2245:: Power flow formulation that provides guarantee of uniqueness of solution and independence on initial conditions for electrical distribution systems. The LPF is based on the current injection method (CIM) and applies the Laurent series expansion. The main characteristics of this formulation are its proven numerical convergence and stability, and its computational advantages, showing to be at least ten times faster than the BFS method both in balanced and unbalanced networks. Since it is based on the system's admittance matrix, the formulation is able to consider radial and meshed network topologies without additional modifications (contrary to the compensation-based BFS). The simplicity and computational efficiency of the LPF method make it an attractive option for recursive power flow problems, such as those encountered in time-series analyses, metaheuristics, probabilistic analysis, reinforcement learning applied to power systems, and other related applications.
93:
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
1285:
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.
2239:: 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).
2659:
3238:
2023:
1861:
1284:
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
2226:
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
125:
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
146:
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. Once this information is known, real and reactive power flow on each branch as well as generator reactive
100:
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
150:
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
126:
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
121:
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
1280:
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
92:
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.
50:
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
1415:
1830:
1648:
319:
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
2227:
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.
1242:
696:
2429:
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.
2018:{\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}}}
2233:: 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.
2467:
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.
2454:
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.
522:
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:
1310:
2262:
flows. This method is non-iterative and absolutely convergent but less accurate than AC Load Flow solutions. Direct current load flow is used wherever repetitive and fast load flow estimations are required.
2088:
1652:
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1305:
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:
1067:
2188:
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.
62:
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.
2173:
1074:
528:
2216:: 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.
147:
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.
94:
101:
consideration, there are several approaches that has been used such as probabilistic, possibilistic, information gap decision theory, robust optimization, and interval analysis.
2222:
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
951:
520:
467:
411:
2481:
Seifi, H. &. (2011). Appendix A: DC Load Flow. In H. &. Seifi, Electric power system planning: issues, algorithms and solutions (pp. 245-249). Berlin: Springer
1463:
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317:
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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.
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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.
3183:
2375:
2321:
Aien, Morteza; Hajebrahimi, Ali; Fotuhi-Firuzabad, Mahmud (2016). "A comprehensive review on uncertainty modeling techniques in power system studies".
1301:
which begins with initial guesses of all unknown variables (voltage magnitude and angles at Load Buses and voltage angles at
Generator Buses). Next, a
212:
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
2037:
2230:
151:
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
2297:
47:
parameters, such as voltages, voltage angles, real power and reactive power. It analyzes the power systems in normal steady-state operation.
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There are several different methods of solving the resulting nonlinear system of equations. The most popular is a variation of the
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to represent all voltages, power flows, and impedances, scaling the actual target system values to some convenient base. A system
1410:{\displaystyle {\begin{bmatrix}\Delta \theta \\\Delta |V|\end{bmatrix}}=-J^{-1}{\begin{bmatrix}\Delta P\\\Delta Q\end{bmatrix}}}
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2896:
2767:
2223:
1855:
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2013 IREP Symposium Bulk Power System
Dynamics and Control - IX Optimization, Security and Control of the Emerging Power Grid
2254:
Direct current load flow gives estimations of lines power flows on AC power systems. Direct current load flow looks only at
3143:
3075:
3065:
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1016:
35:
of the flow of electric power in an interconnected system. A power-flow study usually uses simplified notations such as a
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59:
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1825:{\displaystyle \Delta Q_{i}=-Q_{i}+\sum _{k=1}^{N}|V_{i}||V_{k}|(G_{ik}\sin \theta _{ik}-B_{ik}\cos \theta _{ik})}
1643:{\displaystyle \Delta P_{i}=-P_{i}+\sum _{k=1}^{N}|V_{i}||V_{k}|(G_{ik}\cos \theta _{ik}+B_{ik}\sin \theta _{ik})}
1294:
3193:
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58:
Commercial power systems are usually too complex to allow for hand solution of the power flow. Special-purpose
3168:
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The process continues until a stopping condition is met. A common stopping condition is to terminate if the
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1237:{\displaystyle 0=-Q_{i}+\sum _{k=1}^{N}|V_{i}||V_{k}|(G_{ik}\sin \theta _{ik}-B_{ik}\cos \theta _{ik})}
691:{\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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20:
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36:
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Solve the power balance equations using the most recent voltage angle and magnitude values.
