text:
Prime Movers

Authored by: Ion Boldea

Synchronous Generators

Print publication date:  October  2015
Online publication date:  October  2015

Print ISBN: 9781498723565
eBook ISBN: 9781498723558

10.1201/b19310-4

Abstract

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.

3.1  Introduction

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) [13]. 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:

1. With a few generators of different powers in parallel, fair (proportional) power load sharing is provided.
2. When power increases too much, the speed decreases accordingly signaling that the turbine has to be shut off.

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.

Table 3.1   Turbines

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 off-highway vehicles), and autonomous power sources

• Gas turbines
• Diesel engines
• Internal combustion engines
• Stirling engines

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 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.

3.2  Steam Turbines

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.

3.3  Steam Turbine Modeling

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) [13], 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:

3.1 $d W d t = V d ρ d t − Q i n p u t − Q o u t p u t$

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:

3.2 $Q o u t p u t = Q 0 P 0 P$

where

• P is the pressure (kPa)
• P0 and Q0 are the rated pressure and flow rate out of the vessel

As the temperature in the vessel may be considered constant:

3.3 $d ρ d t = ∂ ρ ∂ P ⋅ d P d t$

Steam tables provide (∂ρ/∂P) functions.

Finally, from Equations 3.1 through 3.3:

3.4 $Q i n p u t − Q o u t p u t = T v d Q o u t p u t d t$
(3.5 $T v = P 0 Q 0 V ⋅ ∂ ρ ∂ P$

TV is the time constant of the steam vessel. With d/dts the Laplace form of Equation 3.4 is written as follows:

3.6 $Q o u t p u t Q i n p u t = 1 1 + T v ⋅ s$

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:

3.7 $T m = K m ⋅ Q$

Therefore, the power Pm is

3.8 $P m = T m ⋅ Ω m = K m Q ⋅ 2 π n m$

Example 3.1

The reheater steam volume of a steam turbine is characterized by

• Q0 = 200 kg/s, V = 100 m3, P0 = 4000 kPa, ∂ρ/∂P = 0.004
• Calculate the time constant TR of the reheater and its transfer function.
• We just use Equations 3.4 and 3.5 and, therefore Equation 3.6:

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:

3.9 $F H P + F I P + F L P = 1$

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:

3.10 $Δ T m Δ V G V ≈ ( 1 + s F H P T R H ) ( 1 + s T C H ) ( 1 + s T R H )$

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.

3.4  Speed Governors for Steam Turbines

The governor system of a turbine performs a multitude of functions such as the following:

• Speed (frequency)/load (power) control: mainly through GV
• Overspeed control: mainly through IV
• Overspeed trip: through MSV and RSV
• Start-up and shutdown

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.

3.5  Gas Turbines

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.

3.6  Diesel Engines

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.

3.6.1  Diesel Engine Operation

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

3.11 $T P S = 2 T R E V$

The frequency of power stroke fPS is

3.12 $f P S = 1 T P S$

For an engine with Nc cylinders, the number of cylinders that fire per each revolution, NF, is

3.13 $N F = N c 2$

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

3.14 $θ c = 720 ° N c$

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.

3.6.2  Diesel Engine Modeling

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:

• The actuator (governor) driver that appears as a simple gain K3.
• The actuator (governor) fuel controller that converts the actuator’s driver into an equivalent fuel-flow Φ. This actuator is represented by a gain K2 and a time constant (delay) τ2 which is dependent on oil temperature, and an aging-produced back lash.
• The inertias of engine JE, turbocharger JT and electric generator (alternator) JG.
• The flexible coupling that mechanically connects the diesel engine to the alternator (it might also contain a transmission).

Figure 3.22   Diesel engine with turbocharger and controller.

