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54-497: It has an installed capacity of 840 MW. The generating units of the power plant are very old and are operating at around 10% PLF , generating about 110 MW. As of April 2015, there is plan to float a joint venture company with National Thermal Power Corporation holding 74% and Jharkhand government holding 26%. The new company will set up a 4000 MW (800 MW × 5) Patratu Super Thermal Power Project which will utilize 1500 acres out of 6300 acres available with

108-401: A diffuser is used to collect additional wind flow and direct it through the turbine, more energy can be extracted, but the limit still applies to the cross-section of the entire structure. Betz's law applies to all Newtonian fluids , including wind. If all of the energy coming from wind movement through a turbine were extracted as useful energy, the wind speed afterward would drop to zero. If

162-743: A cylinder of fluid with cross-sectional area S and velocity v 1 is P = C P ⋅ 1 2 ρ S v 1 3 . {\displaystyle P=C_{\text{P}}\cdot {\tfrac {1}{2}}\rho Sv_{1}^{3}.} The reference power for the Betz efficiency calculation is the power in a moving fluid in a cylinder with cross-sectional area S and velocity v 1 : P wind = 1 2 ρ S v 1 3 . {\displaystyle P_{\text{wind}}={\tfrac {1}{2}}\rho Sv_{1}^{3}.} The power coefficient C P (= P / P wind )

216-704: A given fluid speed v 1 and a given area S , one finds the maximum or minimum value for P {\displaystyle P} . The result is that P {\displaystyle P} reaches maximum value when v 2 v 1 = 1 3 {\displaystyle {\tfrac {v_{2}}{v_{1}}}={\tfrac {1}{3}}} . Substituting this value results in P max = 16 27 ⋅ 1 2 ρ S v 1 3 . {\displaystyle P_{\text{max}}={\tfrac {16}{27}}\cdot {\tfrac {1}{2}}\rho Sv_{1}^{3}.} The power obtainable from

270-542: A given installation is defined as that due to its continuous operation at full nameplate capacity over the relevant period. The capacity factor can be calculated for any electricity producing installation, such as a fuel consuming power plant or one using renewable energy , such as wind, the sun or hydro-electric installations. The average capacity factor can also be defined for any class of such installations, and can be used to compare different types of electricity production. The actual energy output during that period and

324-547: A higher capacity factor than many other power sources, and geothermal resources are generally available all the time. According to the US Energy Information Administration (EIA), from 2013 to 2017 the capacity factors of utility-scale generators were as follows: However, these values often vary significantly by month. The following figures were collected by the Department of Energy and Climate Change on

378-429: A large-scale photovoltaic system (PV system). An inherent limit to its capacity factor comes from its requirement of daylight , preferably with a sun unobstructed by clouds, smoke or smog , shade from trees and building structures. Since the amount of sunlight varies both with the time of the day and the seasons of the year, the capacity factor is typically computed on an annual basis. The amount of available sunlight

432-412: A law based solely on energy and flux conservation laws lurks in the seemingly modest assumption of transverse uniformity of the axial wind profile within the stream lines. For example, the aforementioned dual actuator wind turbine has, downstream, a transverse wind profile that has two distinct velocities and thus is not bound by the limits of the single actuator disk. Mathematically, the derivation for

486-442: A more relaxed higher bound of 2 3 {\displaystyle {\tfrac {2}{3}}} can be achieved when the "unneeded assumptions" in the Betz's law derivation are removed. Most real wind turbines are aerodynamically "thin" making them approximate the assumptions of Betz law. To the extent that a typical wind turbine approximates the assumptions in Betz law, then Betz limit places an approximate upper bound on

540-404: A nameplate capacity of 2080 MW and an annual generation averaging 4.2 TW·h. (The annual generation has varied between a high of 10.348 TW·h in 1984, and a low of 2.648 TW·h in 1956. ). Taking the average figure for annual generation gives a capacity factor of: At the low range of capacity factors is the photovoltaic power station , which supplies power to the electricity grid from

594-426: A nameplate capacity of 25.7 MW and an actual average annual production of 26.98 GWh/year it has a capacity factor of 12.0%. There are several reasons why a plant would have a capacity factor lower than 100%. These include technical constraints, such as availability of the plant, economic reasons, and availability of the energy resource. A plant can be out of service or operating at reduced output for part of

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648-404: A nameplate capacity of 3,942 MW. In 2010 its annual generation was 31,200,000 MWh, leading to a capacity factor of: Each of Palo Verde’s three reactors is refueled every 18 months, with one refueling every spring and fall. In 2014, a refueling was completed in a record 28 days, compared to the 35 days of downtime that the 2010 capacity factor corresponds to. In 2019, Prairie Island 1

