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Electric generators convert mechanical energy into electrical energy. Mechanical energy is produced by prime movers. Prime movers are mechanical machines that convert primary energy of a fuel or fluid into mechanical energy. They are also called "turbines or engines." The fossil fuels commonly used in prime movers are coal, gas, oil, and nuclear fuel.
Electric generators convert mechanical energy into electrical energy. Mechanical energy is produced by prime movers. Prime movers are mechanical machines that convert primary energy of a fuel or fluid into mechanical energy. They are also called "turbines or engines." The fossil fuels commonly used in prime movers are coal, gas, oil, and nuclear fuel.
Essentially, the fossil fuel is burned in a combustor and thus thermal energy is produced.
Thermal energy is then taken by a working fluid and converted into mechanical energy in the prime mover itself.
Steam is the working fluid for coal or nuclear fuel turbines, but it is the gas or oil in combination with air, in gas turbines, diesel, or internal combustion engines.
On the other hand, the potential energy of water from an upper-level reservoir may be converted into kinetic energy that hits the runner of a hydraulic turbine, changes momentum and direction, and produces mechanical work at the turbine shaft as it rotates against the "braking" torque of the electric generator under electric load.
Wave energy is similarly converted into mechanical work in special tidal hydraulic turbines. Wind kinetic energy is converted into mechanical energy by wind turbines.
A complete classification of prime movers is very difficult due to so many variations in construction, from topology to control. However, a simplified one is shown in Table 3.1.
In general, a prime mover or turbine drives an electric generator directly, or through a transmission (Figure 3.1) [1–3]. The prime mover is necessarily provided with a so-called speed governor (in fact, a speed control and protection system) that properly regulates the speed, according to electric generator frequency/power curves (Figure 3.2).
Note that the turbine is provided with a servomotor that activates one or a few control valves that regulate the fuel (or fluid) flow in the turbine, thus controlling the mechanical power at the turbine shaft.
The speed at turbine shaft is measured rather precisely and compared with the reference speed. The speed controller then acts on the servomotor to open or close control valves and control speed. The reference speed is not constant. In AC power systems, with generators in parallel, a speed droop of 2%–3% is allowed with power increased to rated value.
The speed droop is required for two reasons as follows:
The point A of intersection between generator power and turbine power in Figure 3.2 is statically stable as any departure from it would provide the conditions (through motion equation) to return to it.
Working Fluid
Power Range
Main Applications
Coal or nuclear fuel
Up to 1500 MW/unit
Electric power systems
Steam turbines
High speed
Gas or oil
Gas (oil) + air
From watts to hundreds of MW/unit
Large and distributed power systems, automotive applications (vessels, trains and highway and off-highway vehicles), and autonomous power sources
With rotary but also linear reciprocating motion
Water energy
Up to 1000 MW/unit
Large and distributed electric power systems, autonomous power sources
Hydraulic turbines
Medium- and low speed >75 rpm
Wind energy
Up to 10 MW/unit
Distributed power systems, autonomous power sources
Wind or wave turbines
Speed down to 10 rpm
Figure 3.1 Basic prime mover generator system.
Figure 3.2 The reference speed (frequency)/power curve.
With synchronous generators operating in a rather constant voltage and frequency power system, the speed droop is very small, which implies strong strains on the speed governor—due to large inertia, etc. It also leads to not-so-fast power control. On the other hand, the use of doubly fed synchronous generators, or of AC generators with full power electronics between them and the power system, would allow for speed variation (and control) in larger ranges (±20% and more). That is, smaller speed reference for smaller power. Power sharing between electric generators would then be done through power electronics in a much faster and more controlled manner. Once these general aspects of prime mover requirements have been clarified, let us deal in some detail with prime movers in terms of principle, steady-state performance and models for transients. The main speed governors and their dynamic models are also included for each main type of prime movers investigated here.
Coal, oil, or nuclear fuels are burned to produce high pressure (HP), high temperature, steam in a boiler. The potential energy in the steam is then converted into mechanical energy in the so-called axial flow steam turbines.
The steam turbines contain stationary and rotating blades grouped into stages—HP, intermediate pressure (IP), low pressure (LP). The HP steam in the boiler is let to enter—through the main emergency stop valves (MSVs) and the governor valves (GVs)—the stationary blades where it is accelerated as it expands to a lower pressure (LP) (Figure 3.3). Then the fluid is guided into the rotating blades of the steam turbine where it changes momentum and direction thus exerting a tangential force on the turbine rotor blades. Torque on the shaft and, thus, mechanical power, are produced. The pressure along the turbine stages decreases and thus the volume increases. Consequently, the length of the blades is lower in the HP stages than in the lower power stages.
Figure 3.3 Single-reheat tandem-compound steam turbine.
The two, three, or more stages (HP, IP, and LP) are all, in general, on same shaft, working in tandem. Between stages the steam is reheated; its enthalpy is increased and thus the overall efficiency is improved, up to 45% for modern coal-burn steam turbines.
Nonreheat steam turbines are built below 100 MW, while single-reheat and double-reheat steam turbines are common above 100 MW, in general.
The single-reheat tandem (same shaft) steam turbine is shown in Figure 3.3.
There are three stages in Figure 3.3: HP, IP, and LP. After passing through MSV and GV, the HP steam flows through the HP stage where its experiences a partial expansion. Subsequently, the steam is guided back to the boiler and reheated in the heat exchanger to increase its enthalpy. From the reheater, the steam flows through the reheat emergency stop valve (RSV) and intercept valve (IV) to the IP stage of the turbine where again it expands to do mechanical work. For final expansion, the steam is headed to the crossover pipes and through the LP stage where more mechanical work is done.
Typically, the power of the turbine is divided as 30% in the HP, 40% in the IP, and 30% in the LP stages.
The governor controls both the GV in the HP stage, and the IV in the IP stage, to provide fast and safe control.
During steam turbine starting—toward synchronous generator synchronization—the MSV is fully open while the GV and IV are controlled by the governor system to regulate the speed and power. The governor system contains a hydraulic (oil) or an electrohydraulic servomotor to operate the GV and IV but also to control the fuel–air mix admission and its parameters in the boiler.
The MSV and RSV are used to stop quickly and safely the turbine under emergency conditions.
Turbines with one shaft are called tandem-compound while those with two shafts (eventually at different speeds) are called cross-compound. In essence, the LP stage of the turbine is attributed to a separate shaft (Figure 3.4).
Figure 3.4 Single-reheat cross-compound (3600/1800 rpm) steam turbine.
Figure 3.5 Typical nuclear steam turbine.
Controlling the speeds and powers of two shafts is rather difficult though it brings more flexibility. Also shafts are shorter. Tandem-compound (single shaft) configurations are more often used. Nuclear units have in general tandem-compound (single-shaft) configurations and run at 1800 (1500) rpm for 60 (50) Hz power systems.They contain one HP and three LP stages (Figure 3.5).
The HP exhaust passes through the moisture reheater (MSR) before entering the LP (LP 1,2,3) stages, to reduce steam moisture losses and erosion.The HP exhaust is also reheated by the HP steam flow.
The governor acts upon GV and IV 1,2,3 to control the steam admission in the HP and LP 1,2,3 stages while MSV, RSV 1,2,3 are used only for emergency tripping of the turbine.
In general, the governor (control) valves are of the plug-diffuser type, while the IV may be either plug or butterfly type (Figure 3.6). The valve characteristics are partly nonlinear and, for better control, are often "linearized" through the control system.
The complete model of a multiple-stage steam turbine is rather involved. This is why we present here first the simple steam vessel (boiler, reheater) model (Figure 3.7) [1–3], and derive the power expression for the single-stage steam turbine.
V is the volume (m3), Q is the steam mass flow rate (kg/s), ρ is the density of steam (kg/m3), and W is the weight of the steam in the vessel (kg).
The mass continuity equation in the vessel is written as follows:
Figure 3.6 Steam valve characteristics: (a) plug-diffuser valve and (b) butterfly-type valve.
Figure 3.7 The steam vessel.
Let us assume that the flow rate out of the vessel Qoutput is proportional to the internal pressure in the vessel:
As the temperature in the vessel may be considered constant:
Steam tables provide (∂ρ/∂P) functions.
Finally, from Equations 3.1 through 3.3:
TV is the time constant of the steam vessel. With d/dt → s the Laplace form of Equation 3.4 is written as follows:
The first-order model of the steam vessel has been obtained. The shaft torque Tm in modern steam turbines is proportional to the flow rate:
Therefore, the power Pm is
The reheater steam volume of a steam turbine is characterized by
Now consider the rather complete model of a single-reheat, tandem-compound steam turbine (Figure 3.8). We will follow the steam journey through the turbine, identifying a succession of time delays/time constants.
The MVS and RSV (stop) valves are not shown in Figure 3.8 because they intervene only in emergency conditions.
The governor (control) valves modulate the steam flow through the turbine to provide for the required (reference) load (power)/frequency (speed) control.
The GV has a steam chest where substantial amounts of steam are stored, also in the inlet piping. Consequently, the response of steam flow to a change in GV opening exhibits a time delay due to the charging time of the inlet piping and steam chest. This time delay is characterized by a time constant TCH in the order of 0.2–0.3 s.
The IV are used for rapid control of mechanical power (they handle 70% of power) during overspeed conditions, and thus their delay time may be neglected in a first approximation.