• The diesel engine is represented by the steady-state gain K1, constant for low fuel–flow Φ and saturated for large Φ, multiplied by the equivalence ratio factor (erf) and by a time constant τ1.
• The erf depends on engine equivalence ratio (eer) which in turn is the ratio of fuel/air normalized by its stoichiometric value. A typical variation of erf with eer is also shown in Figure 3.22. In essence erf is reduced, because when the ratio fuel/air increases, incomplete combustion occurs, leading to low torque and smoky exhaust.
• The dead time of the diesel engine comprises three delays: the time elapsed until the actuator output actually injects fuel in the cylinder, fuel burning time to produce torque, and time until all cylinders produce torque at engine shaft:
3.15 $τ 1 ≈ A + B n E + C n E 2$
where nE is the engine speed

The turbocharger acts upon the engine in the following ways:

• It draws energy from the exhaust to run its turbine; the more fuel in engine the more exhaust is available.
• It compresses air at a rate that is a nonlinear function of speed; the compressor is driven by the turbine, and thus the turbine speed and ultimate erf in the engine is influenced by the air flow rate.
• The turbocharger runs freely at high speed, but it is coupled through a clutch to the engine at low speeds, to be able to supply enough air at all speeds; the system inertia changes thus at low speeds, by including the turbocharger inertia.

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.

3.7  Stirling Engines

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.

3.7.1  Summary of Thermodynamic Basic Cycles

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:

• 1 → 2 The isochoric compression (1–1′) followed by isothermal expansion (1′–2). The hot steam enters the cylinder through the open valve at constant volume; then, it expands at constant temperature.
• 2 → 3 Isotropic expansion: once the valve is closed the expansion goes on till the maximum volume (3).
• 3 → 1 Isochoric heat regeneration (3–3′) and isothermal compression (3′–4)—the pressure drops at constant volume and then the steam is compressed at constant temperature.
• 4 → 1 Isentropic compression takes place after the valve is closed and the gas is mechanically compressed.

An approximate formula for thermal efficiency ηth is [13]:

3.16 $η th = 1 − ρ K − 1 ( K − 1 ) ( 1 + ln ρ ) ε K − 1 ( x − 1 ) + ( K − 1 ) ln ρ$

where

• ε = V3/V1 is the compression ratio
• ρ = V2/V1 = V3/V4 is the partial compression ratio
• $x = p ′ 1 / p 1$ is the pressure ratio

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:

• 1 → 2 Isentropic compression
• 2 → 3 Isobaric input of thermal energy
• 3 → 4 Isentropic expansion (work generation)
• 4 → 1 Isobaric thermal energy loss

Similarly, with T1/T4 = T2/T3 for the isentropic steps and the injection ratio ρ = T3/T2, the thermal efficiency ηth is

3.17 $η th ≈ 1 − 1 ρ T 4 T 2$

With ideal, complete, heat recirculation,

3.18 $η th ≈ 1 − 1 ρ$

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:

• 1 → 2—Isentropic compression
• 2 → 3—Isochoric input of thermal energy
• 3 → 4—Isentropic expansion (kinetic energy output)
• 4 → 1—Isochoric heat loss

Figure 3.25   Spark ignition engines: (a) PV diagram and (b) TS diagram.

The thermal efficiency ηth is

3.19 $η th = 1 − 1 ε K − 1 ; ε = V 1 V 2$

where

3.20 $T 4 T 3 = T 1 T 2 = ( V 3 V 4 ) Κ − 1 = 1 ε K − 1$

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):

3.21

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.

3.7.2  Stirling Cycle Engine

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

3.22 $η th i = 1 − T min T max$

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: $η th < η th i K th$

(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].

3.7.3  Free-Piston Linear-Motion Stirling Engine Modeling

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

3.23 $M d X ¨ d + D d X ˙ d = A d ( P p − P )$

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:

3.24 $M p X ¨ p + D p X ˙ p + F e l m + K p X p + ( A − A d ) ∂ P ∂ x d X d = 0$

for the piston, where

• Ad is the displacer rod area in m2
• Dd is the displacer damping constant in N/ms
• Pd is the gas spring pressure in N/m2
• P is the working gas pressure in N/m2
• Dp is the piston damping constant (N/ms)
• Xd is the displacer position (m)

Figure 3.30   Linear Stirling engine with free-piston displacer mover.