702-429: A plant is only needed during the day, for example, even if it operates at full power output from 8 am to 8 pm every day (12 hours) all year long, it would only have a 50% capacity factor. Due to low capacity factors, electricity from peaking power plants is relatively expensive because the limited generation has to cover the plant fixed costs. A third reason is that a plant may not have the fuel available to operate all of

756-415: A single actuator disk implicitly embeds the assumption that the wind does not change velocity as it transits the "infinitely thin" actuator ; in contrast, in the dual actuator hybrid, the wind does change velocity as it transits, invalidating the derivation's key step requiring constant velocity. A single infinitely thin actuator cannot change the velocity because it would otherwise not conserve flux, but in

810-433: A stopped condition to full power in just a few minutes. Wind farms are variable, due to the natural variability of the wind. For a wind farm, the capacity factor is determined by the availability of wind, the swept area of the turbine and the size of the generator . Transmission line capacity and electricity demand also affect the capacity factor. Typical capacity factors of current wind farms are between 25 and 45%. In

864-410: A wind turbine operating at the Betz maximum efficiency has a non-zero wind velocity wake. Any actuator disk placed downstream of the first will extract added power and so the combined dual actuator complex exceeds Betz limit. The second actuator disk could be, but need not be, in the far field wind zone (parallel streamline) for this consideration to hold. The reason for this surprising exception to

918-705: Is generally the availability of the energy source. The plant may be capable of producing electricity, but its "fuel" ( wind , sunlight or water ) may not be available. A hydroelectric plant's production may also be affected by requirements to keep the water level from getting too high or low and to provide water for fish downstream. However, solar, wind and hydroelectric plants do have high availability factors , so when they have fuel available, they are almost always able to produce electricity. When hydroelectric plants have water available, they are also useful for load following, because of their high dispatchability . A typical hydroelectric plant's operators can bring it from

972-416: Is mostly determined by the latitude of the installation and the local cloud cover. The actual production is also influenced by local factors such as dust and ambient temperature, which ideally should be low. As for any power station, the maximum possible power production is the nameplate capacity times the number of hours in a year, while the actual production is the amount of electricity delivered annually to

1026-425: Is named after its actual discoverer. The Betz Limit is the maximum possible energy that can be extracted by an infinitely thin rotor from a fluid flowing at a certain speed. In order to calculate the maximum theoretical efficiency of a thin rotor (of, for example, a wind turbine ), one imagines it to be replaced by a disc that removes energy from the fluid passing through it. At a certain distance behind this disc,

1080-993: Is the dimensionless ratio of the extractable power P to the kinetic power P wind available in the undistributed stream. It has a maximum value C P  max  = 16/27 = 0.593 (or 59.3%; however, coefficients of performance are usually expressed as a decimal, not a percentage). The resulting expression is: C P ( v 2 v 1 ) = 1 2 ( 1 + ( v 2 v 1 ) − ( v 2 v 1 ) 2 − ( v 2 v 1 ) 3 ) {\displaystyle C_{P}\left({\frac {v_{2}}{v_{1}}}\right)={\tfrac {1}{2}}\left(1+\left({\frac {v_{2}}{v_{1}}}\right)-\left({\frac {v_{2}}{v_{1}}}\right)^{2}-\left({\frac {v_{2}}{v_{1}}}\right)^{3}\right)} Modern large wind turbines achieve peak values for C P in

1134-408: Is the speed in the front of the rotor, v 2 is the speed downstream of the rotor, v is the speed at the fluid power device, ρ is the fluid density, S {\displaystyle S} is the area of the turbine, and A 1 {\displaystyle A_{1}} and A 2 {\displaystyle A_{2}} are the areas of the fluid before and after

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1188-465: Is unrelated to Betz's coefficient of 16/27 ≈ {\displaystyle \approx } 59.3%, which limits production vs. energy available in the wind. As of 2017 Three Gorges Dam in China is, with its nameplate capacity of 22,500 MW, the largest power generating station in the world by installed capacity. In 2015 it generated 87 TWh, for a capacity factor of: Hoover Dam has

1242-457: The south-western United States , although in some locations solar PV does not reduce the need for generation of network upgrades given that air conditioner peak demand often occurs in the late afternoon or early evening when solar output is reduced. SolarPACES states that by using thermal energy storage systems the operating periods of solar thermal power (CSP) stations can be extended to become dispatchable (load following). Geothermal has

