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 upperlevel 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 socalled 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.

Fuel 
Working Fluid 
Power Range 
Main Applications 
Type 
Observation 

1. 
Coal or nuclear fuel 
Steam 
Up to 1500 MW/unit 
Electric power systems 
Steam turbines 
High speed 
2. 
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 offhighway vehicles), and autonomous power sources 

With rotary but also linear reciprocating motion 
3. 
Water energy 
Water 
Up to 1000 MW/unit 
Large and distributed electric power systems, autonomous power sources 
Hydraulic turbines 
Medium and low speed >75 rpm 
4. 
Wind energy 
Air 
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 notsofast 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, steadystate 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 socalled 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 Singlereheat tandemcompound 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 coalburn steam turbines.
Nonreheat steam turbines are built below 100 MW, while singlereheat and doublereheat steam turbines are common above 100 MW, in general.
The singlereheat 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 tandemcompound while those with two shafts (eventually at different speeds) are called crosscompound. In essence, the LP stage of the turbine is attributed to a separate shaft (Figure 3.4).
Figure 3.4 Singlereheat crosscompound (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. Tandemcompound (single shaft) configurations are more often used. Nuclear units have in general tandemcompound (singleshaft) 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 plugdiffuser 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 multiplestage 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 singlestage steam turbine.
V is the volume (m^{3}), Q is the steam mass flow rate (kg/s), ρ is the density of steam (kg/m^{3}), 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) plugdiffuser valve and (b) butterflytype valve.
Figure 3.7 The steam vessel.
Let us assume that the flow rate out of the vessel Q_{output} is proportional to the internal pressure in the vessel:
where
As the temperature in the vessel may be considered constant:
Steam tables provide (∂ρ/∂P) functions.
Finally, from Equations 3.1 through 3.3:
T_{V} is the time constant of the steam vessel. With d/dt → s the Laplace form of Equation 3.4 is written as follows:
The firstorder model of the steam vessel has been obtained. The shaft torque T_{m} in modern steam turbines is proportional to the flow rate:
Therefore, the power P_{m} is
The reheater steam volume of a steam turbine is characterized by
Now consider the rather complete model of a singlereheat, tandemcompound 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 T_{CH} 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 T_{RM} of 5–10 s is characteristic to this delay.
The crossover piping also introduces a delay that may be characterized by another time constant T_{CO}.
Figure 3.8 Singlereheat tandemcompound steam turbine.
We should also consider that the HP, IP, LP stages produce F_{HP}, F_{IP}, F_{LP} 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: F_{HP} = F_{IP} = 0.3, F_{LP} = 0.4, T_{CH} ≈ 0.2–0.3 s, T_{RH} = 5–9 s, T_{CO} = 0.4–0.6 s.
In a nuclear–fuel steam turbine, the IP stage is missing (F_{IP} = 0, F_{LP} = 0.7) and T_{RH} and T_{CH} are notably smaller.
As T_{CH} is largest, reheat turbines tend to be slower than nonreheat turbines.
Figure 3.9 Structural diagram of singlereheat tandemcompound steam turbine.
Figure 3.10 Steam turbine response to 0.1 (p.u.) 1 s ramp change of GV opening.
After neglecting T_{CO} 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 T_{CH} = 8 s, F_{HP} = 0.3, and T_{CH} = T_{CO} = 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.
Reheattype 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 startup, 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 springloaded 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, K_{SR} = 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 turbinegenerator 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 hotcompressed 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 combinedgas and steamcycle 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 SO_{2} 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 dieselengine 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 fourcycle internal combustion engines [7] and diesel engine is one of them, with the period of one shaft revolution T_{REV} = 1/n (nshaft speed in rev/s), the period of one engine power stroke T_{PS} is
The frequency of power stroke f_{PS} is
For an engine with N_{c} cylinders, the number of cylinders that fire per each revolution, N_{F}, is
The cylinders are arranged symmetrically on the crankshaft such that the firing of the N_{F} cylinders is uniformly spaced in angle terms.
Consequently, the angular separation (θ_{c}) between successive firings in a fourcycle engine is
The firing angles for a 12 cylinder diesel engine are illustrated in Figure 3.17a, while the tworevolution sequence is: intake (I), compression (C), power (P), and exhaust (E) (see Figure 3.17b).
The 12cylinder 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 12cylinder fourcycle diesel engine: (a) configuration and (b) sequence.
Figure 3.18 The 12cylinder engine timing.
Figure 3.19 p.u. Torque/angle for one cylinder.
Figure 3.20 p.u. Torque versus shaft angle in a 12cylinder 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 higherorder 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, sparkignited 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]:
where
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 T_{1}/T_{4} = T_{2}/T_{3} for the isentropic steps and the injection ratio ρ = T_{3}/T_{2}, 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 sparkignited (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
where
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 sparkignited engines for same ε. Lower compression ratios (ε) than for sparkignited 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 T_{max}, while the other volume is the compression space kept at low temperature T_{min}. 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 T_{min} to T_{max}. 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 T_{min} 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 Stirlingtype cycles depart from the ideal one. Practical efficiency of Stirling cycle engines is much lower: ${\eta}_{\text{th}}<{\eta}_{\text{th}}^{i}{K}_{\text{th}}$
(K_{th} < 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 steadystate operation point. There are many variants for rotarymotion Stirling engines [14].
Freepiston linearmotion 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 freepiston 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 freepiston Stirlingengine model in Equation 3.25 is a fourthorder system, with ${X}_{d},{\dot{X}}_{d},{X}_{p},{\dot{X}}_{p}$
as 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 standalone or powergridconnected electric generator operation modes.The merits and demerits of Stirling engines are as follows:
Figure 3.31 Freepiston Stirling engine dynamics model.
Thermal Engine 
Parameter 
Combustion Type 
Efficiency 
Quietness 
Emissions 
Fuel Type 
Starting 
Dynamic Response 