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70:
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In addition to a power-flow study, computer programs perform related calculations such as
40:
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86:
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feasible, and a linear (but less accurate) DC power-flow model is used instead.
114:
2334:
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2633:
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2194:
Linearize the system around the most recent voltage angle and magnitude values
2456:
2416:
2408:
2702:
2697:
2583:
2543:
152:
2443:
2373:
Andersson, G: Lectures on Modelling and Analysis of Electric Power Systems
2280:
Low, S. H. (2013). "Convex relaxation of optimal power flow: A tutorial".
2028:
The linearized system of equations is solved to determine the next guess (
2816:
130:
analysis, another simplification. A further simplification is to use the
44:
2203:
Check the stopping conditions, if met then terminate, else go to step 2.
2737:
2727:
2488:
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is known. For the Slack Bus, it is assumed that the voltage magnitude
2732:
2439:
127:
52:
2469:
2687:
69:
fault analysis, stability studies (transient and steady-state),
2492:
2391:
Stott, B.; Alsac, O. (May 1974). "Fast Decoupled Load Flow".
158:
In the power-flow problem, it is assumed that the real power
2184:
A rough outline of solution of the power-flow problem is:
2181:
of the mismatch equations is below a specified tolerance.
2083:{\displaystyle \theta _{m+1}=\theta _{m}+\Delta \theta \,}
55:, and the real and reactive power flowing in each line.
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1380:
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is a model used in electrical engineering to analyze
1062:{\displaystyle \theta _{ik}=\theta _{i}-\theta _{k}}
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3054:
3016:
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2807:
2718:
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Solve for the change in voltage angle and magnitude
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2393:IEEE Transactions on Power Apparatus and Systems
95:optimal operations of groups of generating units
85:, the conditions which give the lowest cost per
2032:+ 1) of voltage magnitude and angles based on:
953:is the difference in voltage angle between the
2168:{\displaystyle |V|_{m+1}=|V|_{m}+\Delta |V|\,}
1854:is a matrix of partial derivatives known as a
2504:
859:is the imaginary part of the element in the Y
8:
1069:). The reactive power balance equation is:
2670:
2511:
2497:
2489:
2457:https://doi.org/10.1016/j.epsr.2022.108326
1273:is the net reactive power injected at bus
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16:Numerical analysis of electric power flow
2323:Renewable and Sustainable Energy Reviews
727:is the net active power injected at bus
2444:https://www.mdpi.com/1996-1073/14/4/897
2272:
2200:Update the voltage magnitude and angles
761:is the real part of the element in the
2231:Holomorphic embedding load flow method
7:
2348:Grainger, J.; Stevenson, W. (1994).
111:alternating current power-flow model
43:, and focuses on various aspects of
2237:Backward-Forward Sweep (BFS) method
1465:are called the mismatch equations:
77:. In particular, some programs use
3189:Renewable energy commercialization
2440:https://doi.org/10.3390/en14040897
2148:
2073:
1987:
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1474:
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1383:
1332:
1322:
1297:. The Newton-Raphson method is an
14:
2470:https://doi.org/10.1109/59.192932
3237:
3236:
2657:
2243:Laurent Power Flow (LPF) method
2220:Fast-decoupled-load-flow method
2160:
2152:
2135:
2126:
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1289:Newton–Raphson solution method
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142:Power-flow problem formulation
1:
3184:Renewable Energy Certificates
3144:Cost of electricity by source
3066:Arc-fault circuit interrupter
2942:High-voltage shore connection
3199:Spark/Dark/Quark/Bark spread
2997:Transmission system operator
2957:Mains electricity by country
2534:Automatic generation control
946:{\displaystyle \theta _{ik}}
515:{\displaystyle 2(N-1)-(R-1)}
462:{\displaystyle 2(N-1)-(R-1)}
406:{\displaystyle 2(N-1)-(R-1)}
3263:Electric power distribution
3224:List of electricity sectors
3219:Electric energy consumption
2937:High-voltage direct current
2912:Electric power transmission
2902:Electric power distribution
2579:Energy return on investment
360:generators, there are then
3284:
3139:Carbon offsets and credits
2857:Three-phase electric power
2335:10.1016/j.rser.2015.12.070
416:In order to solve for the
239:and the voltage magnitude
3232:
3194:Renewable Energy Payments
2683:Fossil fuel power station
2655:
2352:. New York: McGraw–Hill.