The steam flow in the IP and LP stages may be changed with the increase of pressure in the reheater. As the reheater holds a large amount of steam, its response time delay is larger. An equivalent larger time constant TRM of 5–10 s is characteristic to this delay.
The crossover piping also introduces a delay that may be characterized by another time constant TCO.
Figure 3.8 Single-reheat tandem-compound steam turbine.
We should also consider that the HP, IP, LP stages produce FHP, FIP, FLP fractions of total turbine power such that:
We may integrate these aspects of steam turbine model into a structural diagram as in Figure 3.9.
Typically, as already stated: FHP = FIP = 0.3, FLP = 0.4, TCH ≈ 0.2–0.3 s, TRH = 5–9 s, TCO = 0.4–0.6 s.
In a nuclear–fuel steam turbine, the IP stage is missing (FIP = 0, FLP = 0.7) and TRH and TCH are notably smaller.
As TCH is largest, reheat turbines tend to be slower than nonreheat turbines.
Figure 3.9 Structural diagram of single-reheat tandem-compound steam turbine.
Figure 3.10 Steam turbine response to 0.1 (p.u.) 1 s ramp change of GV opening.
After neglecting TCO and considering GV as linear, the simplified transfer function may be obtained:
The transfer function in Equation 3.10 clearly shows that the steam turbine has a straightforward response to GV opening.
A typical response in torque (in PU)—or in power—to 1 s ramp of 0.1 (p.u.) change in GV opening is shown in Figure 3.10 for TCH = 8 s, FHP = 0.3, and TCH = TCO = 0.
Enhanced steam turbine models involving various details, such as IV more rigorous representation counting for the (fast) pressure difference across the valve, may be required to better model various intricate transient phenomena.
The governor system of a turbine performs a multitude of functions such as the following:
The speed/load (frequency/power) control (Figure 3.2) is achieved through the control of the GV to provide linearly decreasing speed with load, with a small speed droop of 3%–5%. This function allows for paralleling generators with adequate load sharing.
Following a reduction in electrical load, the governor system has to limit the overspeed to a maximum of 120%, in order to preserve the turbine integrity.
Reheat-type steam turbines have two separate valving groups (GV and IV) to rapidly control the steam flow to the turbine.
The objective of the overspeed control is set to about 110%–115% of rated speed to prevent overspeed tripping of the turbine in case a load rejection condition occurs.
The emergency tripping (through MSV and RSV; Figures 3.3 and 3.5) is a protection solution in case normal and overspeed controls fail to limit the speed below 120%.
A steam turbine is provided with four or more governor (control) valves that admit steam through nozzle sections distributed around the periphery of the HP stage.
In normal operation, the GVs are open sequentially to provide better efficiency at partial load. During the start-up, all the GVs are fully open and stop valves control the steam admission.
Governor systems for steam turbines evolved continuously, from mechanical–hydraulic to electrohydraulic ones [4].
In some embodiments, the main governor systems activate and control the GV while an auxiliary governor system operates and controls the IV [4]. A mechanical–hydraulic governor contains, in general, a centrifugal speed governor (controller) whose effect is amplified through a speed relay to open the steam valves. The speed relay contains a pilot valve (activated by the speed governor) and a spring-loaded servomotor (Figure 3.11).
In electrohydraulic turbine governor systems, the speed governor and speed relay are replaced by electronic controls and an electric servomotor that finally activates the steam valve.
In large turbines, an additional level of energy amplification is needed. Hydraulic servomotors are used for the scope (Figure 3.12).
Combining the two stages—the speed relay and the hydraulic servomotor—the basic turbine governor is obtained (Figure 3.13).
For a speed droop of 4% at rated power, KSR = 25 (Figure 3.13). A similar structure may be used to control the IV [2].
Electrohydraulic governor systems perform similar functions; but by using electronics control in the lower power stages, they bring more flexibility and faster and more robust response.
They are, in general, provided with acceleration detection and load power unbalance relay compensation.
The structure of a generic electrohyraulic governor system is shown in Figure 3.14.
We should notice the two stages in actuation: the electrohydraulic converter plus the servomotor and the electronic speed controller.
Figure 3.11 Speed relay: (a) configuration and (b) transfer function.
Figure 3.12 Hydraulic servomotor structural diagram.
Figure 3.13 Basic turbine governor.
Figure 3.14 Generic electrohydraulic governing system.
The development of modern nonlinear control (adaptive, sliding mode, fuzzy, neural networks, H∞, etc.) has led to a wide variety of recent electronic speed controllers or total steam turbine-generator controllers [5,6]. However, they fall beyond our scope here.
Gas turbines burn gas, whose thermal energy is converted into mechanical work. Air is used as the working fluid. There are many variations in gas turbine topology and operation [1], but the most commonly used seems to be the open regenerative cycle type (Figure 3.15).
Figure 3.15 Open regenerative cycle gas turbine.
The gas turbine in Figure 3.14 consists of an air compressor (C) driven by the turbine itself (T) and the combustion chamber (CH).
The fuel enters the CH through the GV where it is mixed with the hot-compressed air from compressor. The combustion product is then directed into the turbine where it expands and transfers energy to the moving blades of the gas turbine. The exhaust gas heats the air from compressor in the heat exchanger. The typical efficiency of a gas turbine is 35%. More complicated cycles such as compressor intercooling and reheating or intercooling with regeneration and recooling are used for further (slight) improvements in performance [1].
The combined-gas and steam-cycle gas turbines have been proven recently to deliver an efficiency of 55% or even slightly more. The generic combined cycle gas turbine is shown in Figure 3.16.
The exhaust heat from the gas turbine is directed through the heat recovery boiler (HRB) to produce steam, which, in turn, is used to produce more mechanical power through a steam turbine section on same shaft.
With the gas exhaust exiting the gas turbine above 500°C and additional fuel burning, the HRB temperature may rise further the temperature of the HP steam and thus increase efficiency more.
Additionally, some steam for home (office) heating or process industries may be delivered.
Already in the tens of megawatts, combined cycle gas turbines are becoming popular for cogeneration and in distributed power systems in the megawatt or even tenth and hundreds of kilowatt per unit.
Besides efficiency, the short construction time, low capital cost, low SO2 emission, little staffing, and easy fuel (gas) handling are all main merits of combined cycle gas turbines.
Their construction at very high speeds (tens of krpm) up to 10 MW range, with full power electronics between the generator and the distributed power grid, or in standalone operation mode at 50(60) Hz, make the gas turbines a way of the future in this power range.
Figure 3.16 Combined cycle unishaft gas turbine.
Distributed electric power systems, with distribution feeders at 12 kV (or around it), standby power sets ready for quich intervention in case of emergency or on vessels, locomotives or series or parallel hybrid vehicles, or for power leveling systems in tandem with wind generators make use of diesel (or internal combustion) engines as prime movers for their electric generators. The power per unit varies from a few tenth of kilowatts to a few megawatts.
As for steam or gas turbines, the speed of diesel-engine generator set is controlled through a speed governor. The dynamics and control of fuel–air mix admission is very important to the quality of the electric power delivered to the local power grid or to the connected loads, in standalone applications.
In four-cycle internal combustion engines [7] and diesel engine is one of them, with the period of one shaft revolution TREV = 1/n (n-shaft speed in rev/s), the period of one engine power stroke TPS is
The frequency of power stroke fPS is
For an engine with Nc cylinders, the number of cylinders that fire per each revolution, NF, is
The cylinders are arranged symmetrically on the crankshaft such that the firing of the NF cylinders is uniformly spaced in angle terms.
Consequently, the angular separation (θc) between successive firings in a four-cycle engine is
The firing angles for a 12 cylinder diesel engine are illustrated in Figure 3.17a, while the two-revolution sequence is: intake (I), compression (C), power (P), and exhaust (E) (see Figure 3.17b).
The 12-cylinder timing is shown in Figure 3.18.
There are 3 cylinders out of 12 firing simultaneously at steady state.
The resultant shaft torque of one cylinder varies with shaft angle as in Figure 3.19.
The compression torque is negative while during power cycle it is positive.
With 12 cylinders, the torque will have much smaller pulsations, with 12 peaks over 720° (period of power engine stroke); Figure 3.20.
Any misfire in one or a few of the cylinders would produce severe pulsations in the torque that would reflect as flicker in the generator output voltage [8].
Large diesel engines are provided, in general, with a turbocharger (Figure 3.21) which influences notably the dynamic response to perturbations by its dynamics and inertia [9].
The turbocharger is essentially an air compressor that is driven by a turbine that runs on the engine exhaust gas.
The compressor provides compressed air to the engine cylinders. The turbocharger works as an energy recovery device with about 2% power recovery.
Figure 3.17 The 12-cylinder four-cycle diesel engine: (a) configuration and (b) sequence.
Figure 3.18 The 12-cylinder engine timing.
Figure 3.19 p.u. Torque/angle for one cylinder.
Figure 3.20 p.u. Torque versus shaft angle in a 12-cylinder ICE (internal combustion angle).
Figure 3.21 Diesel engine with turbocharger.
The general structure diagram of a diesel engine with turbocharger and control is shown in Figure 3.22.
The most important components are as follows:
Figure 3.22 Diesel engine with turbocharger and controller.