• Xp is the power piston position (m)
• A is the cylinder area (m2)
• Md is the displacer mass (kg)
• Mp is the power piston mass (kg)
• Felm is the electromagnetic force (of linear electric generator) (N)

Equations 3.23 and 3.24 may be linearized as follows:

3.25 $M d X ¨ d + D d X ˙ d = − K d X p − α p X p M p X ¨ p + D p X ˙ p + F e l m = − K p X p − α T X d$
3.26 $K d = − A d ∂ P d ∂ X d − ∂ P ∂ X d ; α p = ∂ P ∂ X d A d K p = ( A − A d ) ∂ P ∂ X p ; α T = ( A − A d ) ∂ P ∂ X d ; F e l m = K e I$

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, with $X d , X ˙ d , X p , X ˙ p$

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 stand-alone or power-grid-connected electric generator operation modes.

The merits and demerits of Stirling engines are as follows:

• Independence of heat source: fossil fuels, solar energy
• Very quiet
• High theoretical efficiency; not so large in practice yet, but still 35%–40% for Tmax = 800°C and Tmin = 40°C

Figure 3.31   Free-piston Stirling engine dynamics model.

Table 3.2   Thermal Engines

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

Reduced

Independent

Easy

Poor

Stirling engines

Continuous

High in theory, lower so far

Very good

Very low

Independent

NA

Good

Spark-ignited engines

Discontinuous

Moderate

Still large

One type

Fast

Very good

Diesel engines

Discontinuous

Good

Larger

One type

Rather fast

Good

• Reduced emissions of noxious gases
• High initial costs
• Conduction and storage of heat are difficult to combine in the regenerator
• Materials have to be heat resistant
• For high efficiency, a heat exchanger is needed for the cooler
• Not easy to stabilize

A general comparison of thermal engines is summarized in Table 3.2.

3.8  Hydraulic Turbines

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.

Table 3.3   Hydraulic Turbines

Turbine

Type

Inventor

Trajectory

Tangential

Impulse

>300

Pelton (P)

Designed in the transverse plane

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.

3.8.1  Basics of Hydraulic Turbines

The terminology in hydraulic turbines is related to variables and characteristics [16]. The main variables are of geometrical and functional type:

• Rotor diameter Dr (m)
• General sizes of the turbine
• Turbine gross head: HT (m)
• Specific energy YT = gHT (J/kg)
• Turbine input flow rate: Q (m3/s)
• Turbine shaft torque: TT (N m)
• Turbine shaft power: PT, W (kW, MW)
• Liquid (rotor properties): density ρ (kg/m3), cinematic viscosity ν (m2/s), temperature T (°C), and elasticity module E (N/m2)

The main characteristics of a hydraulic turbine are, in general, as follows

• Efficiency
3.27 $η T = P T P h = T T ⋅ Ω r ρ g H T Q$
• Specific speed ns
3.28 $n s = n P T ⋅ 0.736 H T 5 / 4 , rpm$
where
• n is the rotor speed in rpm
• PT in kW
• HT in m
The specific speed corresponds to a turbine that for a head of 1 m produces 1 HP (0.736 kW)
• the characteristic speed nc:
3.29 $n c = n Q H T 3 / 4 , rpm$
where
• n is the rotor speed in rpm
• Q is the flow rate in m3/s
• HT in m
• Reaction rate γ
3.30 $γ = p 1 − p 2 ρ g H T$
where
• p1, p2 are the water pressure right before and after turbine rotor
• γ = 0 for Pelton turbines (p1 = p2)—zero reaction (impulse) turbine and 0 < γ < 1 for radial axial and axial turbines (Francis, Kaplan turbines)
• Cavitation coefficient σT:
3.31 $σ T = Δ h i H T$
where Δhi is the net positive suction head

It is good for σt to be small, σt = 0.01–0.1. It increases with ns and decreases with HT

• Specific weight Gsp:
3.32 $G s p = G T P T , N/kW$
where GT is the turbine mass × g, in N.