2290:10.1109/IREP.2013.6629391
2977:Single-wire earth return
2917:Electrical busbar system
2574:Energy demand management
2409:10.1109/tpas.1974.293985
2208:Other power-flow methods
1458:{\displaystyle \Delta Q}
1435:{\displaystyle \Delta P}
469:unknowns, there must be
3108:Residual-current device
3098:Power system protection
3088:Generator interlock kit
2395:. PAS-93 (3): 859–869.
312:{\displaystyle \theta }
2892:Distributed generation
2564:Electric power quality
2169:
2084:
2019:
1848:
1826:
1708:
1644:
1526:
1459:
1436:
1411:
1267:
1238:
1120:
1063:
1007:
1006:{\displaystyle k_{th}}
977:
976:{\displaystyle i_{th}}
947:
917:
916:{\displaystyle k_{th}}
887:
886:{\displaystyle i_{th}}
853:
852:{\displaystyle B_{ik}}
823:
822:{\displaystyle k_{th}}
793:
792:{\displaystyle i_{th}}
755:
754:{\displaystyle G_{ik}}
721:
692:
574:
516:
463:
407:
354:
334:
313:
293:
263:
233:
206:
179:
3164:Fossil fuel phase-out
2932:Electricity retailing
2927:Electrical substation
2907:Electric power system
2350:Power System Analysis
2170:
2085:
2020:
1849:
1827:
1688:
1645:
1506:
1460:
1437:
1412:
1295:Newton–Raphson method
1268:
1266:{\displaystyle Q_{i}}
1239:
1100:
1064:
1008:
978:
948:
918:
888:
863:corresponding to the
854:
824:
794:
769:corresponding to the
763:bus admittance matrix
756:
722:
720:{\displaystyle P_{i}}
693:
554:
517:
464:
408:
355:
335:
314:
294:
264:
234:
232:{\displaystyle P_{G}}
207:
205:{\displaystyle Q_{D}}
180:
178:{\displaystyle P_{D}}
2520:Electricity delivery
2093:
2038:
1862:
1838:
1653:
1471:
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1423:
1311:
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1075:
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529:
473:
420:
364:
344:
324:
303:
273:
243:
216:
189:
162:
3129:Availability factor
3081:Sulfur hexafluoride
2962:Overhead power line
2862:Virtual power plant
2837:Induction generator
2790:Sustainable biofuel
2599:Home energy storage
2589:Grid energy storage
2554:Droop speed control
2401:1974ITPAS..93..859S
2258:flows and neglects
2214:Gauss–Seidel method
292:{\displaystyle |V|}
262:{\displaystyle |V|}
185:and reactive power
3003:Transmission tower
2614:Nameplate capacity
2378:2017-02-15 at the
2165:
2080:
2015:
2009:
2005:
1968:
1939:
1902:
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1407:
1401:
1350:
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1234:
1059:
1003:
973:
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913:
883:
849:
819:
789:
751:
717:
688:
512:
459:
403:
350:
330:
309:
299:and voltage phase
289:
259:
229:
202:
175:
83:optimal power flow
79:linear programming
33:numerical analysis
3268:Power engineering
3250:
3249:
3154:Environmental tax
3034:Cascading failure
2803:
2802:
2639:Utility frequency
2299:978-1-4799-0199-9
2284:. pp. 1–06.
