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 T_REV = 1/n (n-shaft 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 four-cycle engine is

The firing angles for a 12 cylinder diesel engine are illustrated in Figure 3.17a, while the two-revolution sequence is: intake (I), compression (C), power (P), and exhaust (E) (see Figure 3.17b).

The 12-cylinder timing is shown in Figure 3.18.

There are 3 cylinders out of 12 firing simultaneously at steady state.

The resultant shaft torque of one cylinder varies with shaft angle as in Figure 3.19.

The compression torque is negative while during power cycle it is positive.

With 12 cylinders, the torque will have much smaller pulsations, with 12 peaks over 720° (period of power engine stroke); Figure 3.20.

Any misfire in one or a few of the cylinders would produce severe pulsations in the torque that would reflect as flicker in the generator output voltage [8].

Large diesel engines are provided, in general, with a turbocharger (Figure 3.21) which influences notably the dynamic response to perturbations by its dynamics and inertia [9].

The turbocharger is essentially an air compressor that is driven by a turbine that runs on the engine exhaust gas.

The compressor provides compressed air to the engine cylinders. The turbocharger works as an energy recovery device with about 2% power recovery.

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 J_E, turbocharger J_T and electric generator (alternator) J_G.
• 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 $${\tau }_1\approx A+\frac{B}{n_E}+\frac{C}{n_E^2}$$

  
  where n_E 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.

3.7.1  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]:

where

• ε = V₃/V₁ is the compression ratio
• ρ = V₂/V₁ = V₃/V₄ is the partial compression ratio
• $x={p^{\prime }}_1/p_1$
  is the pressure ratio

For ρ = 2, x = 10, K = 1.4, ε = 3, and ηth = 31%.

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

With ideal, complete, heat recirculation,

Gas turbines are more compact than other thermal machines; they are easy to start, have low vibrations, have low efficiency at low loads (ρ small), and tend to have poor behavior during transients.

The spark-ignited (Otto) engines work on the cycle in Figure 3.25.

The four steps are as follows:

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

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.

During the downward movement of the piston, an isobaric state change takes place by controlled injection of fuel (2 → 3):

Efficiency decreases when load ρ increases, in contrast to spark-ignited engines for same ε. Lower compression ratios (ε) than for spark-ignited engine are characteristic for diesel engines to obtain higher thermal efficiency.

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 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 Stirling-type 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.

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

For the normal displacer and:

for the piston, where

• A_d is the displacer rod area in m²
• D_d is the displacer damping constant in N/ms
• P_d is the gas spring pressure in N/m²
• P is the working gas pressure in N/m²
• D_p is the piston damping constant (N/ms)
• X_d is the displacer position (m)

• X_p is the power piston position (m)
• A is the cylinder area (m²)
• M_d is the displacer mass (kg)
• M_p is the power piston mass (kg)
• F_elm is the electromagnetic force (of linear electric generator) (N)

Equations 3.23 and 3.24 may be linearized as follows:

where I is the generator current

The electric circuit correspondent of Equation 3.25 is shown in Figure 3.31.

The free-piston Stirling-engine model in Equation 3.25 is a fourth-order system, 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 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 T_max = 800°C and T_min = 40°C

• 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.

3.8.1  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 D_r (m)
• General sizes of the turbine
• Turbine gross head: H_T (m)
• Specific energy Y_T = gH_T (J/kg)
• Turbine input flow rate: Q (m³/s)
• Turbine shaft torque: T_T (N m)
• Turbine shaft power: P_T, W (kW, MW)
• Rotor speed Ω_T (rad/s)
• 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 $${\eta }_T=\frac{P_T}{P_h}=\frac{T_T\cdot {\Omega }_r}{\rho g{H}_{T}Q}$$

  
• Specific speed n_s
  3.28 $$n_s=n\frac{\sqrt{P_T\cdot 0.736}}{H_T^{5/4}},\text{rpm}$$

  
  where
  • n is the rotor speed in rpm
  • P_T in kW
  • H_T 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 n_c:
  3.29 $$n_c=\frac{n\sqrt{Q}}{H_T^{3/4}},\text{rpm}$$

  
  where
  • n is the rotor speed in rpm
  • Q is the flow rate in m³/s
  • H_T in m
• Reaction rate γ
  3.30 $$\gamma =\frac{p_1-p_2}{\rho g{H}_T}$$

  
  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 σ_T:
  3.31 $${\sigma }_T=\frac{\Delta h_i}{H_T}$$

  
  where Δh_i is the net positive suction head

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

• Specific weight G_sp:
  3.32 $$G_{sp}=\frac{G_T}{P_T},\text{N/kW}$$

  
  where G_T is the turbine mass × g, in N.

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^–3–10³ 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 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.

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 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).

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

where

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

Linearizing this equation and normalizing it to rated quantities $\left(U_0=K_u{G}_0\sqrt{H_0}\right)$

yields

The turbine mechanical power P_m writes

After normalization (P_m0 = K_pH₀U₀) and linearization, Equation 3.35 becomes

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

and finally

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

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:

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 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.

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:

where

• q is the volume flow
• h is the net head
• n is the turbine speed
• z is the gate opening
• m_t 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 a₁₂ ≈ a₂₂ ≈ 0 and, with constant a_ij 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]:

where

• F is the friction factor
• T_e is the elastic time constant of the conduit

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. 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), G₁(s) and G₂(s) for T_e = 0.25 s and T_w = 1 s.

The second-order 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 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.

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]:

where

• T_ec is the elastic time constant of the tunnel
• T_WC is the water starting time in the tunnel
• q_c is the the surge tank friction coefficient
• h_S is the surge tank head
• U_p is the upper penstock water speed
• T_S is the surge tank riser time (T_S ≈ 600–900 s)

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

where

• Z_p is the hydraulic impedance of the penstock (Z_p = T_Wp/T_ep)
• q_p is thefriction coefficient in the penstock
• T_ep is the penstock elastic time
• T_Wp is the penstock water starting time
• h_r is the riser head
• h_t is the turbine head

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 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, U_NL is the no-load 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 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 (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 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.

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 R_T and reset time T_R is essential; they are related to water starting time constant T_W and mechanical (inertia) time constant of the turbine/generator set T_M. Also the gain K_V should be high.
• According to Reference 2,
  3.59 $$\begin{array}{lll}\hfill R_T& =\hfill & \left[2.3-\left(T_W-1.0\right)0.15\right]\frac{T_W}{T_M}\hfill \\ \hfill T_R& =\hfill & \left[5.0-\left(T_W-1.0\right)0.5\right]T_W\hfill \\ \hfill T_M& =\hfill & 2H;\hfill \\ \hfill H& =\hfill & \frac{J{\omega }_0^2}{2S_0}\left(s\right)\hfill \end{array}$$

  
  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
   • Radial–axial (Russel Dam) (Figure 3.43)
   • Axial (Annapolis) (Figure 3.44)
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
   • Unidirectional/operation mode (Figure 3.43)
   • Bidirectional/operation mode (Figure 3.44)

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.

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.

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.

3.9.1  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 ω_r (rad/s)
• Rated wind speed U_R
• Tip speed ratio:
  3.64 $$\lambda =\frac{{\omega }_r\cdot D_r/2}{U}=\frac{\text{rotor blade tip speed}}{\text{wind speed}}$$

  

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

• The power efficiency coefficient C_p:
  3.65 $$C_p=\frac{8P}{\rho \pi D_r^2\cdot U^3}<>1$$

  

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:

• 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 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 self-start 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 Z₁ 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.

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 self-starting 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, u₁ > u_∞ and thus A₁ < 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_∞/U₁ = 1/3) with η_imax = 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 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

The local blade pitch θ is

where

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

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

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

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 steady-state 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):

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 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 > U_rated):

• Stall control
• 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 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.

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.

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.