The turbocharger acts upon the engine in the following ways:
Any load change leads to transients in the system pictured in Figure 3.22 that may lead to oscillations due to the nonlinear effects of fuel–air flow erf inertia. As a result, there will be either less or too much air in the fuel mix. In the first case, smoky exhaust will be apparent while in the second situation not enough torque will be available for the electric load, and the generator may pull out of synchronism.
This situation indicates that PI controllers of engine speed are not adequate and nonlinear controllers (adaptive, variable structure, etc.) are required.
A higher-order model may be adopted both for the actuator [11,12] and for the engine [13] to better simulate in detail the diesel engine performance for transients and control.
Stirling engines are part of the family of thermal engines: steam turbines, gas turbines, spark-ignited engines, and diesel engines. They all have already been described briefly in this chapter, but it is now time to dwell a little on the thermodynamic engines cycles to pave the way to Stirling engines.
The steam engine, invented by James Watt, is a continuous combustion machine. Subsequently, the steam is directed from the boiler to the cylinders (Figure 3.23).
The typical four steps of the steam engine (Figure 3.23) are as follows:
An approximate formula for thermal efficiency ηth is [13]:
For ρ = 2, x = 10, K = 1.4, ε = 3, and ηth = 31%.
Figure 3.23 The steam engine "cycle": (a) the four steps and (b) PV diagram.
Figure 3.24 Brayton cycle for gas turbines: (a) PV diagram and (b) TS diagram.
The gas turbine engine fuel is also continuously combusted in combination with precompressed air. The gas expansion turns the turbine shaft to produce mechanical power.
The gas turbines work on a Brayton cycle (Figure 3.24).
The four steps include the following:
Similarly, with T1/T4 = T2/T3 for the isentropic steps and the injection ratio ρ = T3/T2, the thermal efficiency ηth is
With ideal, complete, heat recirculation,
Gas turbines are more compact than other thermal machines; they are easy to start, have low vibrations, have low efficiency at low loads (ρ small), and tend to have poor behavior during transients.
The spark-ignited (Otto) engines work on the cycle in Figure 3.25.
The four steps are as follows:
Figure 3.25 Spark ignition engines: (a) PV diagram and (b) TS diagram.
The thermal efficiency ηth is
for isentropic processes.
With a high compression ratio (say ε = 9) and the adiabatic coefficient K = 1.5 and ηth = 0.66.
The diesel engine cycle is shown in Figure 3.26.
Figure 3.26 The diesel engine cycle.
During the downward movement of the piston, an isobaric state change takes place by controlled injection of fuel (2 → 3):
Efficiency decreases when load ρ increases, in contrast to spark-ignited engines for same ε. Lower compression ratios (ε) than for spark-ignited engine are characteristic for diesel engines to obtain higher thermal efficiency.
The Stirling engine (born in 1816) is a piston engine with continuous heat supply (Figure 3.27).
In some respect, Stirling cycle is similar to Carnot cycle (with its two isothermal steps). It contains two opposed pistons and a regenerator in between.
The regenerator is made in the form of strips of metal. One of the two volumes is the expansion space kept at the high temperature Tmax, while the other volume is the compression space kept at low temperature Tmin. Thermal axial conduction is considered negligible. Let us suppose that the working fluid (all of it) is in the cold compression space.
During compression (1–2), the temperature is kept constant because heat is extracted from the compression space cylinder to the surroundings.
During the transfer step (2–3), both pistons move simultaneously; the compression piston moves toward the regenerator, while the expansion piston moves away from it. Thus, the volume stays constant. The working fluid is consequently transferred through the porous regenerator from compression to expansion space, and is heated from Tmin to Tmax. An increase in pressure takes place also from 2 to 3. In the expansion step 3–4, the expansion piston still moves away from the regenerator, but the compression piston stays idle at inner dead point. The pressure decreases and the volume increases, but the temperature stays constant because heat is added from an external source. Then, again, a transfer step (4–1) occurs, with both pistons moving simultaneously to transfer the working fluid (at constant volume) through the regenerator from the expansion to the compression space. Heat is transferred from the working fluid to the regenerator, which cools at Tmin in the compression space.
The ideal thermal efficiency ηth is
Therefore, it is heavily dependent on the maximum and minimum temperatures as the Carnot cycle is. Practical Stirling-type cycles depart from the ideal one. Practical efficiency of Stirling cycle engines is much lower:(Kth < 0.5, in general).
Stirling engines may use any heat source and can use various working fuels such as air, hydrogen, or helium (with hydrogen the best and air the worst).
Typical total efficiencies versus HP/liter density are shown in Figure 3.28 [14] for three working fluids at various speeds.
Figure 3.27 The Stirling engine: (a) mechanical representation and (b) and (c) the thermal cycle.
As the power and speed go up, the power density decreases. Methane may be a good replacement for air for better performance.
Typical power/speed curves of Stirling engines with pressure p are shown in Figure 3.29a. While the power of a potential electric generator, with speed, and voltage V as parameter, appear in Figure 3.29b.
The intersection at A of Stirling engine and electric generator power/speed curves looks clearly like a stable steady-state operation point. There are many variants for rotary-motion Stirling engines [14].
Free-piston linear-motion Stirling engines were rather recently developed (by Sunpower and STC companies) for linear generators for spacecraft or for home electricity production (Figure 3.30) [15].
The dynamic equations of the Stirling engine (Figure 3.30) are
Figure 3.28 Efficiency/power density of Stirling engines.
Figure 3.29 Power/speed curves: (a) the Stirling engine and (b) the electric generator.
For the normal displacer and:
for the piston, where
Figure 3.30 Linear Stirling engine with free-piston displacer mover.
Equations 3.23 and 3.24 may be linearized as follows:
where I is the generator current
The electric circuit correspondent of Equation 3.25 is shown in Figure 3.31.
The free-piston Stirling-engine model in Equation 3.25 is a fourth-order system, withas variables. Its stability when driving a linear PM generator will be discussed in Chapter 20 dedicated to linear reciprocating electric generators. It suffices to say here that at least in the kilowatt range such a combination has been proven stable in stand-alone or power-grid-connected electric generator operation modes.
The merits and demerits of Stirling engines are as follows:
Figure 3.31 Free-piston Stirling engine dynamics model.
Thermal Engine
Combustion Type
Fuel Type
Dynamic Response
Steam turbines
Not so good
Gas turbines
Good at full loads, low at low load
Stirling engines
High in theory, lower so far
Very good
Very low
Spark-ignited engines
Rather bad
Still large
One type
Very good
Diesel engines
One type
Rather fast
A general comparison of thermal engines is summarized in Table 3.2.
Hydraulic turbines convert the water energy of rivers into mechanical work at the turbine shaft. River water energy or tidal (wave) sea energy are renewable. They are the results of water circuit in nature, and, respectively, are gravitational (tide energy).
Hydraulic turbines are one of the oldest prime movers that man has used.
The energy agent and working fluid is water: in general, the kinetic energy of water (Figure 3.32). Wind turbines are similar, but the wind air kinetic energy replaces the water kinetic energy. Wind turbines will be treated separately, however, due to their many particularities. Hydraulic turbines are, in general, only prime movers, that is motors. There are also reversible hydraulic machines that may operate either as a turbine or as pump. They are also called hydraulic turbine pumps. There are also hydrodynamic transmissions made of two or more conveniently mounted hydraulic machines in a single frame. They play the role of mechanical transmissions but have active control. Hydrodynamic transmissions fall beyond our scope here.
Figure 3.32 Hydropower plant schematics.
Head (m)
Pelton (P)
Designed in the transverse plane
Francis (F)
Bented into the axial plane
Reaction (propeller)
Kaplan (K), Strafflo (S), Bulb (B)
Bented into the axial plane
Hydraulic turbines are of two main types: impulse turbines for heads above 300–400 m and reaction turbines for heads below 300 m. A more detailed classification is related to main direction of the water particles in the rotor zone: bent axially or transverse to the rotor axis, or related to the inventor (Table 3.3). In impulse turbines, the run is at atmospheric pressure and all pressure drop occurs in the nozzles where potential energy is turned into kinetic energy of water that hits the runner.
In reaction turbines, the pressure in the turbine is above the atmospheric one; water supplies energy in both potential and kinetic form to the runner.
The terminology in hydraulic turbines is related to variables and characteristics [16]. The main variables are of geometrical and functional type:
The main characteristics of a hydraulic turbine are, in general, as follows
It is good for σt to be small, σt = 0.01–0.1. It increases with ns and decreases with HT
In general, Gsp ≈ 70–150 N/kW.
In general the rotor diameter Dr = 0.2–12.0 m, the head HT = 2–2000 m, the efficiency at full load is ηT = 0.8–0.96, the flow rate Q = 10–3–103 m3/s, rotor speed n ≈ 50–1000 rpm.
Typical variations of efficiency [16] with load are given in Figure 3.33.
The maximum efficiency [16] depends on the specific speed ns and on the type of the turbine—Figure 3.34.
The specific speed is a good indicator for of the best type of turbine for a specific hydraulic site. In general, nSopt = 2–64 for Pelton turbines, ns = 50–500 for Francis turbines and ns = 400–1700 for Kaplan turbines. The specific speed ns could be changed by changing the rotor speed n, the total power division in multiple turbines rotors or injectors and the turbine head.
Figure 3.33 Typical efficiency/load for Pelton, Kaplan, and Francis turbines.