2004:
1967:
1938:
1901:
1847:{\displaystyle J}
353:{\displaystyle R}
333:{\displaystyle N}
75:economic dispatch
60:network analyzers
21:power engineering
3275:
3240:
3239:
3149:Energy subsidies
3103:Protective relay
3044:Rolling blackout
2671:
2661:
2629:Power-flow study
2569:Electrical fault
2513:
2506:
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2174:
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2171:
2166:
2163:
2155:
2144:
2143:
2138:
2129:
2121:
2120:
2109:
2100:
2089:
2087:
2086:
2081:
2069:
2068:
2056:
2055:
2024:
2022:
2021:
2016:
2014:
2013:
2006:
2003:
2002:
1994:
1985:
1974:
1969:
1966:
1958:
1947:
1940:
1937:
1936:
1928:
1919:
1908:
1903:
1900:
1892:
1881:
1853:
1851:
1850:
1845:
1831:
1829:
1828:
1823:
1818:
1817:
1799:
1798:
1783:
1782:
1764:
1763:
1748:
1743:
1742:
1733:
1728:
1723:
1722:
1713:
1707:
1702:
1684:
1683:
1668:
1667:
1649:
1647:
1646:
1641:
1636:
1635:
1617:
1616:
1601:
1600:
1582:
1581:
1566:
1561:
1560:
1551:
1546:
1541:
1540:
1531:
1525:
1520:
1502:
1501:
1486:
1485:
1464:
1462:
1461:
1456:
1441:
1439:
1438:
1433:
1416:
1414:
1413:
1408:
1406:
1405:
1374:
1373:
1355:
1354:
1347:
1339:
1299:iterative method
1272:
1270:
1269:
1264:
1262:
1261:
1243:
1241:
1240:
1235:
1230:
1229:
1211:
1210:
1195:
1194:
1176:
1175:
1160:
1155:
1154:
1145:
1140:
1135:
1134:
1125:
1119:
1114:
1096:
1095:
1068:
1066:
1065:
1060:
1058:
1057:
1045:
1044:
1032:
1031:
1012:
1010:
1009:
1004:
1002:
1001:
982:
980:
979:
974:
972:
971:
952:
950:
949:
944:
942:
941:
922:
920:
919:
914:
912:
911:
892:
890:
889:
884:
882:
881:
858:
856:
855:
850:
848:
847:
828:
826:
825:
820:
818:
817:
798:
796:
795:
790:
788:
787:
760:
758:
757:
752:
750:
749:
726:
724:
723:
718:
716:
715:
697:
695:
694:
689:
684:
683:
665:
664:
649:
648:
630:
629:
614:
609:
608:
599:
594:
589:
588:
579:
573:
568:
550:
549:
521:
519:
518:
513:
468:
466:
465:
460:
412:
410:
409:
404:
359:
357:
356:
351:
339:
337:
336:
331:
318:
316:
315:
310:
298:
296:
295:
290:
288:
280:
268:
266:
265:
260:
258:
250:
238:
236:
235:
230:
228:
227:
211:
209:
208:
203:
201:
200:
184:
182:
181:
176:
174:
173:
136:one-line diagram
119:nonlinear system
117:. It provides a
37:one-line diagram
25:power-flow study
3283:
3282:
3278:
3277:
3276:
3274:
3273:
3272:
3253:
3252:
3251:
3246:
3228:
3212:
3210:
3203:
3134:Capacity factor
3122:
3120:
3113:
3093:Numerical relay
3071:Circuit breaker
3059:
3057:
3050:
3012:
2952:Load management
2922:Electrical grid
2887:Demand response
2880:
2875:
2866:
2847:Microgeneration
2799:
2714:
2662:
2653:
2649:Vehicle-to-grid
2522:
2517:
2487:
2486:
2479:
2475:
2466:
2462:
2453:
2449:
2428:
2424:
2390:
2389:
2385:
2380:Wayback Machine
2371:
2367:
2360:
2347:
2346:
2342:
2320:
2319:
2315:
2300:
2279:
2278:
2274:
2269:
2252:
2210:
2133:
2104:
2091:
2090:
2060:
2041:
2036:
2035:
2008:
2007:
1986:
1975:
1970:
1959:
1948:
1942:
1941:
1920:
1909:
1904:
1893:
1882:
1872:
1860:
1859:
1836:
1835:
1806:
1787:
1771:
1752:
1734:
1714:
1675:
1659:
1651:
1650:
1624:
1605:
1589:
1570:
1552:
1532:
1493:
1477:
1469:
1468:
1444:
1443:
1421:
1420:
1400:
1399:
1390:
1389:
1376:
1362:
1349:
1348:
1329:
1328:
1315:
1309:
1308:
1291:
1253:
1248:
1247:
1218:
1199:
1183:
1164:
1146:
1126:
1087:
1073:
1072:
1049:
1036:
1020:
1015:
1014:
990:
985:
984:
960:
955:
954:
930:
925:
924:
900:
895:
894:
870:
865:
864:
862:
836:
831:
830:
806:
801:
800:
776:
771:
770:
768:
738:
733:
732:
707:
702:
701:
672:
653:
637:
618:
600:
580:
541:
527:
526:
471:
470:
418:
417:
362:
361:
342:
341:
322:
321:
301:
300:
271:
270:
241:
240:
219:
214:
213:
192:
187:
186:
165:
160:
159:
144:
132:per-unit system
107:
71:unit commitment
41:per-unit system
29:load-flow study
17:
12:
11:
5:
3281:
3279:
3271:
3270:
3265:
3255:
3254:
3248:
3247:
3245:
3244:
3233:
3230:
3229:
3227:
3226:
3221:
3215:
3213:
3209:Statistics and
3208:
3205:
3204:
3202:
3201:
3196:
3191:
3186:
3181:
3176:
3171:
3166:
3161:
3159:Feed-in tariff
3156:
3151:
3146:
3141:
3136:
3131:
3125:
3123:
3118:
3115:
3114:
3112:
3111:
3105:
3100:
3095:
3090:
3085:
3084:
3083:
3078:
3068:
3062:
3060:
3055:
3052:
3051:
3049:
3048:
3047:
3046:
3036:
3031:
3026:
3020:
3018:
3014:
3013:
3011:
3010:
3005:
3000:
2994:
2989:
2984:
2979:
2974:
2969:
2964:
2959:
2954:
2949:
2947:Interconnector
2944:
2939:
2934:
2929:
2924:
2919:
2914:
2909:
2904:
2899:
2897:Dynamic demand
2894:
2889:
2883:
2881:
2871:
2868:
2867:
2865:
2864:
2859:
2854:
2849:
2844:
2839:
2834:
2829:
2827:Combined cycle
2824:
2819:
2813:
2811:
2805:
2804:
2801:
2800:
2798:
2797:
2792:
2787:
2782:
2781:
2780:
2775:
2770:
2765:
2760:
2750:
2745:
2740:
2735:
2730:
2724:
2722:
2716:
2715:
2713:
2712:
2707:
2706:
2705:
2700:
2695:
2690:
2679:
2677:
2668:
2664:
2663:
2656:
2654:
2652:
2651:
2646:
2641:
2636:
2631:
2626:
2621:
2616:
2611:
2606:
2604:Load-following
2601:
2596:
2591:
2586:
2581:
2576:
2571:
2566:
2561:
2559:Electric power
2556:
2551:
2546:
2541:
2536:
2530:
2528:
2524:
2523:
2518:
2516:
2515:
2508:
2501:
2493:
2485:
2484:
2473:
2460:
2447:
2422:
2383:
2365:
2358:
2340:
2313:
2298:
2271:
2270:
2268:
2265:
2260:reactive power
2251:
2248:
2247:
2246:
2240:
2234:
2228:
2217:
2209:
2206:
2205:
2204:
2201:
2198:
2195:
2192:
2189:
2162:
2158:
2154:
2150:
2147:
2142:
2137:
2132:
2128:
2124:
2119:
2116:
2113:
2108:
2103:
2099:
2078:
2075:
2072:
2067:
2063:
2059:
2054:
2051:
2048:
2044:
2012:
2001:
1997:
1993:
1989:
1984:
1981:
1978:
1971:
1965:
1962:
1957:
1954:
1951:
1944:
1943:
1935:
1931:
1927:
1923:
1918:
1915:
1912:
1905:
1899:
1896:
1891:
1888:
1885:
1878:
1877:
1875:
1870:
1867:
1843:
1821:
1816:
1813:
1809:
1805:
1802:
1797:
1794:
1790:
1786:
1781:
1778:
1774:
1770:
1767:
1762:
1759:
1755:
1751:
1747:
1741:
1737:
1732:
1727:
1721:
1717:
1712:
1706:
1701:
1698:
1695:
1691:
1687:
1682:
1678:
1674:
1671:
1666:
1662:
1658:
1639:
1634:
1631:
1627:
1623:
1620:
1615:
1612:
1608:
1604:
1599:
1596:
1592:
1588:
1585:
1580:
1577:
1573:
1569:
1565:
1559:
1555:
1550:
1545:
1539:
1535:
1530:
1524:
1519:
1516:
1513:
1509:
1505:
1500:
1496:
1492:
1489:
1484:
1480:
1476:
1454:
1451:
1431:
1428:
1404:
1398:
1395:
1392:
1391:
1388:
1385:
1382:
1381:
1379:
1372:
1369:
1365:
1361:
1358:
1353:
1346:
1342:
1338:
1334:
1331:
1330:
1327:
1324:
1321:
1320:
1318:
1290:
1287:
1260:
1256:
1233:
1228:
1225:
1221:
1217:
1214:
1209:
1206:
1202:
1198:
1193:
1190:
1186:
1182:
1179:
1174:
1171:
1167:
1163:
1159:
1153:
1149:
1144:
1139:
1133:
1129:
1124:
1118:
1113:
1110:
1107:
1103:
1099:
1094:
1090:
1086:
1083:
1080:
1056:
1052:
1048:
1043:
1039:
1035:
1030:
1027:
1023:
1000:
997:
993:
970:
967:
963:
940:
937:
933:
910:
907:
903:
880:
877:
873:
860:
846:
843:
839:
816:
813:
809:
786:
783:
779:
766:
748:
745:
741:
714:
710:
687:
682:
679:
675:
671:
668:
663:
660:
656:
652:
647:
644:
640:
636:
633:
628:
625:
621:
617:
613:
607:
603:
598:
593:
587:
583:
578:
572:
567:
564:
561:
557:
553:
548:
544:
540:
537:
534:
511:
508:
505:
502:
499:
496:
493:
490:
487:
484:
481:
478:
458:
455:
452:
449:
446:
443:
440:
437:
434:
431:
428:
425:
402:
399:
396:
393:
390:
387:
384:
381:
378:
375:
372:
369:
349:
329:
308:
287:
283:
279:
257:
253:
249:
226:
222:
199:
195:
172:
168:
143:
140:
106:
103:
15:
13:
10:
9:
6:
4:
3:
2:
3280:
3269:
3266:
3264:
3261:
3260:
3258:
3243:
3235:
3234:
3231:
3225:
3222:
3220:
3217:
3216:
3214:
3206:
3200:
3197:
3195:
3192:
3190:
3187:
3185:
3182:
3180:
3179:Pigouvian tax
3177:
3175:
3172:
3170:
3167:
3165:
3162:
3160:
3157:
3155:
3152:
3150:
3147:
3145:
3142:
3140:
3137:
3135:
3132:
3130:
3127:
3126:
3124:
3116:
3109:
3106:
3104:
3101:
3099:
3096:
3094:
3091:
3089:
3086:
3082:
3079:
3077:
3076:Earth-leakage
3074:
3073:
3072:
3069:
3067:
3064:
3063:
3061:
3053:
3045:
3042:
3041:
3040:
3037:
3035:
3032:
3030:
3027:
3025:
3022:
3021:
3019:
3017:Failure modes
3015:
3009:
3006:
3004:
3001:
2998:
2995:
2993:
2990:
2988:
2985:
2983:
2980:
2978:
2975:
2973:
2970:
2968:
2967:Power station
2965:
2963:
2960:
2958:
2955:
2953:
2950:
2948:
2945:
2943:
2940:
2938:
2935:
2933:
2930:
2928:
2925:
2923:
2920:
2918:
2915:
2913:
2910:
2908:
2905:
2903:
2900:
2898:
2895:
2893:
2890:
2888:
2885:
2884:
2882:
2879:
2874:
2869:
2863:
2860:
2858:
2855:
2853:
2852:Rankine cycle
2850:
2848:
2845:
2843:
2840:
2838:
2835:
2833:
2832:Cooling tower
2830:
2828:
2825:
2823:
2820:
2818:
2815:
2814:
2812:
2810:
2806:
2796:
2793:
2791:
2788:
2786:
2783:
2779:
2776:
2774:
2771:
2769:
2766:
2764:
2761:
2759:
2756:
2755:
2754:
2751:
2749:
2746:
2744:
2741:
2739:
2736:
2734:
2731:
2729:
2726:
2725:
2723:
2721:
2717:
2711:
2708:
2704:
2701:
2699:
2696:
2694:
2691:
2689:
2686:
2685:
2684:
2681:
2680:
2678:
2676:
2675:Non-renewable
2672:
2669:
2665:
2660:
2650:
2647:
2645:
2642:
2640:
2637:
2635:
2632:
2630:
2627:
2625:
2622:
2620:
2617:
2615:
2612:
2610:
2607:
2605:
2602:
2600:
2597:
2595:
2594:Grid strength
2592:
2590:
2587:
2585:
2582:
2580:
2577:
2575:
2572:
2570:
2567:
2565:
2562:
2560:
2557:
2555:
2552:
2550:
2549:Demand factor
2547:
2545:
2542:
2540:
2537:
2535:
2532:
2531:
2529:
2525:
2521:
2514:
2509:
2507:
2502:
2500:
2495:
2494:
2491:
2482:
2477:
2474:
2471:
2464:
2461:
2458:
2451:
2448:
2445:
2441:
2437:
2433:
2426:
2423:
2418:
2414:
2410:
2406:
2402:
2398:
2394:
2387:
2384:
2381:
2377:
2374:
2369:
2366:
2361:
2359:0-07-061293-5
2355:
2351:
2344:
2341:
2336:
2332:
2329:: 1077–1089.