1296-426: The Betz limit, the rotor extracts 8 9 {\displaystyle {\tfrac {8}{9}}} of 2 3 {\displaystyle {\tfrac {2}{3}}} , or 16 27 , {\displaystyle {\tfrac {16}{27}},} of the incoming kinetic energy. Because the cross-sectional area of wind flowing through the rotor changes, there must be some flow of air in

1350-844: The United Kingdom during the five year period from 2011 to 2019 the annual capacity factor for wind was over 30%. Solar energy is variable because of the daily rotation of the earth, seasonal changes, and because of cloud cover. For example, the Sacramento Municipal Utility District observed a 15% capacity factor in 2005. However, according to the SolarPACES programme of the International Energy Agency (IEA), solar power plants designed for solar-only generation are well matched to summer noon peak loads in areas with significant cooling demands, such as Spain or

1404-429: The air stream flowing through an idealized " actuator disk" that extracts energy from the wind stream. According to Betz's law, no wind turbine of any mechanism can capture more than 16/27 (59.3%) of the kinetic energy in wind. The factor 16/27 (0.593) is known as Betz's coefficient . Practical utility-scale wind turbines achieve at peak 75–80% of the Betz limit. The Betz limit is based on an open-disk actuator . If

1458-483: The airflow must distribute to a wider area. As a result, geometry limits the maximum efficiency of any turbine. British scientist Frederick W. Lanchester derived the same maximum in 1915. The leader of the Russian aerodynamic school, Nikolay Zhukowsky , also published the same result for an ideal wind turbine in 1920, the same year as Betz. It is thus an example of Stigler's law , which posits that no scientific discovery

1512-452: The annual energy that can be extracted at a site. Even if a hypothetical wind blew consistently for a full year, any wind turbine well approximated by the actuator disk model can extract no more than the Betz limit of the energy contained in that year's wind could be extracted. Essentially increasing system economic efficiency results from increased production per unit, measured per square meter of vane exposure. An increase in system efficiency

1566-433: The capacity factor vary greatly depending on a range of factors. The capacity factor can never exceed the availability factor , or uptime during the period. Uptime can be reduced due to, for example, reliability issues and maintenance, scheduled or unscheduled. Other factors include the design of the installation, its location, the type of electricity production and with it either the fuel being used or, for renewable energy,

1620-495: The capacity factors for various types of plants in UK grid: Betz%27s law In aerodynamics , Betz's law indicates the maximum power that can be extracted from the wind, independent of the design of a wind turbine in open flow. It was published in 1919 by the German physicist Albert Betz . The law is derived from the principles of conservation of mass and momentum of

1674-402: The current power need, conserving its stored water for later usage. Other reasons that a power plant may not have a capacity factor of 100% include restrictions or limitations on air permits and limitations on transmission that force the plant to curtail output. For renewable energy sources such as solar power , wind power and hydroelectricity , the main reason for reduced capacity factor

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1728-484: The directions perpendicular to the axis of the rotor. Any kinetic energy associated with this radial flow has no effect on the calculation because the calculation considers only the initial and final states of the air in the system. Although it is often touted (e.g. ) as the definitive upper bound on energy extraction by any possible wind turbine, it is not. Despite the misleading title of his article, Betz (nor Lanchester) never made such an unconditional claim. Notably,

1782-433: The electricity is not needed or because the price of electricity is too low to make production economical. This accounts for most of the unused capacity of peaking power plants and load following power plants . Peaking plants may operate for only a few hours per year or up to several hours per day. Many other power plants operate only at certain times of the day or year because of variation in loads and electricity prices. If

1836-423: The existing power plant. http://www.juvnl.org.in/ptps.html 23°38′28″N 85°17′38″E  /  23.641°N 85.294°E  / 23.641; 85.294 Plant Load Factor The net capacity factor is the unitless ratio of actual electrical energy output over a given period of time to the theoretical maximum electrical energy output over that period. The theoretical maximum energy output of

1890-482: The fluid that has passed through the disc has a reduced, but nonzero, velocity. Applying conservation of mass to the control volume, the mass flow rate (the mass of fluid flowing per unit time) is m ˙ = ρ A 1 v 1 = ρ S v = ρ A 2 v 2 , {\displaystyle {\dot {m}}=\rho A_{1}v_{1}=\rho Sv=\rho A_{2}v_{2},} where v 1

1944-406: The force may be written d E = F d x , {\displaystyle dE=F\,dx,} and the power (rate of work done) of the wind is P = d E d t = F d x d t = F v . {\displaystyle P={\frac {dE}{dt}}=F{\frac {dx}{dt}}=Fv.} Substituting the force F computed above into