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.

3.8.2  First-Order Ideal Model for Hydraulic Turbines

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:

1. Water velocity U equation in the penstock
2. Turbine shaft (mechanical) power equation
3. Acceleration of water volume equation

According to the above assumptions the water velocity in the penstock U is

3.33 $U = K u G H$

where

• G is the gate opening
• H is the net head at the gate

Linearizing this equation and normalizing it to rated quantities $( U 0 = K u G 0 H 0 )$

yields

3.34 $Δ U U 0 = Δ H 2 H 0 + Δ G G 0$

The turbine mechanical power Pm writes

3.35 $P m = K p H U$

After normalization (Pm0 = KpH0U0) and linearization, Equation 3.35 becomes

3.36 $Δ P m P 0 = Δ H H 0 + Δ U U 0$

Substituting ΔH/H0 or ΔU/U0 from Equation 3.34 into Equation 3.36 yields

3.37 $Δ P m P 0 = 1.5 Δ H H 0 + Δ G G 0$

and finally

3.38 $Δ P m P 0 = 3 Δ U U 0 − 2 Δ G G 0$

The water column that accelerates due to change in head at the turbine is described by its motion equation:

3.39 $ρ L A d Δ U d t = − A ( ρ g ) Δ H$

where

• ρ is the mass density
• L is the conduit length
• A is the pipe area, g is the acceleration of gravity

By normalization Equation 3.39 becomes:

3.40 $T w d d t Δ U U 0 = − Δ H H 0$

where

3.41 $T w = L U 0 g H 0$

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:

3.42 $Δ U / U 0 Δ G / G 0 = 1 1 + ( T W / 2 ) s$
3.43 $Δ P m / P 0 Δ G / G 0 = 1 − T w ⋅ s 1 + ( T w / 2 ) s$

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

3.44 $Δ P m P 0 ( 0 ) = lim s → ∞ s 1 s 1 − T w s 1 + ( 1 / 2 ) T w s = − 2$
3.45 $Δ P m P 0 ( ∞ ) = lim s → ∞ s 1 s 1 − T w s 1 + ( 1 / 2 ) T w s = 1.0$

The time response to such a gate step opening is

3.46 $Δ P m P 0 ( t ) = ( 1 − 3 e − 2 t / T w ) Δ G G 0$

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.

3.8.3  Second- and Higher-Order Models of Hydraulic Turbines

We start with a slightly more general small deviation linear model of the hydraulic turbine:

3.47 $q = a 11 h + a 12 n + a 13 z m t = a 21 h + a 22 n + a 23 z$

where

• q is the volume flow
• h is the net head
• n is the turbine speed
• z is the gate opening
• mt is the shaft torque, all in p.u. values

As expected, the coefficients a11, a12, a13, a21, a22, a23 vary with load, etc. To a first approximation a12a22 ≈ 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]:

3.48 $h ( s ) q ( s ) = − T w T e tan h ( T e ⋅ s + F )$

where

• F is the friction factor
• Te is the elastic time constant of the conduit
3.49
3.50 $α = ρ g ( 1 K + D E f )$

where

• ρ is the water density
• g is the acceleration of gravity
• f is the thickness of conduit wall
• D is the conduit diameter
• K is the bulk modulus of water compression
• E is the Young’s modulus of elasticity for the pipe material

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:

3.51

Alternatively, from Equation 3.47, we obtain the following:

3.52

where qp is the friction

With F = qp = 0, Equations 3.51 and 3.52 degenerate into the first order model provided tan h TesTes, that is for very low frequencies:

3.53 $G 1 ( s ) = 1 − T W ⋅ s 1 + ( T W / 2 ) ⋅ s$

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]:

3.54

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]:

3.55

where

• Tec is the elastic time constant of the tunnel
• TWC is the water starting time in the tunnel
• qc is the the surge tank friction coefficient
• hS is the surge tank head
• Up is the upper penstock water speed
• TS is the surge tank riser time (TS ≈ 600–900 s)

Now for the penstock, the wave equation yields (in p.u.)