Steam turbines 
Continuous 
Poor 
Not so good 
Low 
Multifuel 
Slow 
Slow 

Gas turbines 
Continuous 
Good at full loads, low at low load 
Good 
Reduced 
Independent 
Easy 
Poor 

Stirling engines 
Continuous 
High in theory, lower so far 
Very good 
Very low 
Independent 
NA 
Good 

Sparkignited engines 
Discontinuous 
Moderate 
Rather bad 
Still large 
One type 
Fast 
Very good 

Diesel engines 
Discontinuous 
Good 
Bad 
Larger 
One type 
Rather fast 
Good 
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.
Turbine 
Type 
Head (m) 
Inventor 
Trajectory 

Tangential 
Impulse 
>300 
Pelton (P) 
Designed in the transverse plane 
Radial–axial 
Reaction 
<50 
Francis (F) 
Bented into the axial plane 
Axial 
Reaction (propeller) 
<50 
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 σ_{t} to be small, σ_{t} = 0.01–0.1. It increases with n_{s} and decreases with H_{T}
In general, G_{sp} ≈ 70–150 N/kW.
In general the rotor diameter D_{r} = 0.2–12.0 m, the head H_{T} = 2–2000 m, the efficiency at full load is η_{T} = 0.8–0.96, the flow rate Q = 10^{–3}–10^{3} m^{3}/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 n_{s} 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, n_{Sopt} = 2–64 for Pelton turbines, n_{s} = 50–500 for Francis turbines and n_{s} = 400–1700 for Kaplan turbines. The specific speed n_{s} 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 n_{S} 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 highvelocity water jets by a set of fixed nozzles. The highspeed water jets hit the bowlshaped 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
where
Linearizing this equation and normalizing it to rated quantities $\left({U}_{0}={K}_{u}{G}_{0}\sqrt{{H}_{0}}\right)$
yieldsThe turbine mechanical power P_{m} writes
After normalization (P_{m0} = K_{p}H_{0}U_{0}) and linearization, Equation 3.35 becomes
Substituting ΔH/H_{0} or ΔU/U_{0} 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:
where
By normalization Equation 3.39 becomes:
where
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 firstorder 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:
where
As expected, the coefficients a_{11}, a_{12}, a_{13}, a_{21}, a_{22}, a_{23} vary with load, etc. To a first approximation a_{12} ≈ a_{22} ≈ 0 and, with constant a_{ij} coefficients, the firstorder 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 shortcircuited at forebay.
Finally, the incremental head and volume flow rate h(s)/q(s) transfer function of the turbine is [2]:
where
where
Typical values of α are around 1200 m/s for steel conduits and around 1400 m/s for rock tunnels. T_{e} 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 ΔP_{m}(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 q_{p} is the friction
With F = q_{p} = 0, Equations 3.51 and 3.52 degenerate into the first order model provided tan h T_{es} ≈ T_{es}, 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), G_{1}(s) and G_{2}(s) for T_{e} = 0.25 s and T_{w} = 1 s.
The secondorder transfer function performs quite well to and slightly beyond the first maximum that occurs at ω = π/2T_{e} = 6.28 rad/s in our case (Figures 3.37 and 3.38).
It is, however, clear that well beyond this frequency a higherorder approximation is required.
Such models can be obtained with advanced curve fitting methods applied to G_{2}(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 higherorder model.
Figure 3.37 Higherorder hydraulic turbine frequency response.
Figure 3.38 The secondorder 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 F_{1}(s) [2]:
where
Now for the penstock, the wave equation yields (in p.u.)
where
Figure 3.40 Nonlinear model of hydraulic turbine with hammer and surge tank effects.
The overall water velocity U_{p} to head at turbine h_{t} 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 g_{fL} and g_{NL} are the fullload and noload actual gate openings in p.u.
Also h_{0} is the normalized turbine head, U_{0} is the normalized water speed at turbine, U_{NL} is the noload water speed at turbine, ω_{r} is the shaft speed, P_{m} is the shaft power, and m_{t} is the shaft torque differential in p.u. values.