1998-1856: The front and back of the interaction region, fixing the airspeed of the actuator disk to be the average. (Removing that restriction may allow higher performance than Betz law allows, but other radial effects must also be considered. This constant velocity effect is distinct from the radial kinetic energy loss that is also ignored. ) Returning to the previous expression for power based on kinetic energy: P = 1 2 m ˙ ( v 1 2 − v 2 2 ) = 1 2 ρ S v ( v 1 2 − v 2 2 ) = 1 4 ρ S ( v 1 + v 2 ) ( v 1 2 − v 2 2 ) = 1 4 ρ S v 1 3 ( 1 + ( v 2 v 1 ) − ( v 2 v 1 ) 2 − ( v 2 v 1 ) 3 ) . {\displaystyle {\begin{aligned}P&={\tfrac {1}{2}}{\dot {m}}(v_{1}^{2}-v_{2}^{2})\\&={\tfrac {1}{2}}\rho Sv(v_{1}^{2}-v_{2}^{2})\\&={\tfrac {1}{4}}\rho S(v_{1}+v_{2})(v_{1}^{2}-v_{2}^{2})\\&={\tfrac {1}{4}}\rho Sv_{1}^{3}\left(1+\left({\frac {v_{2}}{v_{1}}}\right)-\left({\frac {v_{2}}{v_{1}}}\right)^{2}-\left({\frac {v_{2}}{v_{1}}}\right)^{3}\right).\end{aligned}}} By differentiating P {\displaystyle P} with respect to v 2 v 1 {\displaystyle {\tfrac {v_{2}}{v_{1}}}} for

2052-510: The grid. For example, Agua Caliente Solar Project , located in Arizona near the 33rd parallel and awarded for its excellence in renewable energy has a nameplate capacity of 290 MW and an actual average annual production of 740 GWh/year. Its capacity factor is thus: A significantly lower capacity factor is achieved by Lauingen Energy Park located in Bavaria , near the 49th parallel. With

2106-506: The hybrid pair, flux can be shed (outside the crossection) between the actuators allowing a different final outlet velocity than the inlet velocity. Physical multi-coaxial-rotor wind turbines have been analyzed. Although these do not exceeded Betz limit in practice, this may be attributable to the fact that rotors not only have losses but must also obey angular momentum and the blade element momentum theory which limits their efficiency below Betz limit. Modern research has suggested that

2160-473: The incoming air which eventually travels through the rotor. The last step in calculating the Betz efficiency C p is to divide the calculated power extracted from the flow by a reference power. As its reference power, the Betz analysis uses the power of air upstream moving at V 1 through the cross-sectional area S of the rotor. Since A 1 = 2 3 S {\displaystyle A_{1}={\tfrac {2}{3}}S} at

2214-472: The lifetime of the power source, both while operational and after decommissioning. A capacity factor can also be expressed and converted to full load hours . Nuclear power plants are at the high end of the range of capacity factors, ideally reduced only by the availability factor , i.e. maintenance and refueling. The largest nuclear plant in the US, Palo Verde Nuclear Generating Station has between its three reactors

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2268-436: The local weather conditions. Additionally, the capacity factor can be subject to regulatory constraints and market forces , potentially affecting both its fuel purchase and its electricity sale. The capacity factor is often computed over a timescale of a year, averaging out most temporal fluctuations. However, it can also be computed for a month to gain insight into seasonal fluctuations. Alternatively, it can be computed over

2322-1513: The mass flow rate from the continuity equation yields P = 1 2 ρ S v ( v 1 2 − v 2 2 ) . {\displaystyle P={\tfrac {1}{2}}\rho Sv(v_{1}^{2}-v_{2}^{2}).} Both of these expressions for power are valid; one was derived by examining the incremental work, and the other by the conservation of energy. Equating these two expressions yields P = 1 2 ρ S v ( v 1 2 − v 2 2 ) = ρ S v 2 ( v 1 − v 2 ) . {\displaystyle P={\tfrac {1}{2}}\rho Sv(v_{1}^{2}-v_{2}^{2})=\rho Sv^{2}(v_{1}-v_{2}).} The density can't be zero for any v and S, so 1 2 ( v 1 2 − v 2 2 ) = 1 2 ( v 1 − v 2 ) ( v 1 + v 2 ) = v ( v 1 − v 2 ) , {\displaystyle {\tfrac {1}{2}}(v_{1}^{2}-v_{2}^{2})={\tfrac {1}{2}}(v_{1}-v_{2})(v_{1}+v_{2})=v(v_{1}-v_{2}),} or v = 1 2 ( v 1 + v 2 ) . {\displaystyle v={\tfrac {1}{2}}(v_{1}+v_{2}).} The constant wind velocity across