Figure 3.34 Maximum efficiency versus specific speed.
The tendency is to increase nS in order to reduce turbine size, by increasing rotor speed, at the costs of higher cavitation risk.
As expected, the efficiency of all hydraulic turbines tends to be high at rated load. At part load Pelton turbines show better efficiency. The worst at part load is the Francis turbine. It is thus the one more suitable for variable speed operation. Basic topologies for Pelton, Francis, and Kaplan turbines are shown in Figure 3.35a–c.
In the high head, impulse (Pelton) turbine, the HP water is converted into high-velocity water jets by a set of fixed nozzles. The high-speed water jets hit the bowl-shaped buckets placed around the turbine runner and thus mechanical torque is produced at turbine shaft.
The area of the jet is controlled by a needle placed in the center of the nozzle. The needle is actuated by the turbine governor (servomotor).
In the event of sudden load reduction, the water jet is deflected from the buckets by a jet deflector (Figure 3.35a).
Figure 3.35 Hydraulic turbine topology: (a) Pelton type, (b) Francis type, and (c) Kaplan type.
In contrast, reaction (radial–axial) or Francis hydraulic turbines (Figure 3.35b) use lower head, high volumes of water and run at lower speeds.
The water enters the turbine from the intake passage or penstock, through a spiral chamber, passes then through the movable wicket gates onto the turbine runner and then, through the draft tube, to the tail water reservoir.
The wicket gates have their axis parallel to the turbine axis. In Francis turbines, the upper ends of the rotor blades are tightened to a crown and the lower ends to a band.
At even lower head, in Kaplan hydraulic turbines, the rotor blades are adjustable through an oil servomotor placed within the main turbine shaft.
Usually, in system stability studies [17], with the turbine coupled to an electrical generator connected to a power grid, a simplified, classical model of the hydraulic turbine is used. Such a lossless model assumes that water is incompressible, the penstock is inelastic, the turbine power is proportional to the product of head and volume flow (volume flow rate), while the velocity of water varies with the gate opening and with the square root of net head.
There are three fundamental equations to consider:
According to the above assumptions the water velocity in the penstock U is
Linearizing this equation and normalizing it to rated quantitiesyields
The turbine mechanical power Pm writes
After normalization (Pm0 = KpH0U0) and linearization, Equation 3.35 becomes
Substituting ΔH/H0 or ΔU/U0 from Equation 3.34 into Equation 3.36 yields
and finally
The water column that accelerates due to change in head at the turbine is described by its motion equation:
By normalization Equation 3.39 becomes:
is the water starting time. It depends on load, and it is in the order of 0.5–5 s for full load.
Replacing d/dt with the Laplace operator, from Equations 3.34 and 3.40 one obtains:
The transfer functions in Equations 3.42 and 3.43 are shown in Figure 3.36.
The power/gate opening transfer function of Equation 3.43 has a zero in the right s plane. It is a nonminimum phase system whose identification may not be completed by investigating only its amplitude from its amplitude/frequency curve.
For a step change in gate opening, the initial and final value theorems yield
The time response to such a gate step opening is
After a unit step increase in gate opening the mechanical power goes first to −2 p.u. value and only then increases exponentially to the expected steady state value of 1 p.u. This is due to water inertia.
Practice has shown that this first-order model hardly suffices when the perturbation frequency is higher than 0.5 rad/s. The answer is to investigate the case of the elastic conduit (penstock) and compressible water where the conduit of the wall stretches at the water wave front.
Figure 3.36 The linear ideal model of hydraulic turbines in p.u.
We start with a slightly more general small deviation linear model of the hydraulic turbine:
As expected, the coefficients a11, a12, a13, a21, a22, a23 vary with load, etc. To a first approximation a12 ≈ a22 ≈ 0 and, with constant aij coefficients, the first-order model is reclaimed.
Now, if the conduit is considered elastic and water as compressible, the wave equation in the conduit may be modeled as an electric transmission line that is open circuited at the turbine end and short-circuited at forebay.
Finally, the incremental head and volume flow rate h(s)/q(s) transfer function of the turbine is [2]:
Typical values of α are around 1200 m/s for steel conduits and around 1400 m/s for rock tunnels. Te is the order of fractions of a second, larger for larger penstocks (Pelton turbines).
If we now introduce Equations 3.42 and 3.43 into Equation 3.48, then the power ΔPm(s) to gate opening Δz(s) in p.u. transfer function can be obtained as follows:
Alternatively, from Equation 3.47, we obtain the following:
where qp is the friction
With F = qp = 0, Equations 3.51 and 3.52 degenerate into the first order model provided tan h Tes ≈ Tes, that is for very low frequencies:
The frequency response (s = jω) of Equation 3.51 with F = 0 is shown in Figure 3.36.
Now, we may approximate the hyperbolic function with truncated Taylor series [18]:
Figure 3.37 shows comparative results for G(s), G1(s) and G2(s) for Te = 0.25 s and Tw = 1 s.
The second-order transfer function performs quite well to and slightly beyond the first maximum that occurs at ω = π/2Te = 6.28 rad/s in our case (Figures 3.37 and 3.38).
It is, however, clear that well beyond this frequency a higher-order approximation is required.
Such models can be obtained with advanced curve fitting methods applied to G2(s) for the frequency range of interest [19,20].
The presence of a surge tank (Figure 3.39) in some hydraulic plants calls for a higher-order model.
Figure 3.37 Higher-order hydraulic turbine frequency response.
Figure 3.38 The second-order model of hydraulic turbines (with zero friction).
Figure 3.39 Hydraulic plant with surge tank.
The wave (transmission) line equations apply now both for tunnel and penstock. Finally, the tunnel and surge tank can be approximated to F1(s) [2]:
Now for the penstock, the wave equation yields (in p.u.)
Figure 3.40 Nonlinear model of hydraulic turbine with hammer and surge tank effects.
The overall water velocity Up to head at turbine ht ratio is [2]
The power differential is written as follows:
F(s) represents now the hydraulic turbine with wave (hammer) and surge tank effects considered.
If we now add Equation 3.33 that ties the speed at turbine head and gate opening to Equations 3.57 and 3.58, the complete nonlinear model of the hydraulic turbine with penstock and surge tank effects included (Figure 3.40) is obtained.
Notice that gfL and gNL are the full-load and no-load actual gate openings in p.u.
Also h0 is the normalized turbine head, U0 is the normalized water speed at turbine, UNL is the no-load water speed at turbine, ωr is the shaft speed, Pm is the shaft power, and mt is the shaft torque differential in p.u. values.
The nonlinear model in Figure 3.40 may be reduced to a high-order (three or more) linear model through various curve fittings applied to the theoretical model with given parameters. Alternatively, test frequency response tests may be fitted to a third-, fourth-, and even higher-order linear system for preferred frequency bands [21].
As the nonlinear complete model is rather involved, the question arises as of when it is to be used. Fortunately, only in long-term dynamic studies it is mandatory.
For governor timing studies, as the surge tank natural period (Ts) is of the order of minutes, its consideration is not necessary. Further on, the hammer effect should be considered in general, but the second-order model suffices.
In transient stability studies, again, the hammer effect should be considered.
For small signal stability studies linearization of the turbine penstock model (second-order model) may be also adequate, especially in plants with long penstocks.
In principle, hydraulic turbine governors are similar to those used for steam and gas turbines. They are mechanohydraulic or electrohydraulic.
In general, for large power levels, they have two stages: a pilot valve servomotor and a larger power gate-servomotor.
An example of a rather classical good performance system with speed control is shown in Figure 3.41.
Figure 3.41 Typical (classical) governor for hydraulic turbines. TPV is the pilot valve with servomotor time constant (0.05 s), TGV is the main (gate) servomotor time constant (0.2 s), KV is the servo (total) gain (5), Rmax open is the maximum gate opening rate ≈ 0.15 p.u./s, Rmax close is the maximum closing rate ≈ 0.15 p.u./s, TR is the reset time (5.0 s), RP is the permanent droop (0.04), and RT is the transient droop (0.4).
Numbers in parenthesis are sample data [2] given only for getting a feeling of magnitudes.
A few remarks on model in Figure 3.41 are in order:
Figure 3.42 Coordinated turbine governor–generator–exciter control system.
The governing system thus becomes more involved. The availability of high-performance nonlinear motion controllers (adaptive, variable structure, fuzzy-logic, or artificial neural networks) and of various powerful optimization methods [22] puts the governor system control into a new perspective (Figure 3.42).
Though most such advanced controllers have been tried on thermal prime movers and especially on power system stabilizers that usually serve only the electric generator excitation, the time for comprehensive digital on line control of the whole turbine generator system seems ripe [23,24].
Still, problems with safety could delay their aggressive deployment; not for a long time, though, we think.
Reversible hydraulic machines are in fact turbines that work part time as pumps, especially in pump-storage hydropower plants.
Pumping may be required either for irrigation or for energy storage during off-peak electric energy consumption hours. It is also a safety and stability improvement vehicle in electric power systems in the presence of fast variations of loads over the hours of the day.
As up to 400 MW/unit pump storage hydraulic turbine pumps have been already in operation [25], their "industrial" deployment seems near. Pump-storage plants with synchronous (constant) speed generators (motors) are a well-established technology.
A classification of turbine-pumps is in order:
There are many topological variations in existing turbine pumps; it is also feasible to design the machine for pumping and then check the performance for turbining, when the direction of motion is reversible.