The nonlinear model in Figure 3.40 may be reduced to a highorder (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 higherorder 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 longterm dynamic studies it is mandatory.
For governor timing studies, as the surge tank natural period (T_{s}) is of the order of minutes, its consideration is not necessary. Further on, the hammer effect should be considered in general, but the secondorder model suffices.
In transient stability studies, again, the hammer effect should be considered.
For small signal stability studies linearization of the turbine penstock model (secondorder 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 gateservomotor.
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 highperformance nonlinear motion controllers (adaptive, variable structure, fuzzylogic, 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 pumpstorage hydropower plants.
Pumping may be required either for irrigation or for energy storage during offpeak 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. Pumpstorage plants with synchronous (constant) speed generators (motors) are a wellestablished technology.
A classification of turbinepumps 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 tidalwave 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 ω_{p} ≈ (1.12–1.18) ω_{T}. 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 tidalwave turbine/pump.
The turbine governor and electric machine control schemes are rather specific for generating electric power (turbining) and for pumping [25].
In tidalwave 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 (u^{3}) up to a design limit, u_{max}. Above u_{max} (P_{rated}) 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 socalled 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 U_{ave} 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 t_{max} (1 year) is then obtained from the integral:
where E(u) = u^{3}f(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 f_{max}, which in turn is smaller than the maximum energy per unit speed range E_{max}.
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 slowspeed wind turbines and λ > 1 for highspeed wind turbines.
In general, C_{p} 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 λ_{opt} ≈ 1, but their maximum power coefficient C_{pmax}(λ_{opt}) is moderate (C_{pmax} ≈ 0.3). In contrast, high speed axial wind turbines selfstart at higher speed (above 5 m/s wind speed) but, for an optimum tip speed ratio λ_{opt} ≈ 7, they have maximum power coefficient C_{pmax} ≈ 0.4. That is, a higher energy conversion ratio (efficiency).
For each location, the average wind speed U_{ave} is known. The design wind speed U_{R} is in general around 1.5 U_{ave}.
In general, the optimum tip speed ratio λ_{opt} increases as the number of rotor blades Z_{1} decreases:
Three or two blades are typical for rapid axial wind turbines.
The rotor diameter D_{r}, may be, to a first approximation, calculated from Equation 3.65 for rated (design conditions): P_{rated}, λ_{opt}, C_{popt}, U_{R} 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 (λ_{opt} < 1) the lift subtype works at high speeds (λ_{opt} > 1). Slow wind turbines have a higher selfstarting torque but a lower power efficiency coefficient C_{pmax}.
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, u_{1} > u_{∞} and thus A_{1} < A_{∞}.
The wind power P_{wind} in front of the wind turbine is the product of mass flow to speed squared per 2:
The power extracted from the wind, P_{turbine} is
Let us assume:
The efficiency limit, η_{ideal}, is
With Equation 3.61, η_{ideal} becomes:
The maximum ideal efficiency is obtained for ∂η_{i}/∂Ψ = 0 at Ψ_{opt} = 2/3 (U_{∞}/U_{1} = 1/3) with η_{imax} = 0.593.
This ideal maximum efficiency is known as the Betz limit [26].
The steadystate 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 crosssectional airfoil element of the blade is shown in Figure 3.54.
