4203:
4197:
4193:
4186:
4178:
4172:
4168:
4161:
4153:
4147:
4143:
4139:
4135:
4129:
4127:
4122:
4112:
4109:
4107:
4104:
4102:
4099:
4097:
4094:
4092:
4089:
4088:
4084:
4083:Energy portal
4078:
4073:
4066:
4064:
4059:
4054:
4048:
4044:
4018:
4012:
4008:
4004:
3999:
3995:
3991:
3988:
3984:
3978:
3974:
3970:
3965:
3961:
3956:
3951:
3947:
3944:
3939:
3936:
3931:
3927:
3921:
3917:
3913:
3908:
3904:
3900:
3897:
3893:
3887:
3883:
3879:
3874:
3870:
3865:
3860:
3856:
3853:
3848:
3845:
3836:
3828:
3823:
3817:
3813:
3809:
3806:
3801:
3797:
3792:
3788:
3784:
3778:
3774:
3770:
3767:
3762:
3758:
3753:
3748:
3744:
3741:
3736:
3733:
3728:
3724:
3719:
3713:
3709:
3705:
3702:
3697:
3693:
3688:
3684:
3680:
3674:
3670:
3666:
3663:
3658:
3654:
3649:
3644:
3640:
3637:
3632:
3629:
3620:
3608:
3604:
3600:
3597:
3592:
3588:
3581:
3578:
3570:
3566:
3562:
3559:
3554:
3550:
3543:
3540:
3537:
3525:
3524:
3523:
3496:
3490:
3484:
3478:
3473:
3468:
3465:
3462:
3458:
3452:
3449:
3444:
3435:
3427:
3426:
3425:
3423:
3403:
3391:
3385:
3378:
3372:
3365:
3361:
3353:
3349:
3344:
3335:
3331:
3327:
3322:
3318:
3313:
3308:
3299:
3291:
3290:
3289:
3285:
3282:
3265:
3259:
3236:
3230:
3203:
3197:
3190:
3184:
3181:
3175:
3169:
3162:
3161:
3160:
3140:
3135:
3118:
3112:
3109:
3104:
3087:
3083:
3080:
3073:
3072:
3071:
3069:
3053:
3044:
3042:
3038:
3025:
2988:
2977:
2960:
2954:
2946:
2936:
2919:
2915:
2898:
2880:
2876:
2868:
2860:
2858:
2853:
2846:
2843:
2841:
2836:
2825:
2824:
2823:
2801:
2796:
2779:
2773:
2770:
2765:
2748:
2744:
2727:
2709:
2705:
2702:
2699:
2696:
2689:
2688:
2687:
2685:
2669:
2666:
2663:
2649:
2633:
2629:
2606:
2602:
2569:
2565:
2557:
2554:
2551:
2542:
2538:
2530:
2527:
2520:
2517:
2507:
2503:
2487:
2483:
2474:
2471:
2466:
2462:
2454:
2453:
2452:
2436:
2432:
2405:
2400:
2396:
2390:
2375:
2358:
2347:
2336:
2327:
2317:
2306:
2297:
2287:
2276:
2266:
2261:
2250:
2239:
2230:
2220:
2209:
2200:
2190:
2179:
2169:
2163:
2160:
2153:
2152:
2151:
2149:
2144:
2140:
2118:
2114:
2101:
2097:
2086:
2082:
2071:
2064:
2059:
2042:
2012:
2008:
1997:
1993:
1982:
1975:
1972:
1961:
1957:
1946:
1942:
1931:
1922:
1917:
1913:
1905:
1904:
1903:
1881:
1877:
1864:
1860:
1849:
1845:
1834:
1827:
1822:
1805:
1778:
1769:
1753:
1744:
1728:
1719:
1714:
1710:
1702:
1701:
1700:
1698:
1694:
1690:
1686:
1682:
1678:
1667:
1665:
1661:
1599:
1551:
1515:
1464:
1428:
1375:
1374:
1373:
1368:
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1355:
1351:
1346:
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1328:
1326:
1315:
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1189:
1183:
1175:
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1143:
1140:
1137:
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1128:
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1120:
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884:
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868:
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847:
846:
845:
826:
822:
814:
804:
792:
782:
771:
768:
763:
753:
745:
741:
737:
734:
726:
718:
715:
708:
707:
706:
684:
680:
673:
663:
652:
649:
644:
634:
626:
623:
616:
615:
614:
612:
608:
603:
600:
579:
555:
551:
542:
538:
534:
530:
526:
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487:
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478:
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471:
468:
461:
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457:
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439:
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433:
429:
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425:
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421:
417:
413:
409:
405:
401:
397:
394:
390:
386:
381:
379:
375:
371:
367:
363:
358:
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333:
327:
324:
321:
315:
309:
306:
303:
300:
292:
288:
285:
282:
278:
274:
270:
267:
264:
260:
256:
255:Complex power
253:
250:
246:
242:
239:
236:
232:
228:
224:
221:
220:
219:
202:
194:
189:
188:|S|
185:
180:
176:
175:Complex power
171:
167:
162:
158:
151:
150:
145:
141:
137:
135:
131:
127:
122:
117:
115:
114:
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98:
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85:
83:
79:
75:
71:
66:
64:
60:
56:
52:
48:
39:
33:
19:
5081:Net metering
5028:and policies
4946:Power outage
4915:Utility pole
4879:Pumped hydro
4785:distribution
4780:Transmission
4729:Cogeneration
4723:
4531:Power factor
4365:
4361:
4341:. Retrieved
4337:the original
4327:
4318:
4312:
4301:. Retrieved
4294:the original
4277:
4262:
4258:
4242:
4238:
4227:. Retrieved
4213:
4205:
4191:
4185:
4166:
4160:
4133:
4052:
4046:
4042:
4038:
3520:
3419:
3286:
3283:
3222:
3158:
3045:
3018:
3016:
2821:
2655:
2594:
2423:
2145:
2141:
2138:
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1688:
1680:
1673:
1657:
1370:
1347:
1343:
1329:
1321:
1312:
1297:
1243:power factor
1240:
1237:Power factor
1231:Power factor
1226:
1159:
1117:
1108:
1106:
1047:
908:
903:
902:
843:
704:
606:
604:
598:
540:
532:
524:
522:
453:
431:
427:
423:
419:
389:volt-amperes
382:
377:
373:
369:
365:
361:
359:
356:
290:
286:
280:
272:
268:
258:
254:
244:
240:
230:
226:
223:Active power
222:
217:
192:
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187:
183:
182:
178:
174:
173:
169:
165:
164:
160:
157:Active power
156:
155:
148:
147:
138:
134:power factor
118:
112:
109:
91:
81:
77:
74:active power
73:
67:
44:
5076:Load factor
4931:Black start
4899:Transformer
4600:Natural gas
4551:Variability
4526:Peak demand
4516:Merit order
4446:Backfeeding
4101:Transformer
2682:(using the
1693:three phase
1683:(1888) and
1660:system gain
609:, the load
263:volt-ampere
70:AC waveform
5164:Categories
5118:production
4963:Protective
4894:Super grid
4889:Smart grid
4716:Generation
4650:Geothermal
4541:Repowering
4343:2015-04-29
4303:2008-02-18
4229:2015-04-29
4117:References
231:real power
101:sinusoidal
78:real power
59:capacitors
18:Real power
5026:Economics
4749:Micro CHP
4627:Renewable
4610:Petroleum
4605:Oil shale
4491:Grid code
4451:Base load
4272:, page 23
4005:−
3975:ω
3971:−
3962:ω
3948:
3884:ω
3871:ω
3857:
3798:ω
3789:−
3759:ω
3745:
3694:ω
3655:ω
3641:
3589:ω
3582:
3551:ω
3544:
3459:∑
3345:∫
3328:−
3041:reactance
2558:
2552:−
2531:
2521:
1685:Steinmetz
1350:inductors
1283:ϕ
1280:
1257:φ
1113:reactance
1027:×
946:φ
942:
823:×
742:φ
738:
685:∗
611:impedance
583:¯
556:∗
507:φ
504:∠
483:∗
458:form is:
328:
322:−
310:
301:φ
277:magnitude
203:φ
105:resistive
55:inductors
5170:AC power
5149:Category
4936:Brownout
4724:AC power
4434:Concepts
4248:Archived
4223:Archived
4069:See also
1111:denotes
428:j Q
398:and RMS
351:quadrant
130:blackout
113:reactive
4965:devices
4675:Thermal
4670:Osmotic
4665:Current
4645:Biomass
4635:Biofuel
4617:Nuclear
4574:Sources
3068:modulus
3039:is the
1677:Stanley
434:is the
430:(where
400:current
396:voltage
275:|: the
4660:Marine
4640:Biogas
4268:
4198:
4173:
4148:
3223:where
3017:where
1618:
1579:
1531:
1483:
1450:
1411:
1107:where
531:, and
523:where
456:phasor
251:(var);
94:linear
5017:(GFI)
4906:(TSO)
4692:Solar
4680:Tidal
4655:Hydro
4297:(PDF)
4286:(PDF)
1658:The "
1300:700 W
1247:phase
265:(VA);
229:, or
4783:and
4702:Wind
4685:Wave
4595:Coal
4290:FERC
4266:ISBN
4196:ISBN
4171:ISBN
4146:ISBN
4041:cos(
3252:and
570:(or
385:watt
237:(W);
235:watt
63:watt
57:and
4370:doi
4138:doi
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3854:cos
3742:cos
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3304:avg
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1687:'s
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939:sin
735:cos
539:of
529:RMS
438:).
393:RMS
325:arg
307:arg
271:, |
136:).
121:RMS
76:or
5166::
4364:.
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4288:.
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34:.
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