With pumping and turbining in both directions of fluid flow, the axial turbine pump in Figure 3.43 may be adequate for tidal-wave power plants.
The passing from turbine to pump mode implies the emptying of the turbine chamber before the machine is started by the electric machine as motor to prepare for pumping. This transition takes time.
Figure 3.43 Radial–axial turbine/pump with reversible speed.
Figure 3.44 Axial turbine pump.
More complicated topologies are required to secure unidirectional rotation for both pumping and turbining, though the time to switch from turbining to pumping mode is much shorter.
In order to preserve high efficiency in pumping the speed in the pumping regime has to be larger than the one for turbining. In effect the head is larger and the volume flow lower in pumping. A typical ratio for speed would be ωp ≈ (1.12–1.18) ωT. Evidently such a condition implies adjustable speed and thus power electronics control on the electric machine side. Typical head/volume flow characteristics [16] for a radial–axial turbine/pumps are shown in Figure 3.45.
They illustrate the fact that pumping is more efficient at higher speed than turbining and at higher heads, in general.
Similar characteristics portray the output power versus static head for various wicket gate openings [25] (Figure 3.46).
Power increases with speed, and higher speeds are typical for pumping. Only wicket gate control by governor system is used, as adjustable speed is practiced through instantaneous power control in the generator rotor windings, through power electronics.
Figure 3.45 Typical characteristics pumping of radial–axial turbine + pumps.
Figure 3.46 Turbine/pump system power/static head curves at various speeds: (a) turbining and (b) pumping.
Figure 3.47 Head/time of the day in a tidal-wave turbine/pump.
The turbine governor and electric machine control schemes are rather specific for generating electric power (turbining) and for pumping [25].
In tidal-wave turbine/pumps, to produce electricity, a special kind of transit takes place from turbining to pumping in one direction of motion and in the other direction of motion in a single day.
The static head changes from 0% to 100% and reverses sign (Figure 3.47).
These large changes in head are expected to produce large electric power oscillations in the electric power delivered by the generator/motor driven by the turbine/pump. Discontinuing operation between pumping and turbining occurs and electric solutions based on energy storage are to be used to improve the quality of power delivered to the electrical power system.
Air pressure gradients along the surface of the earth produce wind whose direction and speed are highly variable.
Uniformity and strength of the wind are dependent on location, height above the ground, and size of local terrain irregularities. In general, wind air flows may be considered turbulent.
In a specific location, variation of wind speed along the cardinal directions may be shown as in Figure 3.48a.
This is an important information as it leads to the optimum directioning of the wind turbine, in the sense of extracting the largest energy from wind per year.
Wind speed increases with height and becomes more uniform. Designs with higher height/turbine diameter lead to more uniform flow and higher energy extraction. At the price of more expensive towers subjected to increased structural vibrations.
With constant energy conversion ratio, the turbine power increases approximately with cubic wind speed (u3) up to a design limit, umax. Above umax (Prated) the power of the turbine is kept constant by some turbine governor control to avoid structural or mechanical inadmissible overload (Figure 3.49).
For a given site, the wind is characterized by the so-called speed deviation (in p.u. per year). For example,
Figure 3.48 Wind speed vs. (a) location and (b) height.
Figure 3.49 Wind turbine power vs. wind speed.
The slope of this curve is called speed/frequency curve f:
The speed/duration is monotonous (as speed increases its time occurrence decreases), but the speed/frequency curve experiences a maximum, in general. The average speed Uave is defined in general as follows:
Other mean speed definitions are also used.
Figure 3.50 Sample time/speed frequency/speed (f) and energy E/speed.
The energy content of the wind E, during tmax (1 year) is then obtained from the integral:
where E(u) = u3f(u), the energy available at speed u. Figure 3.50 illustrates this line of thinking.
The time average speed falls below the frequency/speed maximum fmax, which in turn is smaller than the maximum energy per unit speed range Emax.
In addition, the adequate speed zone for efficient energy extraction is apparent in Figure 3.50.
It should be kept in mind that these curves, or their approximations, depend heavily on location. In general, inland sites are characterized by large variations of speed over the day, month, while winds from the sea tend to have smaller variations in time.
Good extraction of energy over a rather large speed span as in Figure 3.50 implies operation of the wind turbine over a pertinent speed range. The electric generator has to be capable to operate at variable speed in such locations.
There are constant speed and variable speed wind turbines.
For centuries, wind mills have been operated in countries like Holland, Denmark, Greece, Portugal, and so forth. The best locations are situated either in the mountains or by the sea or by the ocean shore (or offshore).
Wind turbines are characterized by the following:
The tip speed ratio λ < 1 for slow-speed wind turbines and λ > 1 for high-speed wind turbines.
In general, Cp is a single maximum function of λ that strongly depends on the type of the turbine. A classification of wind turbines is thus in order:
The axial wind turbines may be slow (Figure 3.51a) and rapid (Figure 3.51b).
The shape of the rotor blades and their number are quite different for the two configurations.
The slow axial wind turbines have a good starting torque and the optimum tip speed ratio λopt ≈ 1, but their maximum power coefficient Cpmax(λopt) is moderate (Cpmax ≈ 0.3). In contrast, high speed axial wind turbines self-start at higher speed (above 5 m/s wind speed) but, for an optimum tip speed ratio λopt ≈ 7, they have maximum power coefficient Cpmax ≈ 0.4. That is, a higher energy conversion ratio (efficiency).
For each location, the average wind speed Uave is known. The design wind speed UR is in general around 1.5 Uave.
In general, the optimum tip speed ratio λopt increases as the number of rotor blades Z1 decreases:
Three or two blades are typical for rapid axial wind turbines.
The rotor diameter Dr, may be, to a first approximation, calculated from Equation 3.65 for rated (design conditions): Prated, λopt, Cpopt, UR with the turbine speed from Equation 3.64.
Tangential—vertical shaft—wind turbines have been built in quite a few configurations. They are of two subtypes: drag type and lift type. The axial (horizontal axis) wind turbines are all of lift subtype.
Some of the tangential wind turbine configurations are shown in Figure 3.52.
Figure 3.51 Axial wind turbines: (a) slow (multiblade) and (b) rapid (propeller).
Figure 3.52 Tangential—vertical axis—wind turbines: (a) drag subtype and (b) lift subtype.
Figure 3.53 Basic wind turbine speed and pressure variation.
While drag subtype works at slow speeds (λopt < 1) the lift subtype works at high speeds (λopt > 1). Slow wind turbines have a higher self-starting torque but a lower power efficiency coefficient Cpmax.
The efficiency limit (Betz limit) may be calculated by portraying the ideal wind speed and pressure profile before and after the turbine (Figure 3.53).
The wind speed decreases immediately before and after the turbine disk plane while also a pressure differential takes place.
The continuity principle shows that
If the speed decreases along the direction of the wind speed, u1 > u∞ and thus A1 < A∞.
The wind power Pwind in front of the wind turbine is the product of mass flow to speed squared per 2:
The power extracted from the wind, Pturbine is
Let us assume:
The efficiency limit, ηideal, is
With Equation 3.61, ηideal becomes:
The maximum ideal efficiency is obtained for ∂ηi/∂Ψ = 0 at Ψopt = 2/3 (U∞/U1 = 1/3) with ηimax = 0.593.
This ideal maximum efficiency is known as the Betz limit [26].
The steady-state behavior of wind turbines is carried out usually through the blade element momentum (BEM) model. The blade is divided into a number of sections whose geometrical, mechanical, and aerodynamic properties are given as functions of local radius from the hub.
At the local radius the cross-sectional airfoil element of the blade is shown in Figure 3.54.
The local relative velocity Urel(r) is obtained by superimposing the axial velocity U(1 − a) and the rotation velocity rωr(1 + a′) at the rotor plane.
The induced velocities (−aU and a′rωr) are produced by the vortex system of the machine.
The local attack angle α is
Figure 3.54 Blade element with pertinent speed and forces.
The local blade pitch θ is
The lift force Flift is rectangular to Urel, while the drag force Fdrag is parallel to it.
The lift and drag forces Flift and Fdrag may be written as follows:
From lift and drag forces, the normal force (thrust), FN, and tangential force FT (along X,Y on the blade section plane) are simply
Various additional corrections are needed to account for the finite number of blades (B), especially for large values of a (axial induction factor).
Now the total thrust FN per turbine is
Similarly, the mechanical power PT is as follows:
Now, with the earlier definition of Equation 3.65, the power efficiency Cp may be calculated.
This may be done using a set of airfoil data for the given wind turbine, when a, a′, CL, and CD are determined first.
A family of curves Cp–λ–β is thus obtained. This in turn may be used to investigate the steady-state performance of the wind turbine for various wind speeds U and wind turbine speeds ωr.
As the influence of blade global pitch angle β is smaller than the influence of tip speed ratio λ in the power efficiency coefficient Cp, we may first keep β = ct. and vary λ for a given turbine. Typical Cp(λ) curves are shown on Figure 3.55 for three values of β.
For the time being let β = ct. and rewrite formula Equation 3.65 by using λ (tip speed ratio):
Figure 3.55 Typical Cp–n(λ)–β curves.
Adjusting turbine speed ωr, that is λ, the optimum value of λ corresponds to the case when Cp is maximum, Cpmax (Figure 3.55).