2328:
2324:
2317:
2314:
2309:
2305:
2301:
2295:
2291:
2287:
2283:
2276:
2273:
2266:
2264:
2261:
2257:
2250:DC power-flow
2249:
2244:
2241:
2238:
2235:
2232:
2229:
2225:
2221:
2218:
2215:
2212:
2211:
2207:
2202:
2199:
2196:
2193:
2190:
2187:
2186:
2185:
2182:
2180:
2175:
2156:
2145:
2140:
2130:
2122:
2117:
2114:
2111:
2101:
2076:
2070:
2065:
2061:
2057:
2052:
2049:
2046:
2042:
2033:
2031:
2026:
2010:
1995:
1982:
1963:
1955:
1929:
1916:
1897:
1889:
1873:
1868:
1865:
1857:
1841:
1832:
1814:
1811:
1807:
1803:
1800:
1795:
1792:
1788:
1784:
1779:
1776:
1772:
1768:
1765:
1760:
1757:
1753:
1739:
1735:
1719:
1715:
1704:
1699:
1696:
1693:
1689:
1685:
1680:
1676:
1672:
1669:
1664:
1660:
1632:
1629:
1625:
1621:
1618:
1613:
1610:
1606:
1602:
1597:
1594:
1590:
1586:
1583:
1578:
1575:
1571:
1557:
1553:
1537:
1533:
1522:
1517:
1514:
1511:
1507:
1503:
1498:
1494:
1490:
1487:
1482:
1478:
1466:
1452:
1429:
1417:
1402:
1396:
1386:
1377:
1370:
1367:
1363:
1359:
1356:
1351:
1340:
1325:
1316:
1306:
1304:
1303:Taylor Series
1300:
1296:
1288:
1286:
1282:
1278:
1276:
1258:
1254:
1244:
1226:
1223:
1219:
1215:
1212:
1207:
1204:
1200:
1196:
1191:
1188:
1184:
1180:
1177:
1172:
1169:
1165:
1151:
1147:
1131:
1127:
1116:
1111:
1108:
1105:
1101:
1097:
1092:
1088:
1084:
1081:
1078:
1070:
1054:
1050:
1046:
1041:
1037:
1033:
1028:
1025:
1021:
998:
995:
991:
968:
965:
961:
938:
935:
931:
908:
905:
901:
878:
875:
871:
844:
841:
837:
814:
811:
807:
784:
781:
777:
764:
746:
743:
739:
730:
712:
708:
698:
680:
677:
673:
669:
666:
661:
658:
654:
650:
645:
642:
638:
634:
631:
626:
623:
619:
605:
601:
585:
581:
570:
565:
562:
559:
555:
551:
546:
542:
538:
535:
532:
524:
506:
503:
500:
494:
488:
485:
482:
476:
453:
450:
447:
441:
435:
432:
429:
423:
414:
397:
394:
391:
385:
379:
376:
373:
367:
347:
327:
306:
281:
251:
224:
220:
197:
193:
170:
166:
156:
154:
148:
141:
139:
137:
133:
129:
123:
120:
116:
112:
104:
102:
98:
96:
90:
88:
87:kilowatt hour
84:
80:
76:
72:
68:
67:short-circuit
63:
61:
56:
54:
48:
46:
42:
38:
34:
30:
26:
22:
3174:Net metering
3121:and policies
3039:Power outage
3008:Utility pole
2972:Pumped hydro
2878:distribution
2873:Transmission
2822:Cogeneration
2628:
2624:Power factor
2476:
2463:
2450:
2435:
2431:
2425:
2392:
2386:
2368:
2349:
2343:
2326:
2322:
2316:
2281:
2275:
2256:active power
2253:
2183:
2176:
2034:
2029:
2027:
1833:
1467:
1418:
1307:
1292:
1283:
1279:
1274:
1245:
1071:
728:
699:
525:
415:
157:
149:
145:
124:
110:
108:
99:
91:
82:
81:to find the
64:
57:
49:
28:
24:
18:
3169:Load factor
3024:Black start
2992:Transformer
2693:Natural gas
2644:Variability
2619:Peak demand
2609:Merit order
2539:Backfeeding
923:column and
115:power grids
89:delivered.