3.56

where

• Zp is the hydraulic impedance of the penstock (Zp = TWp/Tep)
• qp is thefriction coefficient in the penstock
• Tep is the penstock elastic time
• TWp is the penstock water starting time
• hr is the riser head
• ht is the turbine head

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]

3.57

The power differential is written as follows:

3.58

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.

3.8.4  Hydraulic Turbine Governors

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:

• The pilot valve servomotor (lower power stage of governor) may be mechanical or electric; electric servomotors tend to provide faster and more controllable response.
• Water is not very compressible, and thus the gate motion has to be gradual; near the full closure even slower motion is required.
• Dead band effects are considered in Figure 3.41, but their identification is not an easy task.
• Stable operation during system islanding (stand-alone operation mode of the turbine—generator system) and acceptable response quickness and robustness under load variations are the main requirements that determine the governor settings.
• The presence of transient compensation droop is mandatory for stable operation.
• For islanding operation, the choice of temporary droop RT and reset time TR is essential; they are related to water starting time constant TW and mechanical (inertia) time constant of the turbine/generator set TM. Also the gain KV should be high.
• According to Reference 2,
3.59 $R T = [ 2.3 − ( T W − 1.0 ) 0.15 ] T W T M T R = [ 5.0 − ( T W − 1.0 ) 0.5 ] T W T M = 2 H ; H = J ω 0 2 2 S 0 ( s )$
where
• J (kg m2) is the turbine/generator inertia
• ω0 is the rated angular speed (rad/s)
• S0 is the rated apparent power (VA) of the electrical generator

Figure 3.42   Coordinated turbine governor–generator–exciter control system.

• In hydraulic turbines where wickets gates (Figure 3.41) are also used, the governor system has to control their motion also, based on an optimization criterion.

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.

3.8.5  Reversible Hydraulic Machines

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:

1. By topology
2. By direction of motion
• With speed reversal for pumping (Figures 3.43 and 3.44)
• Without speed reversal for pumping
3. By direction of fluid flow/operation mode

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.

3.9  Wind Turbines

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,

3.60 $t t max = e − U 4$

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:

3.61 $f ( U ) = − d ( t / t max ) d U$

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:

3.62 $U a v e = ∫ 0 ∞ U f ( U ) d U$

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:

3.63

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.

3.9.1  Principles and Efficiency of 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:

• Mechanical power P (W)
• Shaft torque (N m)
• Rotor speed n (rpm) or ωr (rad/s)
• Rated wind speed UR
• Tip speed ratio:
3.64

The tip speed ratio λ < 1 for slow-speed wind turbines and λ > 1 for high-speed wind turbines.

• The power efficiency coefficient Cp:
3.65 $C p = 8 P ρ π D r 2 ⋅ U 3 < > 1$

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:

• Axial (with horizontal shaft)
• Tangential (with vertical shaft)

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 Cpmaxopt) 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:

3.66 $( λ o p t , Z 1 ) = ( 1 , 8 − 24 ; 2 , 6 − 12 ; 3 , 3 − 6 ; 4 , 2 − 4 ; 5 , 2 − 3 ; > 5 , 2 )$

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

3.67 $u 1 A 1 = u ∞ A ∞$

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:

3.68 $P w i n d = ρ U 1 A ⋅ 1 2 ⋅ U 1 2 = 1 2 ρ A U 1 3$

The power extracted from the wind, Pturbine is

3.69 $P t u r b i n e = ρ U A ( U 1 2 2 − U ∞ 2 2 )$

Let us assume:

3.70 $U ≈ U 1 − Δ U ∞ / 2 ; U ∞ = U 1 − Δ U ∞ ; ψ = Δ U ∞ U 1$

The efficiency limit, ηideal, is

3.71 $η i d e a l = P n u r b i n e P w i n d = ( 1 / 2 ) ρ U A ( U 1 2 − U ∞ 2 ) ( 1 / 2 ) ρ A U 1 3$

With Equation 3.61, ηideal becomes:

3.72 $η i = ( 1 − Ψ 2 ) [ 1 − ( 1 − Ψ ) 2 ]$

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].

3.9.2  Steady-State Model of Wind Turbines

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 arωr) are produced by the vortex system of the machine.

The local attack angle α is

3.73 $α − ϕ − θ$

with

3.74 $tan ϕ = U ( 1 − a ) r ω r ( 1 + a ′ )$

Figure 3.54   Blade element with pertinent speed and forces.

The local blade pitch θ is

3.75 $θ = τ + β$

where

• τ is the local blade twist angle
• β is the global pitch angle

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:

3.76
3.77 $F d r a g = 1 2 ρ U r e l ⋅ C ⋅ C D$

where

• C is the local chord of the blade section
• CL and CD are lift and drag coefficients, respectively, known for a given blade section [26,27]

From lift and drag forces, the normal force (thrust), FN, and tangential force FT (along X,Y on the blade section plane) are simply

3.78 $F N ( r ) = F l i f t cos Φ + F d r a g sin Φ$
3.79 $F T ( r ) = F l i f t sin Φ − F d r a g cos Φ$

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

3.80 $F T = B ∫ 0 D r / 2 F N ( r ) d r$

Similarly, the mechanical power PT is as follows:

3.81 $P T = B ⋅ ω r ⋅ ∫ 0 D r / 2 r F T ( r ) d r$

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):

3.82 $P M = 1 2 ρ C p π D r 2 U 3 = 1 2 ρ π ( D r 2 ) 5 ⋅ C p λ 3 ω r 3$

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:

3.83 $P M o p t = 1 2 ρ π ( D r 2 ) 5 ⋅ C p max λ o p t 3 ω r 3 = K W ω r 3$

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:

3.84 $ω r = U ⋅ 2 D r ⋅ λ o p t$

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):

• Stall control
• Pitch control

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.

3.9.3  Wind Turbine Models for Control

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.

3.9.3.1 Unsteady Inflow Phenomena in Wind Turbines

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 + aDrωr/2).

The inclusion of a lead-lag filter to simulate the inflow phenomena seems insufficient due to considerable uncertainty in the modeling.

3.9.3.2 Pitch-Servo and Turbine Model

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]:

3.85 $d d t [ | θ | | ω | ] = [ 0 I − [ 2 H − 1 ] [ C ] − [ 2 H − 1 ] D ] ⋅ [ | θ | ω ] + [ 0 2 H − 1 ] [ T ]$

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.