Consequently, from Equation 3.82, we obtain the following:
Therefore, basically the optimal turbine power is proportional to the third power of its angular speed.
Within the optimal power range, the turbine speed ωr should be proportional to wind speed U as follows:
Above the maximum allowable turbine speed, obtained from mechanical or thermal constraints in the turbine and electric generator, the turbine speed remains constant. As expected, in turbines with constant speed—imposed by the generator necessity to produce constant frequency and voltage power output, the power efficiency constant Cp varies with wind speed (ωr = ct) and thus less-efficient wind energy extraction is performed (Figure 3.56).
Typical turbine power versus turbine speed curves are shown in Figure 3.57.
Variable speed operation—which needs power electronics on the generator side—produces considerably more energy only if the wind speed varies considerably in time (inland sites). Not so in on or off-shore sites, where wind speed variations are smaller.
However, the flexibility brought by variable speed in terms of electric power control of the generator and its power quality, with a reduction in mechanical stress in general (especially the thrust and torque reduction) is in favor of variable speed wind turbines.
There are two methods (Figure 3.58) to limit the power during strong winds (U > Urated):
Figure 3.56 Typical optimum turbine/wind speed correlation.
Figure 3.57 Turbine power versus turbine speed for various wind speed u values.
Figure 3.58 Stall and pitch control above rated wind speed Urated: (a) passive stall, (b) active stall, and (c) pitch control.
Stalled blades act as a "wall in the wind." Stall occurs when the angle α between air flow and the blade chord is increased so much that the air flow separates from the airfoil in the suction side to limit the torque-producing force to its rated value.
For passive stall angle β stays constant as no mechanism to turn the blades is provided.
With a mechanism to turn the blades in place above rated wind speed Urated, to enforce stall, the angle β is decreased by a small amount. This is the active stall method that may be used at low speeds also to increase power extraction by increasing power efficiency factor Cp (Figure 3.55).
With the pitch control (Figure 3.58) the blades are turned by notably increasing the angle β. The turbine turns to the position of the "flag in the wind" so that aerodynamic forces are reduced. As expected, the servodrive—for pitch control—to change β has to be designed for higher rating than for active stall.
Besides wind slow variation with day or season time there are also under 1 Hz and over 1 Hz random wind speed variations (Figure 3.59) due to turbulent and wind gusts. Axial turbines (with 2,3 blades) experience 2,3 speed pulsations per revolution when the blades pass in front of the tower.
Figure 3.59 Wind speed typical variation with time.
Sideways tower oscillations also induce shaft speed pulsations.
Mechanical transmission and (or) the elasticity of blades, blades-fixtures, couplings produce additional oscillations. The pitch-servo dynamics has to be considered also.
The wind speed spectrum of wind turbine located in the wake of a neighboring one in a wind park, may also change. Care must be exercised in placing the components of a wind park [29].
Finally, electric load transients or faults are producing again speed variations.
All of the above clearly indicate the intricacy of wind turbine modeling for transients and control.
The BEM model is based in steady state. It presupposes that an instant change of wind profile can take place (Figure 3.59).
Transition from state (1) to state (2) in Figure 3.59 corresponds to an increase of global pitch angle β by the pitch-servo.
Experiments have shown that in reality there are at least two time constants that delay the transition: one related to Dr/U and the other related to 2C/(Drωr) [30].
Time lags are related to the axial/and tangential induced velocities (−aU and + a′Drωr/2).
The inclusion of a lead-lag filter to simulate the inflow phenomena seems insufficient due to considerable uncertainty in the modeling.
The pitch-servo is implemented as a mechanical hydraulic or electrohydraulic governor. A first-order (Figure 3.60) or a second-order model could be adopted.
In Figure 3.60, the pitch-servo is modeled as a simple delay Tservo, while the variation slope is limited between dβmin/dt to dβmax/dt (to take care of inflow phenomena). In addition, the global attack angle β span is limited from βoptimum to βmaximum. βopt is obtained from Cp–λ–β curve family for Cpmax with respect to β (Figure 3.61) [28].
Now from angle β to output power the steady-state model of the wind turbine is used (Figure 3.62) for a constant-speed active-stall wind turbine.
Figure 3.60 Optimum β(U) for variable wind speed.
Figure 3.61 Wind profile transition from state to state.
Figure 3.62 Simplified structural diagram of the constant speed wind turbine with active stall control.
When the turbine produces more than rated power, the switch is in position a and the angle β is increased at the rate of 6°/s to move the blades toward the "flag-in-the-wind" position. When the power is around rated value, the pitch drive stays idle with β = 0, and thus β = constant (position b).
Below rated power, the switch goes to position "C" and a proportional controller (Kp) produces the desired β. The reference value β* corresponds to its optimum value as function of mechanical power, that is maximum power.
This is only a sample of the constant speed turbine model with pitch-servo control for active stall above rated power and βoptimization control below rated power.
As can be seen from Figure 3.62, the model is highly nonlinear. Still, the delays due to inflow phenomena, elasticity of various elements of the turbine are not yet included. Also the model of the pitch-servo is not included. Usually, there is a transmission between the wind turbine and the electric generator. A six-order drive train is shown in Figure 3.63.
Figure 3.63 Six inertia drive train.
Inertias of hub, blades, gearbox, and of generator are denoted by Hi. Each part has also a spring and a dashpot element. The matrix dynamic equation of the drive train is of the following form [31]:
Quite a few "real-world" pulsations in speed or electric power may be detected by such models; resonance conditions may be avoided through design or control measures.


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Welcome to What I've Been Reading Lately, a feature where I'll be giving short reviews of what I'm currently reading:
The Sanatorium by Sarah Pearse
Everyone's in danger. Anyone could be next.
Half-hidden by forest and overshadowed by threatening peaks, Le Sommet has always been a sinister place. Long plagued by troubling rumors, the former abandoned sanatorium has since been renovated into a five-star minimalist hotel.
An imposing, isolated getaway spot high up in the Swiss Alps is the last place Elin Warner wants to be. But Elin's taken time off from her job as a detective, so when her estranged brother, Isaac, and his fianc e, Laure, invite her to celebrate their engagement at the hotel, Elin really has no reason not to accept.
Arriving in the midst of a threatening storm, Elin immediately feels on edge–there's something about the hotel that makes her nervous. And when they wake the following morning to discover Laure is missing, Elin must trust her instincts if they hope to find her. With the storm closing off all access to the hotel, the longer Laure stays missing, the more the remaining guests start to panic.
Elin is under pressure to find Laure, but no one has realized yet that another woman has gone missing. And she's the only one who could have warned them just how much danger they are all in. . .
I'm on a Mystery Committee this year at the library I work at so you are going to see me reading a lot mystery novels (as well as it's being one of may favorite genres). I am intrigued by this one! It has a hint of Laura Purcell's books in modern times.
The Girl Who Became A Tree: A Story Told in Poems by Joseph Coelho and illustrated by Kate Milner
Daphne is unbearably sad and adrift. She feels the painful loss of her father acutely and seeks solace both in the security of her local library and the escape her phone screen provides by blocking out the world around her. As Daphne tries to make sense of what has happened she recalls memories of shared times and stories past, and in facing the darkness she finds a way back from the tangle of fear and confusion, to feel connected once more with her friends and family. The Girl Who Became a Tree sees Joseph Coelho deploy a wide variety of poetic forms with consummate skill in its narration of events. He seamlessly but searingly weaves together the ancient legend of Daphne, who was turned into a tree to avoid the attentions of the god Apollo, and a totally modern tale, mixing real-life and fantasy, in which a latter-day Daphne seeks her own freedom. This a heart-stoppingly imaginative story told in poems, at times bleak and even tragic, which is layered, rich and ultimately a tour de force of poetic skill and energy.
So beautiful and heartbreaking, another novel in verse that will always be in my heart!
Love in Excess by Eliza Haywood
Love in Excess is a well-crafted novel in which the claims of love and ambition are pursued through multiple storylines until the heroine engineers a melodramatic conclusion. Love in Excess and its reception provide a lively and valuable record of the challenge that female desire posed to social decorum. (Credit: Broadview Press Ltd.)
If you enjoy Jane Austen, then you should definitely give this one a try!
Slay in Your Lane Presents: Loud Black Girls edited by Elizabeth Uviebinené and Yomi Adegoke
Despite young black women reading more than any other ethnic group in the UK, they are still largely invisible as published authors. Loud Black Girls seeks to change that by giving black women a voice and a platform. Readers can expect frank, funny and fearless contributions about what matters to black women today, from a range of prominent voices. The book features a contribution by the winner of the submission competition, an introduction by Yomi and Elizabeth, and a foreword by Bernadine Evaristo. (Credit: Fourth Estate)
What I Plan to Read Next:
Concrete Rose by Angie Thomas
If there's one thing seventeen-year-old Maverick Carter knows, it's that a real man takes care of his family. As the son of a former gang legend, Mav does that the only way he knows how: dealing for the King Lords. With this money he can help his mom, who works two jobs while his dad's in prison.
Life's not perfect, but with a fly girlfriend and a cousin who always has his back, Mav's got everything under control.
Until, that is, Maverick finds out he's a father.