The local relative velocity U_{rel}(r) is obtained by superimposing the axial velocity U(1 − a) and the rotation velocity rω_{r}(1 + a′) at the rotor plane.
The induced velocities (−aU and a′rω_{r}) are produced by the vortex system of the machine.
The local attack angle α is
with
Figure 3.54 Blade element with pertinent speed and forces.
The local blade pitch θ is
where
The lift force F_{lift} is rectangular to U_{rel}, while the drag force F_{drag} is parallel to it.
The lift and drag forces F_{lift} and F_{drag} may be written as follows:
where
From lift and drag forces, the normal force (thrust), F_{N}, and tangential force F_{T} (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 F_{N} per turbine is
Similarly, the mechanical power P_{T} is as follows:
Now, with the earlier definition of Equation 3.65, the power efficiency C_{p} may be calculated.
This may be done using a set of airfoil data for the given wind turbine, when a, a′, C_{L}, and C_{D} are determined first.
A family of curves C_{p}–λ–β is thus obtained. This in turn may be used to investigate the steadystate performance of the wind turbine for various wind speeds U and wind turbine speeds ω_{r}.
As the influence of blade global pitch angle β is smaller than the influence of tip speed ratio λ in the power efficiency coefficient C_{p}, we may first keep β = ct. and vary λ for a given turbine. Typical C_{p}(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 ω_{r}, that is λ, the optimum value of λ corresponds to the case when C_{p} is maximum, C_{pmax} (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 ω_{r} 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 C_{p} varies with wind speed (ω_{r} = ct) and thus lessefficient 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 offshore 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 > U_{rated}):
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 torqueproducing 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 U_{rated}, 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 C_{p} (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, bladesfixtures, couplings produce additional oscillations. The pitchservo 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 pitchservo.
Experiments have shown that in reality there are at least two time constants that delay the transition: one related to D_{r}/U and the other related to 2C/(D_{r}ω_{r}) [30].
Time lags are related to the axial/and tangential induced velocities (−aU and + a′D_{r}ω_{r}/2).
The inclusion of a leadlag filter to simulate the inflow phenomena seems insufficient due to considerable uncertainty in the modeling.
The pitchservo is implemented as a mechanical hydraulic or electrohydraulic governor. A firstorder (Figure 3.60) or a secondorder model could be adopted.
In Figure 3.60, the pitchservo is modeled as a simple delay T_{servo}, while the variation slope is limited between dβ_{min}/dt to dβ_{max}/dt (to take care of inflow phenomena). In addition, the global attack angle β span is limited from β_{optimum} to β_{maximum}. β_{opt} is obtained from C_{p}–λ–β curve family for C_{pmax} with respect to β (Figure 3.61) [28].
Now from angle β to output power the steadystate model of the wind turbine is used (Figure 3.62) for a constantspeed activestall 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 “flaginthewind” 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 (K_{p}) 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 pitchservo control for active stall above rated power and β_{optimization} 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 pitchservo is not included. Usually, there is a transmission between the wind turbine and the electric generator. A sixorder 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 H_{i}. 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 “realworld” pulsations in speed or electric power may be detected by such models; resonance conditions may be avoided through design or control measures.