3.10  Summary

• Prime movers are mechanical machines that convert primary energy of a fuel (or fluid) into mechanical energy.
• Prime movers drive electric generators connected to the power grid or operating in isolation.
• Steam and gas turbines and internal combustion engines (spark-ignited or diesel) are burning fossil fuels to produce mechanical work.
• Steam turbines contain stationary and rotating blades grouped into (HP, IP, and LP sections on same shaft in tandem-compound and on two shafts in cross-compound configurations.
• Between stages, steam engines use reheaters; single reheat and double reheat, at most.
• Typically in steam turbines, the power is divided as 30% in the HP, 40% in the IP and 30% in the LP stage.
• Governor valves and IV are used to control the HP and, respectively, LP stages of the steam flow.
• The steam vessel may be modeled by a first-order delay, while the steam turbine torque is proportional to its steam flow rate.
• Three more delays related to inlet and steam chest, to the reheater and to the crossover piping may be identified.
• Speed governor for steam turbines include a speed relay with a first-order delay and a hydraulic servomotor characterized by a further delay.
• Gas turbines burn natural gas in combination with air that is compressed in a compressor driven by the gas turbine itself.
• The 500°C gas exhaust is used to produce steam that drives a steam turbine placed on the same shaft. These combined cycle unishaft gas turbines are credited with a total efficiency above 55%. Combined cycle gas turbines at large powers seem the way of the future. They are also introduced for cogeneration in high-speed small- and medium-power applications.
• Diesel engines are used from the kilowatt range to megawatt range power per unit for cogeneration or for standby (emergency) power sets.
• The fuel injection control in diesel engines is performed by a speed-governing system.
• The diesel engine model contains a nonconstant gain. The gain depends on the eer, which in turn is governed by the fuel/air ratio; a dead time constant dependent on engine speed is added to complete the diesel engine model.
• Diesel engines are provided with a turbocharger that has a turbine “driven” by the fuel exhaust that drives a compressor that provides the hot HP air for the air mix of the main engine. The turbocharger runs freely at high speed but is coupled to the engine at low speed.
• Stirling engines are “old” thermal piston engines with continuous heat supply. Their thermal cycles contains two isotherms. It contains, in a basic configuration, two opposed pistons and a regenerator in between. The efficiency of Stirling engine is temperature limited.
• Stirling engines are independent fuel type; use air, methane, He, or H2 as working fluids. They did not reach commercial success in kinetic type due to problems with the regenerator and stabilization.
• Stirling engines with free piston-displacer mover and linear motion have reached recently the markets in units in the 50 W to a few kilowatts.
• The main merits of Stirling engines are related to their quietness and reduced noxious emissions, but they tend to be expensive and difficult to stabilize.
• Hydraulic turbines convert water energy of rivers into mechanical work. They are the oldest prime movers.
• Hydraulic turbines are of impulse type for heads above 300–400 m and of reaction type (below 300 m). In a more detailed classification, they are tangential (Pelton), radial–axial (Francis) and axial (Kaplan, Bulb, Straflo).
• High head (impulse) turbines use a nozzle with a needle-controller where water is accelerated and then it impacts the bowl-shaped buckets on the water wheel of the turbine. A jet deflector deflects water from runner to limit turbine speed when electric load decreases.
• Reaction turbines—at medium and low head—use wicket gates and rotor blade servomotors to control water flow in the turbine.
• Hydraulic turbines may be modeled by a first-order model if water hammer (wave) and surge effects are neglected. Such a rough approximation does not hold above 0.1 Hz.
• Second-order models for hydraulic turbines with water hammer effect in the penstock considered are valid up to 1 Hz. Higher orders are required above 1 Hz as the nonlinear model has a gain whose amplitude varies periodically. Second- or third-order models may be identified from tests through adequate curve fitting methods.
• Hydraulic turbine governors have one or two power levels. The lower power level may be electric, while the larger (upper) power level is a hydraulic servomotor. The speed controller of the governor has traditionally a permanent droop and a transient droop.
• Modern nonlinear control systems may now be used to control simultaneously the guide vane runner and the blade runner.
• Reversible hydraulic machines are used for pump storage power plants or for tidal-wave power plants. The optimal pumping speed is about 12%–20% above the optimal turbining speed. Variable speed operation is required. Therefore, power electronics on the electric side is mandatory.
• Wind turbines use the wind air energy. Nonuniformity and strength vary with location height and terrain irregularities. Wind speed duration versus speed, speed versus frequency and mean (average) speed using Raleigh or Weibull distribution are used to characterize wind on a location in time. The wind turbine rated wind speed is, in general, 150% of mean wind speed.
• Wind turbines are of two main types: axial (with horizontal shaft) and tangential (with vertical shaft).
• Wind turbines’ main steady-state parameter is the power efficiency coefficient Cp, which is dependent on blade tip speed Rωr to wind speed U (ratio λ). Cp depends on λ and on blade absolute attack angle β.
• The maximum Cp (0.3–0.4) with respect to β is obtained for λopt ≤ 1 for low-speed axial turbines and for λopt ≥ 1 for high-speed turbines.
• The ideal maximum efficiency limit of wind turbines is about 0.6 (Betz limit).
• Wind impacts on the turbine a thrust force and a torque. Only torque is useful. The thrust force and Cp depend on blade absolute attack angle β.
• The optimal power PTopt) is proportional to u3 (u—wind speed).
• Variable speed turbines will collect notably more power from a location if the speed varies significantly with time and season, such that λ may be kept optimum. Above rated wind speed (and power), the power is limited by passive stall, active stall, or pitch-servo control.
• Wind turbine steady-state models are highly nonlinear. Unsteady inflow phenomena show up in fast transients and have to be accounted for by more than lead-lag elements.
• Pitch-servo control is becoming more and more frequent even with variable speed operation, to allow speed limitation during load transients or power grid faults.
• First- or second-order models may be adopted for speed governors. Elastic transmission multimass models have to be added to complete the controlled wind turbine models for transients and control.
• R&D efforts on prime movers’ modeling and control seem rather dynamic [3234].
• Prime mover models will be used in following chapters where electric generators control will be treated in detail.