Suddenly he has a baby, Seven, who depends on him for everything. But it's not so easy to sling dope, finish school, and raise a child. So when he's offered the chance to go straight, he takes it. In a world where he's expected to amount to nothing, maybe Mav can prove he's different.
When King Lord blood runs through your veins, though, you can't just walk away. Loyalty, revenge, and responsibility threaten to tear Mav apart, especially after the brutal murder of a loved one. He'll have to figure out for himself what it really means to be a man. (Credit: Balzer +Bray)
Men Who Hate Women: From Incels to Pickup Artists: The Truth about Extreme Misogyny and How It Affects Us All by Laura Bates
An explosive book examining the rise of secretive, extremist communities who despise women. In this ground-breaking investigation, Laura Bates traces the roots of misogyny across a complex spider's web of groups extending from Men's Rights Activists and Pick up Artists to "Men Going their Own Way" trolls and the Incel movement, in the name of which some men have committed terrorist acts. Drawing parallels with other extremist movements around the world, Bates seeks to understand what attracts men to the movement, how it grooms and radicalizes boys, how it operates, and what can be done to stop it. Most urgently of all, she traces the pathways this extreme ideology has taken from the darkest corners of the internet to emerge covertly in our mainstream media, our playgrounds, and our parliament. Going undercover online and off, Bates provides the first, comprehensive look at this hitherto under-the-radar phenomenon, including fascinating interviews with trolls, former incels, the academics studying this movement, and the men fighting back. (Credit: Simon & Schuster)
Girl, Woman, Other by Bernardine Evaristo
Girl, Woman, Other follows the lives and struggles of twelve very different characters. Mostly women, black and British, they tell the stories of their families, friends and lovers, across the country and through the years.
Joyfully polyphonic and vibrantly contemporary, this is a gloriously new kind of history, a novel of our times: celebratory, ever-dynamic and utterly irresistible. (Credit: Grove Press, Black Cat)

subset: commoncrawl


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The RCF Contact Top Senior Black is a must have to keep training in all seasons.
- Vaposhield wicking technology.
- 600mm water resistant finish.
- Mesh lining.
- Ribbed stretch crew neck.
- Ribbed bound cuffs.
- Contrast white piping detail.
- Twin needle open hem.
- Branding on right chest.
- 100% polyester ripstop.


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USMCA "Trade Agreement", the North American Union, an Article V convention, and Red Flag Laws: Connecting the dots
"The Globalists have long been in the process of setting up a dictatorial and totalitarian oligarchy over the United States. Now they are putting the last pieces in place. That is what is behind the pushes for the USMCA "Trade Agreement," an Article V convention, and red-flag and other laws to disarm the American People. The Globalists want to move the United States into the North American Union."


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At Residus, we take creating garments seriously. Each Residus piece is made with sustainability and ethical practices in mind, taking responsibility for both the environmental impact of our clothes as well as the people that are involved in every step of creating one of our garments.
Learn more about our:
Leftover fabrics
European production and sustainable fibers
Code of conduct


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If you're recently experienced a fire or flood at your home or even at your business, one of the most important things about the cleanup process is ensuring that the place where you reside or where you work is safe. However, there's another element to clean up after a disaster that isn't often mentioned. That is trying to clean and restore the contents of your home or business and your belongings. This can be a challenging task, depending on the nature of the emergency and its intensity. Still, if you need restoration contents cleaning in San Diego, here at Hometown Restoration, we try to do our very best.
When it comes to a flood, we remove water, dehumidify, clean and sanitize, and restore the property. Fire cleanup is similar. We remove water (from putting out the fire), remove smoke odor and damage, clean and sanitize, and then restore the property. Part of restoring the property means restoring the belongings and contents as well. Items that we would try to save within your home would include your electronics, valuables, antiques, appliances, furniture, power tools, and anything else you would like us to restore or save. We will use the same general applications and ideas that we do toward your whole house and try to apply them toward your belongings and sanitize everything at the end.
If you would like to hear more about our process or need immediate help with restoration contents cleaning in San Diego, contact Hometown Restoration today at (760) 653-5400 to speak to a member of our team to make an appointment for an estimate.


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Expertise: Stocks, ETF and Options
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Table of Contents
How to Read Betting Odds: Simple Guide
The essential part of being a successful sports bettor is understanding how to read betting odds when you are placing wagers. You will also want to become familiar with the probabilities that these odds imply. You don't have to be a whiz at math to master the odds of sports betting, but you do have to have a good grasp of what these odds reflect as this is the key to making the right bets. It all starts with understanding what betting odds truly are.
When it comes to odds, you need to understand that they represent the probability of any outcome that will occur from a game of a match and they also represent the return you can earn if you place a winning bet. The probability is typically referred to as the implied probability.
As a rule, you should only place a bet when the probability of an outcome occurring is actually higher than what is implied.
For example, if the NY Giants are playing the Seattle Seahawks in an NFL game and Giants have odds of -110 with a betting line of -3.5 points, the implied probability is -110 x 52.4%.
If you have done research before betting and have learned that this team has a 65% chance of covering the betting line, you have found a value bet.
On the flip side, if their chance of covering the betting line is only 45%, the implied probability is higher than the probability. This is not a value bet and you should avoid placing it. To get a better understanding of how odds work, you will also have to become familiar with the odds formats that are used at leading betting sites.
Here is an example of how you read betting odds +135 when you bet $250 from a great wiki source wikihow.com/read-odds:
Reading Odds Formats
When you start placing wagers at online sportsbooks, you will find you have the ability to view odds in three different formats. The three popular formats include fractional odds, decimal odds, and American odds.
Fractional odds
Fractional odds are more commonly found at UK and Irish bookmakers. They are written using a / or a hyphen and are used by some of the most popular bookmakers online. Let's say a fractional odd is listed at 6/1. This would mean that you would win $6 for every $1 you have wagered, in addition to your original wager amount. So, if you bet $10 at 6/1 odds, you will get a return of $70.
Decimal odds
Decimal odds are also referred to as European odds and these are popular at betting sites located in Europe, Canada, and Australia. These odds are easier to understand as you can quickly spot the favorite and the underdog just by looking at the presented odds. The decimal odds amount will indicate how much you will win for every $1 that is wagered and returns will be a total return instead of the profit. These odds are much easier when trying to calculate your return. All you have to do is multiply your stake by the decimal odds number to determine the total return.
American odds
American odds, also known as moneyline odds, are the popular choice in the US. The favorite team will be indicated by a – sign and the underdog will be represented by a +. When betting on favorites, the amount will indicate how much you need to wager to earn a payout of $100. For underdog betting, the number reflects the amount that can be won for every $100 bet.
If you are just getting started with sports betting, you will want to make sure you understand how all of these odds work. Luckily, the best betting sites will automatically calculate your potential payout when you place a bet. You will also find that most sites will let you choose between the three odds. You can also find free odds calculators online or at bookies that can be used to convert one format to another.
How to use advanced sports betting odds analytics by AI?
How to Read Moneyline Odds and Place Bets
When you are wagering on the major American sports, you will often see a point spread, moneyline, and a total. These three options make up American odds. With moneyline odds, there will be a + or a – in front of the number, which will indicate the underdog and the favorite. There is just one exception, and this is when both betting options are close in value and they both have negative odds. When this happens, the favorite will be the number of higher value. So, the team with -300 odds vs one with -125 odds will mean the -300 is the favorite.
Understanding moneylines is an essential part of being a successful sports bettor since most bookies in the US will use this format. You will know right away which team is the favorite and which is the underdog, so you can quickly make educated decisions on how to wager.
Understand betting odds better, read these guides:
How to Read MMA/UFC Betting Odds and Moneylines? Simple Guide
Which online sportsbooks offer NCAA football money lines?
What does it mean to bet a money line for a particular quarter?
NFL Moneyline Bets, Point Spread and Super Bowl Props Explained
What does money line mean in American football (NFL, NCAA) betting?
Point Spread Bets and Odds
As you become more experienced, you will find that point spreads are the most popular ay to bet on any spot. They often offer higher payouts when betting on a favorite team and they also come with reduced risks that are associated with betting on an underdog. Basically, an underdog does not have to win the game to generate a paying bet with point spreads. They just have to cover the spread.
Just like with moneylines, point spreads are expressed using negative and positive numbers and the negative will indicate the underdog in the game. Let's say the Atlanta Hawks have a point spread of +11 (-110) and the 76ers are -11 (-110). The positive point spread shows that Atlanta is the favorite to win and they have to win or lose by less than 11 points. If you bet on the 76ers, they will have to win by 12 to cover the spread.
Understand betting odds better, read these guides:
NFL Moneyline Bets, Point Spread and Super Bowl Props Explained
How to parlay bet spread and over/under in College Football?
What is the puck line in hockey betting? Is it point spread?
How do injuries affect the point spread in the NBA?
Over/Under Odds
This is also referred to as totals betting and the odds will represent the number of points that will be scored cumulatively by the teams in the game. With this type of betting, you can bet on whether the total points will be over or under the projected amount.
These odds are displayed using (O) and (U) followed by the point spread and the moneyline. So, if the Red Sox are listed as (U) 8 (-115), they are the favorite and will have to win by more or less than 8 runs. The payouts are indicated by the moneyline odds shows, so in this case, $115 would have to be wagered to win $100.
Understand betting odds better, read these guides as well:
What is over under 2.5 goals in Betting?
What does over/under mean in football betting?
6 Insider Tips For Betting Over/Under in Boxing
What is over and under in soccer betting? How to bet on O/U?
Free Betting Odds Calculators
You will find many betting sites to offer a free calculator that can be used to convert odds formats and determine possible payout amounts. Try free betting calculator at AceOdds. These tools are beneficial for new bettors and you can also find them with a simple online search. The best sportsbooks online will automatically convert your odds to your preferred format, but if you are comparing odds at different sites, a calculator can be quite beneficial.
With a betting odds calculator, you will enter your bet amount and your chosen odds format. The calculator will then calculate the payout for the bet that is being placed. You can take the time to calculate your own odds and change formats, but most find this to be time-consuming and they want immediate answers.
Using any free odds calculator will allow you to know what bets are best to place and will also let you know how much you stand to win when you place a bet using any of the supported odds at your chosen online sportsbook.


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Is there any way todo a Tabbed Multi-document just like Solmetra SPAW editor?
A way to have several tabs just like (Used usually for translation but can be used for other stuff)?
I browsed arround and couldn't find a way to do it and couldn't find any plugins. Anybody has any suggestions?
Fri, 09/14/2007 - 15:22#1
Tabbed Multi-document


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This is a guide to the few but interesting and sometimes important differences in language and cultural norms between telephone calls in the UK and USA. For a comprehensive guide to the basics of both, see the e-book Really Learn the Most Important Telephoning Phrases. There is also a more general article on cultural differences in telephoning available.
As with emailing (https://www.usingenglish.com/articles/differences-between-british-american-emails.html) and the English language more generally, telephone calls in the UK and USA are very similar, with most differences being minor and almost everything being mutually understandable. However, there are a few interesting differences in how British and Americans speak on the phone, including some potential misunderstandings. This article lists and explains most of them, starting with beginning and ending British and American calls, and then dealing with the body of such calls and some vocabulary differences.
Starting British and American phone calls
Answering the phone in the UK and USA
The standard way of answering a work phone in English is some part of "Good morning/ afternoon/ evening…. Limited/ Corp. … Department. … Section. … speaking. How can/ may I help you?" The only thing that differs in that is that "How can I help you?" is more common in Britain and "How may I help you?" is more standard in the US.
With a home phone or mobile, just answering the phone with "Hello?" is standard in English. If you can see who is calling (e.g. on your smartphone screen) and know them (well), it has become more common to greet them by name straightaway instead, e.g. "Hi Steve. Thanks for calling me back". More casual greetings in that situation would be "Yo, Steve" or just "What's up?" in the US and "Alright Steve?" in the UK, with Americans perhaps more likely to use these kinds of more informal phrases. Americans are also more likely to use affectionate phrases like "Hi Honey" or "What's up, bro?", with the British equivalent "Alright, mate" being both less common and not actually being so friendly.
Polite language at the beginning of calls in the UK and USA
"Thanks for…" and "Sorry for…" are not only equally likely in different forms of English, but seem to be very popular at the start of phone calls worldwide. The only small UK/USA difference is the rather cute British verb "ring" in "Thanks for ringing me back" and "Sorry to ring again so soon, but…". The more standard versions "Thanks for calling me back" and "Sorry to call again so soon, but…" are used in both the UK and USA.
Small talk in British and American calls
If the caller knows the receiver, in English we usually start with a social question like "How are you?", or generally a more casual and friendlier version. In American English the standard casual form of "How are you?" is "How are you doing?", with Brits more commonly using "How's it going?"/ "How's life?"/ "How are things?" The answers can also vary, with Americans often going for enthusiastic words like "Great" but the British traditionally preferring understatement such as "Not bad" or even "Mustn't grumble".
In both Britain and the US a couple of sentences of small talk before getting down to business is probably enough, with other "How…?" questions like "How was your weekend?" and "How was your trip?" being popular. Note that if an American asks "How was your holiday?" it means a (single) public holiday, but a British person asking "How was your holiday?" means "How was your vacation?" If there was a public holiday on Monday, Brits are more likely to ask "How was your long weekend?"
Other topics of conversation vary a little, with Americans more likely to talk about families, even with work contacts, and the British famously preferring to talk about the weather.
Asking to speak to someone in US and UK
If you want to speak to someone else, there is again a difference in the use of "can" and "may", with Americans often saying "May I speak with…?" and British more often saying "Can I speak to…?" In this case "May I…?" seems too much in the UK, as in British English "May I…?" is asking for permission rather than for making requests. If you need to ask more politely, you can use "Could I speak to…?", which is also fine in US English. Also note the use of "with" in "speak with" in the American example but "speak to" in the British example.
The positive answers to a request to speak to someone also vary, with Brits most commonly using the phrasal verb "put someone through" in "Of course. Please hold the line. I'm putting you through". Americans prefer the more straightforward "I'm transferring your call". "I'll connect you" is fine in both kinds of English. All three expressions are also possible in the questions, with "Can you put me through to…?" being most common of those three for British people.
When it comes to negative answers, English speakers share a cultural need to be very specific about why someone can't speak (right now). You may come across the well-known "engaged"/ "busy" vocabulary difference in that excuse, in "I'm afraid his line is engaged/ busy" in British and American respectively. However, it's more common nowadays to say "I'm afraid he's on another line" in both places.
Getting down to business on the phone in the US and UK
English speakers tend to put more effort into making getting down to business smooth (meaning not sudden) than most other nationalities. The only slight variation among English speakers is the British love of linking the small talk topic and the reason with the call with phrases like "Talking about…, I reason I am phoning is…", "I'm glad you mentioned that, because I need to talk to you about…" and "Connected to that, I need some advice on…". Americans seem to put less effort into this kind of smooth transition (in business meetings as well as phone calls).
When it comes to saying the reason for the call, the British have "I'm ringing about…" to go with the more standard "I'm phoning about…" and "I'm calling about…"
Dictating in British and American English
There are some important differences in how Brits and Americans say letters and numbers that could impact intelligibility when dictating and taking dictation on the phone. For example, the British say "Zed" for the last letter of the alphabet ("Z"), while Americans say "Zee". In addition, in the UK people say "oh" (in phone numbers) or nought for numbers, while Americans almost always say "zero". Americans are also more likely to say "sixteen hundred twenty three" rather than "one thousand and twenty three", with the love of "and" in big numbers being a British peculiarity.
There are also some differences in the names of punctuation, with "." being "full stop" in British and "period" in American. "()" is "brackets" and "parentheses" in the UK and US respectively (both of those words having different meanings in the other variety of English).
Names for what you are dictating can also vary, with a British "mobile (phone) (number)" being an American "cell(phone) (number)" and a "postcode" being a "zip code".
Making arrangements on the phone in British and American English
Again, the similarities (giving detailed reasons why an arrangement is impossible, etc) are more important than the rather minor differences. These tend to be mainly about prepositions ("Monday to Friday" in British against "Monday thru Friday" in American, "at the weekend" vs "on the weekend", etc).
Ending British and American phone calls
Reasons are also often needed when ending a call, as in "Someone has just come in" (British)/ "Someone just came in" (American).
We then often mention future contact. That lovely little British verb "ring someone" can come up again here, in "Can I ring you back later?" and "Please ring again if you have any other questions". The famous "mobile phone"/ "cellphone" distinction might also come up when talking about future contact, as in "Can you ring me again later on my mobile (phone)?"/ "Can you call me later on my cell(phone)?" The same is true of text-based electronic communication. For example, "I'll mail it to you later" needs the full word "email" in British and "I'll text you later" in British means "I'll send you an SMS" in American.
The British sometimes say "Cheers" at the end of a phone call to mean "Thanks (in advance)" and/ or "Bye". This has somewhat caught on in the US, but is usually a conscious "Briticism" and is far from standard. There are also some specifically British ways of saying goodbye like "Ta-rah".
As in face to face situations like shopping, Americans are much more likely to say "Have a nice/ good/ great day" than British people are. They are also more likely to show more genuine feelings, with "Love you, bye" etc to family members and lovers being rare in the UK, at least when other people might be able to overhear you. In British English, expressions that superficially sound affectionate like "Bye love" (popular with old ladies) and "Okay, mate, see you" actually have no special meaning.
Telephone vocabulary in British and American English
This is a summary of the most important vocabulary differences above plus a few more that are rarer to actually say during phone calls but could come up when discussing the topic of telephoning, such as the emergency phone numbers "nine nine nine" (UK) and "nine one one" (USA). Ones with a star (*) are also fine in the other kind of English.
brackets/ parentheses (= ())
cheers/ thanks*
directory enquiries/ directory assistance
email someone*/ mail someone
engaged (tone)/ busy (signal) (= on another line)
free phone (number)/ toll-free (number)
full stop/ period (= .)
give some a ring/ give some a call*
hash key/ pound key (= #)
mobile (phone) (number)/ cell(phone) (number)
nine nine nine/ nine one one (= emergency phone number)
oh/ zero*
oh eight nine eight number/ nine hundred number or one-nine hundred number (= premium rate phone number)
on holiday/ on vacation*
phone box/ phone booth (= public phone)
postcode/ zip code
put someone through/ connect someone* or transfer someone's call
reverse the charges/ a collect call (= the person who receives the call pays)
ring someone/ call someone* (= telephone someone)
text someone/ send someone an SMS
zed/ zee (= Z)
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