3257:Categories
3211:production
3056:Protective
2987:Super grid
2982:Smart grid
2809:Generation
2743:Geothermal
2634:Repowering
2267:References
413:unknowns.
340:buses and
3119:Economics
2842:Micro CHP
2720:Renewable
2703:Petroleum
2698:Oil shale
2584:Grid code
2544:Base load
2417:0018-9510
2149:Δ
2077:θ
2074:Δ
2062:θ
2043:θ
1988:∂
1980:Δ
1977:∂
1964:θ
1961:∂
1953:Δ
1950:∂
1922:∂
1914:Δ
1911:∂
1898:θ
1895:∂
1887:Δ
1884:∂
1808:θ
1804:
1785:−
1773:θ
1769:
1690:∑
1673:−
1657:Δ
1626:θ
1622:
1591:θ
1587:
1508:∑
1491:−
1475:Δ
1450:Δ
1427:Δ
1394:Δ
1384:Δ
1368:−
1360:−
1333:Δ
1326:θ
1323:Δ
1220:θ
1216:
1197:−
1185:θ
1181:
1102:∑
1085:−
1051:θ
1047:−
1038:θ
1022:θ
932:θ
674:θ
670:
639:θ
635:
556:∑
539:−
504:−
495:−
486:−
451:−
442:−
433:−
395:−
386:−
377:−
307:θ
153:slack bus
3242:Category
3029:Brownout
2817:AC power
2527:Concepts
2432:Energies
2376:Archived
2308:14195805
2224:Jacobian
1856:Jacobian
893:row and
829:column,
799:row and
45:AC power
3058:devices
2768:Thermal
2763:Osmotic
2758:Current
2738:Biomass
2728:Biofuel
2710:Nuclear
2667:Sources
2438:, 897.
2397:Bibcode
1013:buses (
31:, is a
27:, or
2753:Marine
2733:Biogas
2434:2021,
2415:
2356:
2306:
2296:
1419:where
1246:where
700:where
128:phasor
23:, the
3110:(GFI)
2999:(TSO)
2785:Solar
2773:Tidal
2748:Hydro
2304:S2CID
105:Model
2876:and
2795:Wind
2778:Wave
2688:Coal
2413:ISSN
2354:ISBN
2294:ISBN
2179:norm
1834:and
1442:and
983:and
73:and
39:and
2405:doi
2331:doi
2286:doi
1801:cos
1766:sin
1619:sin
1584:cos
1213:cos
1178:sin
861:BUS
767:BUS
667:sin
632:cos
109:An
53:bus
19:In
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2337:.
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2310:.
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2161:|
2157:V
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2141:m
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2118:1
2115:+
2112:m
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2071:+
2066:m
2058:=
2053:1
2050:+
2047:m
2030:m
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2000:|
1996:V
1992:|
1983:Q
1956:Q
1934:|
1930:V
1926:|
1917:P
1890:P
1874:[
1869:=
1866:J
1842:J
1820:)
1815:k
1812:i
1796:k
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1746:|
1740:k
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1720:i
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1705:N
1700:1
1697:=
1694:k
1686:+
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1665:i
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1638:)
1633:k
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1568:(
1564:|
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1523:N
1518:1
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1403:]
1397:Q
1387:P
1378:[
1371:1
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1352:]
1345:|
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1317:[
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1132:i
1128:V
1123:|
1117:N
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1109:=
1106:k
1098:+
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1082:=
1079:0
1055:k
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1034:=
1029:k
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999:h
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498:(
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489:1
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480:(
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445:(
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427:(
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389:(
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328:N
286:|
282:V
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256:|
252:V
248:|
225:G
221:P
198:D
194:Q
171:D
167:P
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