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VIB-ROTATIONAL ANALYSIS ON $2\nu_{1} + \nu_{5}, \nu_{1} + \nu_{2} + 3\nu_{5}$, and $2\nu_{2} + 5\nu_{5}$ COMBINATION BANDS OF ACETYLENE USING A GaAs SEMICONDUCTOR LASER

Please use this identifier to cite or link to this item: http://hdl.handle.net/1811/18924

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 Title: VIB-ROTATIONAL ANALYSIS ON $2\nu_{1} + \nu_{5}, \nu_{1} + \nu_{2} + 3\nu_{5}$, and $2\nu_{2} + 5\nu_{5}$ COMBINATION BANDS OF ACETYLENE USING A GaAs SEMICONDUCTOR LASER Creators: Suzuki, M.; Yamashita, T.; Ohtsuka, T.; Yukiya, T.; Nishimiya, N. Issue Date: 1998 Abstract: The vib-rotational spectrum of the $2\nu_{1}+\nu^{i=1}_{5}, \nu_{1}+\nu_{2}+3\nu_{5}^{i=1}$ and $2\nu_{2}+5\nu_{5}^{i=1}$ combination bands of acetylene was measured using a GaAs semiconductor laser (New Focus 6148) at the wavenumber region from 7400 to $7600 cm^{-1}$ and the P-, R- and Q-branches were assigned. The wavenumber measuring system with a wavelength meter (Anritsu MF9630A) and a confocal interferometer of 15 cm was calibrated against the $H_{2}O lines.^{1}$ The formulation describing the spectral line positions by a third order polynomial of $J(J+1)$ with the effective spectroscopic parameters of $\nu_{0}^{eff},B_{\nu}^{eff}$, and $H_{\nu}^{eff}$ was obtained using second order perturbation theory to take the l-type resonance effect into consideration. The spectroscopic constants and the l-type doubling constants were calculated from the effective rotational parameters determined using a least squares fitting procedure for the e- and f-levels by the aid of the molecular constants reported by ref. 2 and 3. URI: http://hdl.handle.net/1811/18924 Other Identifiers: 1998-RF-11

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# Infinite Products!

Calculus Level 3

For $$x > 1$$, define $$\displaystyle f(x) = \prod_{n=2}^{\infty} \left(1 - \dfrac{1}{n^x}\right)$$.

Find the value of $$\dfrac{f(6)}{f^2(3)}$$.

×


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Computing with Data

elgeish
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Classification

Using the iris dataset, we implement a binary classifier that predicts whether a sample is an Iris-Versicolor (denoted by the label 1) or not: