Feedback Control of Dynamic Systems 7th Edition Chapter 3 Solutions
Digital Controllers
David M. Auslander , in Encyclopedia of Physical Science and Technology (Third Edition), 2003
I.B Why Feedback is Hard: Dynamics
Feedback control has its limitations. There is a fundamental limitation imposed by the instrument: the overall performance of the control system cannot be any better than that of the sensor. There is another fundamental limit imposed by physical realizability: because no physical system can change state instantly, there is a limit to how fast a control system can respond to changes in either its command or to changes in disturbances. The "dynamic" behavior of a physical system describes how it changes in response to a change in its environment, purposeful change as when motor input power is changed, or unexpected change as when a "downdraft" causes sudden loss of altitude in a passenger airplane, much to the discomfort of the passengers.
The behavior of a dynamic system depends on the history of what has been done to it. It is not enough to know what its current inputs are—even if the airplane has moved past the downdraft, it will take some time to bring it back to its previous altitude. This is the crux of the feedback control design problem: react too energetically to a deviation from the desired output and the future consequences may be more than was bargained for; react too timidly and it will take altogether too long for the system to get to where it should be (maybe never). The essence of the mathematical side of control engineering is to devise methods for doing the feedback process just right. The practical side of control engineering is in understanding when and how to apply the mathematics and in designing the environment in which these mathematical methods are embedded, software and hardware, so as to build the most effective possible control systems.
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Chemical Storage, Dosing and Control
Malcolm J. Brandt BSc, FICE, FCIWEM, MIWater , ... Don D. Ratnayaka BSc, DIC, MSc, FIChemE, FCIWEM , in Twort's Water Supply (Seventh Edition), 2017
12.28 Feedback Control
In feedback control, the desired value for the associated water quality parameter is entered as a set-point in the controller by the operator. The signal from a water quality monitor downstream of the chemical dosing point is compared with this set-point and the controller adjusts the dosing rate to maintain the set-point.
Feedback process control may be continuous using conventional PID control, described below, or it may be implemented at regular pre-set intervals. This latter method makes corrections to the process at specific intervals, to take account of the time taken for adjustments made to the chemical dose rate to be observed on the downstream water quality monitor. This period is referred to as the control loop time and includes the transit time from the chemical dosing point to the sampling point, the sample transit time from the sampling point to the water quality monitor, the response time of the water quality monitor and the time taken for the changed chemical dose rate to reach the dosing point (Fig. 12.1). The interval between adjustments to the chemical dosing rate is varied with the flow past the dosing point, to take account of the change in time taken for dosed water to reach the sample point. Flow proportional control is maintained during this interval, to maintain the prevailing dose. It is important to minimize the overall control loop time, particularly where PID control is used (Section 12.31). Control loop times should normally be no more than 4 minutes and should not exceed 10 minutes under any circumstances, unless as part of a cascade system (Section 12.29).
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An overview of ground-source heat pump technology
Rao Martand Singh , ... Tony Amis , in Managing Global Warming, 2019
15.4.2 Closed-loop ground-source heat exchangers
The closed-loop systems utilize and circulate constant volume of heat carrier fluid (HCF) in a closed plastic pipe network installed in subsurface structures or buried to some depth beneath the ground surface. The exchange of heat between the ground and the superstructure/infrastructure is through an indirect contact in comparison to the open-loop systems. In closed-loop systems, heat is transferred from the ground through the concrete and plastic pipes, via conduction, and transported to the heat pump through HCF circulation, via convection and conduction.
The plastic loops are made from high-density polyethylene/polypropylene (HDPE/HDPP), polyvinyl-chloride (PVC), polybutylene (PB) plastic pipes [20–23], with a nominal internal diameter that ranges between 20 and 44 mm and an appropriate wall thickness of 2.0–2.3 mm [12,24–29].
Once the plastic loops are fitted and secured in place, each of the loops is individually connected to the header or manifold. This allows room for reliability check of the system, and ensures the safety of the primary casing element especially where structural foundation elements are used. Thus, an energy loop with a defect can be independently disconnected at the manifold block to safeguard the system performance.
Enclosed in the energy exchanging loops, is an HCF that serves as a medium of heat exchange between the ground and the heat pump. Typically water, water plus biocide, antifreeze (glycol) or saline solutions are used as the HCF [15,30–33]. However, using an antifreeze-water-based solution increases the HCF viscosity, decreases its freezing point, resulting in more laminar flow behavior of the fluid, thus requiring more energy input to circulate the HCF in the loops.
The heat pump unit can be connected to different types of closed-loop heat exchanger configurations [14,15,31,34,35], as shown in Fig. 15.5A–F . The choice of which particular configuration to adopt should be based on some factors including installation cost, land area availability, thermal demand, building's foundation type, depth of heterothermal influence zone, and whether the installation of the heat pump system is to be carried out prior to or after building construction and occupancy.
In the configuration shown in Fig. 15.5A, HDPE pipes are laid horizontally in the ground beneath the frost influence zone. This usually occurs at a depth ranging between 1.5 and 2.0 m. Beyond this depth, daily changes to ground temperature are less significant. This type of configuration should be installed in areas where the ground is in direct exposure of the sun to enhance heat transfer due to solar radiation. Also, infiltration of rain water through the ground also has positive effect on the system performance. The pipes are arranged in such a way that a spacing of about 0.5–0.8 m is maintained between the exit and return leg of the pipe network. This reduces the risk of heat exchange between the pipes. In the design of this type of system, it is safe to assume a usable thermal energy output of about 10–40 W per meter length of the loop, depending on soil conditions. The horizontal trench collector requires large area of land to sufficiently supply energy to the building. For example, to heat up a 1 m2 of living space, a power of about 50 W is needed. Thus, an area of about 1.5–5 times, the area of living space is needed to obtain the required energy to heat the space. This type of configuration is not suitable in densely populated areas. However, in places where land is abundant, e.g., in rural areas, horizontal trench configuration can be used to provide heating for a single-family house. They are relatively simple to implement with very low installation cost [35,36].
Fig. 15.5B shows a slinky pipe configuration. This layout is similar to the horizontal trench installation shown in Fig. 15.5A. However, their only difference is that they allow longer loop length to be incorporated in the horizontal trench. The standard installation procedure recommends the loops to be installed in trenches that are 1.2 m wide. Similarly, in situations where more than one trench is required, the spacing between individual trenches should be at least 5 m apart centers to prevent excessive heat extraction. Additionally, the slinky configuration can also be incorporated vertically, in a narrow trench. The slinky type of configuration could provide 1 kW of energy per 10-m linear length of the pipes [35].
In addition to the configuration in Fig. 15.5B, the loops could also be fitted into a vertical trench in a spiral form as shown in Fig. 15.5C. The spring-like loops are installed to a depth ranging between 2 and 5 m. The distance between each spiral probe to another should be about 3–4 m. This type of configuration is ideally suitable where limited land area is available especially considering it does not require specialized heavy drilling equipment. This is a relatively new approach and could also be used where difficult geological conditions exist and drilling becomes nearly impossible. Under favorable ground conditions, it is possible to yield an output of 100–700 W per helix probe depending on the soil type and its degree of saturation [37–40]. A typical example of this spiral-like probe is the RAUGEO Helix probe manufactured by REHAU unlimited polymer solutions.
Fig. 15.5D shows the diagram of boreholes installed in a slanting form. The construction of this type of configuration occurs by the drilling of boreholes at an angle from a single manifold chamber. This ensures that the tips of the boreholes are placed farther apart from each other to maximize and enhance heat exchange of the system. During the construction stages, a single chamber is excavated, which serves as an access point through which the bores can be drilled and installed. The loops are installed at a sloping angle that ranges between 30° and 65°, within any point along the 360° radial plane [32]. These configurations eliminate the limitations of large-area requirement associated with the horizontally laid configurations, i.e., Fig. 15.5A and B. These allow the slanting boreholes to penetrate beneath existing buildings, gardens, driveways, etc. However, there are legal issues allied with drilling beneath other people's properties and close to government underground structures, e.g., tunnels [34].
The configuration shown in Fig. 15.5E depicts the conventional set-up with the energy loops installed in typical vertical borehole heat exchangers. The bored holes are generally filled with bentonite grout mixed with sand to enhance the borehole thermal conductivity. The borehole diameter and length ranges between 100 and 200 mm, and 20–200 m [18,41–43]. Also, the spacing between neighboring boreholes should be at least 4.5 m apart to ensure effective system performance in the long run [44]. It is estimated that on average, 1 m length of a vertical borehole could provide sufficient energy to heat up 1 m2 of living space. This type of configuration has gained popularity because it requires a small land area for installation compared to horizontal trench heat exchangers. Furthermore, it provides greater efficiency because it is installed beyond the depth where annual temperature variation occurs. However, its main limitation is the high initial cost, which is attributed to the bore-hole drilling. A single or few boreholes can be installed to provide thermal comfort of a small-to-medium house unit. However, where high-heating loads exist (office buildings, utilities, and others) a group of boreholes, comprising dozens or even hundreds of boreholes, are commonly used; these are often referred to as geothermal fields.
The installation shown in Fig. 15.5F represents the typical case where the energy loops are installed in structural pile foundation elements. They offer similar advantages to the vertical borehole heat exchangers. Also, they are both installed to a depth where the ground annual temperature is relatively constant. Furthermore, the initial drilling cost associated with boreholes is negated because the piles were already needed for structural support. Also, due to the larger diameter of piles, i.e., 300–1500 mm, they offer more flexible and diversified loops arrangement compared to a conventional borehole [21,45]. These elements are generally referred to as energy piles, thermal piles, or geothermal energy piles. It is expected that piles with diameter of 300–500 mm could yield about 40–60 W of heat energy per meter length. Whereas piles with diameter greater than 600 mm are expected to provide about 35 W per square meter of the earth-contact surface area [12,21,45].
In addition, other alternative structural elements such as diaphragm walls, tunnels, and base slabs could be fitted with energy loops to provide the required thermal demand while meeting their primary need of structural support [12,33,46–49].
The configuration shown in Fig. 15.5G shows the scenario where the loops are placed in small diameter bored-holes (up to 300 mm) made using micropiling technology. This new solution offers new opportunities for existing buildings where micropiles are needed to provide additional strength to the existing foundation elements, and as anchors for retaining walls [50].
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Architectures of Transportation Cyber-Physical Systems
John D. McGregor , ... Eduardo S. Almeida , in Transportation Cyber-Physical Systems, 2018
3 Current canonical cyber-physical system architectures
In this section we survey a couple of CPS architectures and describe some criteria for evaluating architectures.
The canonical feedback/control loop is an architecture that is used in many CPSs. The main objective of such an architecture is to control some physical process. Typically the controlled process is at least partially a physical system such as a jet engine, automated vehicle or chemical reaction. A commonly used example is controlling the temperature in a house. Fig. 2.5 shows the four major constituent pieces to the architecture. Below we define the function of each piece of the basic architecture in the context of the home heating system.
Controlled process – The controlled process is the heating and cooling functions provided by the system. Its operation is physical and controlled by the amount of energy/fuel that is being supplied to the heating or cooling unit. When turned on, the system will blow air into the house. As the system operates, the air temperature will gradually change until it reaches the target temperature. Once the blower shuts off, the unit will continue to radiate hot or cold for some time. Without the unit operating, the temperature will move away from the target temperature toward the ambient temperature due to air circulation in the building. Eventually the deviation from target will be sufficiently great and the system will begin to blow again repeating the cycle.
Controller – The controller is usually a computational engine that takes inputs from sensors and computes the outputs that should be used to command the actuators. Traditionally, the controller for a home furnace simply turns on the current to heat an element or turns it off. When the controller is a computational engine, a more sophisticated controller can be designed that varies the amount of current to be fed to the furnace depending on the mode the system is in. It sends commands to actuators about how much fuel to feed.
Sensor – A sensor measures some characteristics within the system or its environment. For example, inside the house, a temperature sensor, essentially a thermometer, produces an electric current proportional to the temperature of the air. Often this is implemented by a metal spring that expands due to the heat. A digital sensor will convert the current into a discrete value. This value is fed to the controller.
Actuator/effector – The controlled process is modified by one or more actuators. For the house heating example, a rheostat increases/decreases the current flowing in the circuit or a servo motor opens/closes a valve to control the flow of fuel. There are delays (latencies) of varying lengths in opening and closing flows depending on how mechanical the device is.
There are many variations on this architecture. One of the major variations is whether the sensors are analog or digital or a combination. Each signal from an analog sensor runs through an analog to digital converter which sanitises the data, in addition to converting the continuous data signal to a discrete signal. Signals from digital sensors can be fed directly into the controller.
A major issue in these systems is the uncertainty associated with the physical elements of the system. The expansion of metal in a thermostat is not exact and even changes with the repeated expansion and contraction of use over time. Other physical changes also introduce uncertainty. Readings from these sensors must be represented by an interval data value instead of a point value. The interval is sufficiently wide for there to be a 95% probability that the actual answer is within the interval.
An important characteristic of these systems is its sensitivity. Sensitivity is how large a deviation from the target value is permitted before the controller activates an actuator. A larger deviation will take longer to get back on target while a smaller deviation will be quickly corrected but will result in a very choppy operation with system turning on then off again rapidly. The first approach will leave the temperature either above or below the target value longer than the second solution. The second will keep the temperature closer to the target temperature but will produce more wear on the switching circuitry.
One example of a CPS using the control feedback/loop architecture is a collaborating, adaptive cruise control system for a semiautonomous vehicle. We will use this as a continuing example throughout this chapter.
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Process Control Systems
Juergen Hahn , Thomas F. Edgar , in Encyclopedia of Physical Science and Technology (Third Edition), 2003
I.A Feedback Control
The purpose of feedback control is to keep the controlled variable close to its set point. This is achieved by using the difference between the set point and the controlled variable to determine the value of the input to the feedback controller. The feedback controller by its design takes corrective action to reduce the deviation. This action is called negative feedback, because the manipulated variable typically moves in a direction opposite in sign to the error. For example, if you were controlling the temperature in a shower with the cold water flow, an increase in temperature above the set point gives a negative error between set point and controlled variable; hence the manipulated variable should move in the opposite (positive) direction to compensate for the error, which in this case is an increase in cold water flow. Feedback controllers have user-specified parameters that can be adjusted to achieve desirable dynamic performance. The design of feedback controllers is discussed in more detail in Sections III and IV.
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Specialty Polymers & Polymer Processing
Won-Kyoo Lee , L. James Lee , in Comprehensive Polymer Science and Supplements, 1989
13.3.2.3 Feedforward Control
As previously discussed, feedback control strategies are effective in tracking setpoint changes and compensating for low-frequency disturbances. For polymer extrusion, the fluctuations in the intermediate frequencies, ranging from 0.5 to 10 cycles min −1, are the main cause of poor product quality. Feedback control cannot handle such disturbances. Feedforward control has generally been considered to be an advanced control method which can overcome many of the disadvantages of feedback control. The philosophy of feedforward control is to provide an appropriate corrective action before the disturbances affect the process. Figure 6(a) shows the basic structure of the feedforward control.
For effective use of feedforward control, the disturbance must be measured on-line and a reasonably accurate process model is required. This means that feedforward control cannot compensate for unmeasured disturbances and any errors in the process model. Thus, it would be expected that addition of a feedback control may correct for unmeasurable disturbances and modeling errors. This is often referred to as feedback trim. A combined feedforward-feedback control has become an accepted practice in the chemical process industries since the 1960s. 51 This approach is illustrated by Figure 6(b). In polymer extrusion, this advanced control method has received relatively limited attention and there are only a very few studies reported.
A simple feedforward controller can be designed to provide steady-state feedforward control action through the use of a physical model. However, this feedforward design neglects process dynamics, i.e. how fast the controlled variable responds to changes in the load and manipulated variables. Thus, it is often necessary to include a dynamic compensation in the feedforward controller. The most direct method of designing the feedforward dynamic compensator is illustrated in Figure 7. G T represents the disturbance transmitter, G FF is the feedforward controller, G L is the load disturbance, G v is the valve, G P is the process, G m is the output transmitter and G C is the feedback controller.
Using block diagram algebra, the output Y(s) is given by equation (13). For the disturbance rejection [L(s) ≠ 0], Y(s) = 0 is desirable. Solving equation (13) for G FF results in equation (14). The feedforward controller can also be simplified to a lead–lag unit, as shown in equation (15), which can be implemented using a digital computer. In equation (15), a physical model can be used to compute K F. In practice K F, τp and τL can be tuned to improve the performance of the feedforward controller.
(13)
(14)
(15)
A feedforward control strategy was proposed to control melt pressure and temperature. 22 In this proposed scheme, the time-series model was suggested for the prediction of die conditions. A transfer-function model relating die conditions to the speed change was suggested in the calculation of speed correction to prevent any error occurring. Later, this control strategy was implemented successfully on an extrusion line. 52
Recently, the effectiveness of feedforward compensation in PI and Dahlin controllers was investigated for the multivariable control of extrudate thickness and melt pressure. 18 The feedforward compensation was designed using transfer-function models and was added to the thickness-control loop to eliminate the effect of melt pressure control. Experimental results shown in Figure 8 indicate that the thickness variation caused by screw-speed change was reduced.
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Engineering Perspectives in Biotechnology
Silvia Ochoa , in Comprehensive Biotechnology (Third Edition), 2019
2.41.3.3.1 Closed Loop Strategies Based on a Feedback Control Law
Closed loop strategies based on a feedback control law calculate the feeding flow rate required in order to keep a certain process variable in a fixed predefined value (regulatory control) or trajectory (servo control). Fig. 10 shows a general block diagram representing the closed loop strategies based on a feedback control law.
In a feedback control system the controller takes the decision on how to act on the final control element by considering the error between the actual value of the controlled variable and its expected set point value, according to the control algorithm or control law used. The loop is closed once the decision on the final control element is implemented. Simple feedback control loops (as the one shown in Fig. 10) are the most frequently used in the chemical process industries, basically due to its simplicity and low implementation costs. In spite of its simplicity, the feedback control strategy performs well in a wide variety of cases. The main inconvenient for implementing feedback control loops for an optimal operation in fed-batch fermenters is the scarce availability of on-line measurements (sensor-transmitter) for important state variables such as: glucose, biomass and metabolites concentrations, requiring the use of soft sensors.
Any feedback control law calculates the feeding flow rate as a function of the error between the controlled variable and its set point value. The control law for a typical PID algorithm is shown in Eq. (16). Other feedback control algorithms which have been successfully tested in real fed-batch fermentation cases include: adaptive control, model predictive control and fuzzy control. 18 Adaptive control algorithms are characterized because they continuously adapt the control law in order to compensate for the presence of uncertainties (e.g., mismatch between the real process and its model). The three main classes of adaptive control strategies are: gain scheduling, model reference adaptive control, and self-tuning control. The simplest adaptive strategy to implement is gain scheduling, where a typical feedback (PI or PID) control law is used but the controller gain ( in Eq. 16) is adapted following some pre-programmed tuning rules, commonly based on information from previous experiments. On the other hand, Model Predictive Control (MPC) is an optimization-based control technique that uses a model for predicting the future response of the process variables when different profiles are applied for the manipulated variables (feeding flow rate in the fed-batch fermentation case). The model is updated at each sample time with actual information of output variables and/or predicted state variables. Optimization of a cost function over a defined time horizon is used for defining the actual feeding flow rate to be applied. However, only the first step of such optimal solution is actually implemented. Then, the optimization procedure is repeated for the next sample time, obtaining a new set of model predictions. MPC is a robust control strategy but involves calculations which are time-consuming. In comparison with all other control strategies mentioned in this article, MPC has the highest hardware and software requirements. Successful implementation of MPC strongly relies upon the process model used and the formulation and solution of the optimization problem. The final feedback based closed loop strategy for controlling Fed-batch fermentations considered is Fuzzy control. Fuzzy control has been widely applied in the control of fermentation processes and related bioprocesses. 26 Fuzzy control is a control method that uses fuzzy logic (or fuzzy reasoning) in order to establish a control system based upon expert knowledge, such as that of a skilled operator, accumulated over years of experience. With this approach, expert knowledge is translated into a series of rules for controlling the process.
Before ending this section, a brief mention must be done regarding the so-called pH-stat and DO-stat feeding strategies. These two strategies are usually cataloged as indirect feedback control schemes aiming to prevent the accumulation of substrate to inhibitory levels. 14 Both strategies work as an on-off controller, where substrate is only fed when pH (for the pH-stat case) or the dissolved oxygen concentration (for the DO-stat) rise over a pre-defined set-point value. The pH-stat is based on the fact that upon carbon substrate depletion, cells release ammonium ions increasing the pH in the medium. Similarly, the DO-stat is based on the fact that when a certain key substrate is exhausted, oxygen consumption suddenly stops giving rise to a quick increase in dissolved oxygen concentration in the medium. 14 Although very simple to implement, these indirect feedback control strategies are not suitable at all, because the on-off controller will provide an intermittent feeding (bang–bang policy) that would lead the process to an oscillatory behavior, exposing the cells to unnecessary starving and stressful conditions. 20
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Process Control
D.G. Hulbert , C. Aldrich , in Gold Ore Processing (Second Edition), 2016
3.4 Proportional-Integral Control
Proportional-integral (PI) feedback control loops are commonly used on gold plants. Because of the noisy measurements generally present on these plants, derivative action tends to cause too much actuator action and is rarely used.
Many PI controllers are poorly tuned, and few people know or remember how to tune them properly. A "fiddle-and-test" method of tuning often results in too low a gain and relatively too much integral action. This occurs because when a control loop appears to be too active, the operator's natural inclination is to turn things down. This is effective in respect of the gain setting, but not the reset setting, which gives more integral action the lower it is. The integral action is generally not immediate acting, so adverse effects are not seen immediately when it is set too high and the gain is made correspondingly small.
PI control is not always very successful. Problems can occur when there are adverse dynamic characteristics. A more advanced controller can handle the dynamics of long time lags introduced by solid feed belts more appropriately. Multivariable interactions between plant inputs and outputs are often more appropriately dealt with by advanced control.
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GAS CHROMATOGRAPHY | Petrochemical Applications
E.R. Adlard , in Encyclopedia of Analytical Science (Second Edition), 2005
Plant Control
GC can be used to give automatic feedback control of plant and operating systems such as pipelines but is less frequently used on a direct basis than might be expected. Most plant control by GC is effected by manually transporting a sample from the plant to a standard laboratory instrument housed in a safe area within the refinery. The main reason is that it takes a considerable time for any altered conditions such as temperature to have an effect on large plants. The second reason, which follows from the first, is that if rapid analysis near the plant is unnecessary then there is little point in using a plant instrument and exposing it to a hostile environment. Another factor is the difficulty of sampling on a continuous basis directly from a plant stream. It is extremely difficult to ensure a completely clean sample and the presence of a small rust particle is sufficient to destroy the highly polished surfaces of liquid sampling valves. Fluid logic devices have been employed experimentally to overcome this problem but they do not seem to have achieved wide acceptance.
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Small Hydro
Arun Kumar , in Reference Module in Earth Systems and Environmental Sciences, 2021
9.3 Criteria for selection of hydraulic turbine
Criteria for selection of hydraulic turbine based on specific speed as per Indian standard IS:12837 is given in Table 5.
Table 5. Criteria for selection of hydraulic turbine based on specific speed (Indian standard IS: 12837, 1989).
Type of machine | Head variation % of rated head | Load variation % of rated output | Specific speed (m-mhp) | Peak efficiency in % |
---|---|---|---|---|
Pelton | 120–80 | 50–100 | 15–065 | 90 |
Francis | 125–65 | 50–100 | 60–400 | 93 |
Deriaz | 125–65 | 50–100 | 200–400 | 92 |
Kaplan | 125–65 | 40–100 | 300–800 | 92 |
Propeller | 110–90 | 90–100 | 300–800 | 92 |
Bulb | 125–65 | 40–100 | 600–1200 | 92 |
Selection of types of turbines can be made by using the operating range graph given in International Standard IEC 61116 (1992) for small hydropower installation (Fig. 21).
In shp scheme cross flow turbine, relatively of lesser efficiency, are also used for small capacity units since are simple in construction and maintenance. A cross flow turbine is an impulse type turbine with partial air admission. Performance characteristics of this turbine are similar to a Pelton turbine, and consist of a flat efficiency curve over a wide range of flow and head conditions and thus can be used for a wide range of head.
9.3.1 Speed governors
Governor control system for hydro turbines is a feedback control system sensing the speed and power of the generating unit and takes control action for operating the discharge/load controlling devices by adjusting the guide vanes/wicket gates, in accordance with the deviation of actual set point from the reference point. There are several types of governors starting from purely mechanical to mechanical-hydraulic to electrical-hydraulic and mechanical-electrical.
9.3.2 Speed increasers
In many instances, the rotating speed of turbines are lower than the rotating speed of standard generators. For cost effective solution a speed increaser in form of gear box or bevel gear or belt is provided between turbine and generator. With this arrangement the alignment line of turbine and generator are shifted to match with speed increaser. However there shall be power losses in this arrangement.
In most of the low head SHP schemes, hydraulic turbines have lower speed than 500 rpm, thus requiring a speed increaser to match the standard 600–1000 rpm of generator to provide more economical solution compared to fabrication of customized generator.
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Feedback Control of Dynamic Systems 7th Edition Chapter 3 Solutions
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