2376-521: The onshore 1 GW Fosen Vind which as of 2017 is under construction in Norway has a projected capacity factor of 39%. Feasibility calculations may be affected by seasonality. For example in Finland, capacity factor during the cold winter months is more than double compared to July. While the annual average in Finland is 29.5%, the high demand for heating energy correlates with the higher capacity factor during

2430-402: The outgoing air has only ( 1 3 ) 2 = 1 9 {\displaystyle ({\tfrac {1}{3}})^{2}={\tfrac {1}{9}}} the kinetic energy of the incoming air, and that 8 9 {\displaystyle {\tfrac {8}{9}}} of the energy of the incoming air was extracted. This is a correct calculation, but it only considers

2484-724: The power equation yields the power extracted from the wind, P = ρ S v 2 ( v 1 − v 2 ) . {\displaystyle P=\rho Sv^{2}(v_{1}-v_{2}).} However, power can be computed another way, by using the kinetic energy. Applying the conservation of energy equation to the control volume yields P = Δ E Δ t = 1 2 m ˙ ( v 1 2 − v 2 2 ) . {\displaystyle P={\frac {\Delta E}{\Delta t}}={\tfrac {1}{2}}{\dot {m}}(v_{1}^{2}-v_{2}^{2}).} Substituting

2538-632: The range of 0.45 to 0.50, about 75–85% of the theoretically possible maximum. In high wind speed, where the turbine is operating at its rated power, the turbine rotates (pitches) its blades to lower C P to protect itself from damage. The power in the wind increases by a factor of 8 from 12.5 to 25 m/s, so C P must fall accordingly, getting as low as 0.06 for winds of 25 m/s. The speed ratio v 2 v 1 = 1 3 {\displaystyle {\tfrac {v_{2}}{v_{1}}}={\tfrac {1}{3}}} between outgoing and incoming wind implies that

2592-460: The rotor may be taken as the average of the upstream and downstream velocities. This is arguably the most counter-intuitive stage of the derivation of Betz's law. It is a direct consequence of the "axial flow" assumption, which disallows any radial mass flow in the actuator disk region. With no mass escape and a constant diameter in the actuator region, the air speed cannot change in the interaction region. Thus no energy can be extracted other than at

2646-591: The time due to equipment failures or routine maintenance. This accounts for most of the unused capacity of base load power plants . Base load plants usually have low costs per unit of electricity because they are designed for maximum efficiency and are operated continuously at high output. Geothermal power plants , nuclear power plants , coal-fired plants and bioenergy plants that burn solid material are almost always operated as base load plants, as they can be difficult to adjust to suit demand. A plant can also have its output curtailed or intentionally left idle because

2700-426: The time. This can apply to fossil generating stations with restricted fuels supplies, but most notably applies to intermittent renewable resources. Solar PV and wind turbines have a capacity factor limited by the availability of their "fuel", sunshine and wind respectively. A hydroelectricity plant may have a capacity factor lower than 100% due to restriction or scarcity of water, or its output may be regulated to match

2754-783: The turbine (the inlet and outlet of the control volume). The density times the area and speed must be equal in each of the three regions: before the turbine, while going through the turbine, and past the turbine. The force exerted on the wind by the rotor is the mass of air multiplied by its acceleration: F = m a = m d v d t = m ˙ Δ v = ρ S v ( v 1 − v 2 ) . {\displaystyle {\begin{aligned}F&=ma\\&=m{\frac {dv}{dt}}\\&={\dot {m}}\,\Delta v\\&=\rho Sv(v_{1}-v_{2}).\end{aligned}}} The incremental work done by

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2808-405: The wind stopped moving at the exit of the turbine, then no more fresh wind could get in; it would be blocked. In order to keep the wind moving through the turbine, there has to be some wind movement, however small, on the other side with some wind speed greater than zero. Betz's law shows that as air flows through a certain area, and as wind speed slows from losing energy to extraction from a turbine,

2862-470: The winter. Certain onshore wind farms can reach capacity factors of over 60%, for example the 44 MW Eolo plant in Nicaragua had a net generation of 232.132 GWh in 2015, equivalent to a capacity factor of 60.2%, while United States annual capacity factors from 2013 through 2016 range from 32.2% to 34.7%. Since the capacity factor of a wind turbine measures actual production relative to possible production, it

2916-413: Was the US unit with the highest factor and actually reached 104.4%. The Danish offshore wind farm Horns Rev 2 has a nameplate capacity of 209.3 MW. As of January 2017 it has produced 6416 GWh since its commissioning 7 years ago, i.e. an average annual production of 875 GWh/year and a capacity factor of: Sites with lower capacity factors may be deemed feasible for wind farms, for example

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