CHAPTER for load frequency control have been proposed
Load-frequency control is important in electrical power System design and operation. The loading in a Power system is never constant. To ensure the quality of the power supply, it is necessary to design a load frequency Control system which deals with the control of loading of the generators depending on the frequency. In the context of a multi-area power system, each area has to be equipped with one local load-frequency Controller. The purpose of the local controller is to maintain the area’s frequency as well as the power flowing in/out of the area. There has been continuing interest in designing these load-frequency controllers for the past 20 years. Many control strategies for load frequency control have been proposed since the 1970s. if the Problems have been considered, a suitable set of local load-frequency controllers is normally obtained via tedious field testing and tuning.
A robust decentralized control concept 13-16 is utilized for the design of a robust decentralized load-frequency Controller (RDLFC). The RDLFC is thus made-up of N local load-frequency controllers (i.e. no measurements from other areas are required). In our new controller-design approach, the RDLFC is realized by solving N decoupled Riccati equations. The originally derived N Riccati equations are interlinked, and are separated using our proposed technique. The overall power system, using the RDLFC, will be asymptotically stable for all admissible parametric uncertainties.
The problem of controlling the real power output of generating units in response to changes in system frequency and tie-line power interchange within specified limits is known as load frequency control (LFC) 1. Due to the increased complexity of modern power systems, advanced control methods were proposed in LFC, e.g., optimal control 2–4; variable structure control 5; adaptive and self-tuning control 6, 7; intelligent control 8, 9; and robust control 10–14. Recently, LFC under new deregulation market 16, 17, LFC with communication delay 18, and LFC with new energy systems 19, 20 received much attention. See 21 and 22 for a complete review of recent philosophies in AGC. Improved performance might be expected from the advanced control methods, however, these methods require either information on the system states or an efficient online identifier thus may be difficult to apply in practice. A unified method to design and tune PID load frequency controller for power systems with non-reheat, reheat and hydro turbines will be discussed.
1.2 Literature survey
Traditional load frequency control (LFC) employs a dedicated communication channel to transmit measurement and control signals, while the LFC under deregulated environment, such as bilateral contract between generation companies (Gencos) and distribution companies (Discos) for the provision of load following and third-party LFC service, tends to use open communication networks 1–4. Although the constant delay introduced by the dedicated communication channel is normally ignored, the breakdown of the communication channel itself could be converted to time delays 9. With the introduction of open communication channel, constant and time-varying delays will be introduced in the LFC scheme due to packet dropout and disordering 4–6, updating of area control error (ACE) signal 7, and faults of communication channels 9. The delays induced by the communication channels will degrade the dynamic performance and may even cause instability of an LFC scheme 4, 8–10. The instability of the LFC scheme means that the ACE and frequency deviation will deviate far away from zero, which makes the control areas impossibly comply with the control performance standards, CPS1 and CPS2, adopted by North American Electric Reliability Council (NERC) 22.
Analysis/synthesis of the LFC in the presence of communication delays has been addressed, such as robust controllers based on linear matrix inequality (LMI) technique 11 and robust decentralized PI-type LFC based on H? theory 12, 13. Robustness of those controllers against time delays was normally verified by simulation studies only. Recently, authors of this paper 9 investigated the delay-dependent stability of the LFC via calculating delay margins and investigating the relationship between the delay margins and controller parameters. The investigation of 9 focuses on traditional LFC schemes, while the LFC schemes in deregulated environment, which more commonly encounters time delays due to the usage of open communication networks, as well as how to use the delay margins as a performance index to guide the design and operation of the LFC controller, have not been investigated. Moreover, the stability criterion used in 9 is also deserved to be improved from the following two aspects. The first one is that it contains too many decision variables such that the total computation time increases sharply with the increasing of the problem dimension, which results in the difficulty of applying the proposed method to deal with multi-area LFC schemes. Secondly, the conservativeness of the stability criterion used cannot reveal the coupling dynamic between different controls areas which results that the multiple delays of different control areas have been found be independent, i.e., looking like a rectangle or cube in 9.
On the other hand, the practical LFC controllers are operated in discrete mode as the ACE signals of the LFC scheme are usually updated in a period from 2 s–4 s 7. It is found in 14 that the optimal integral controller gains designed in the continuous mode cannot be applied directly in discrete mode, while the simulation studies in 7 revealed that a relatively large sampling period around 20 s can still result in satisfactory results for some special cases. To the authors’ best knowledge, how to select the sampling period and determine the acceptable upper bound of the sampling period (UBSP) for a given LFC controller have not been investigated analytically yet. In fact, a continuous controller with an input delay can be used to model such sample-based controller, in which the input delay is bounded by the sampling period 15. Based on this understanding, the delay margin can be used as the UBSP to guide the choice of the sampling period for an LFC controller.
1.3 PROBLEM FORMULATION
Load frequency control (LFC) has been effectively used in electrical power systems for many years to maintain the system frequency and the power exchange between areas at desired scheduled values 1. In the past four decades, several LFC approaches have been proposed to guarantee the stability, optimality and robustness of power systems 2–4. One of the most applicable approaches is a proportional–integral–derivative (PID) control 5, 6. Besides PIDs, the other modern and extended approaches in load frequency control have been presented in the literature. They are reported as fuzzy logic 7; adaptive control 8; neural network 9; active disturbance rejection control 10, 11; linear matrix inequality (LMI) approach 12, 13; two-level optimal robust control 14; robust H? control 2; and so on.
A LFC system employs dedicated communication channels for transmission of measurements from remote terminal units (RTUs) to the control center. Due to the introduction of open communication channel, constant and time-varying delays have been introduced in the LFC problems 15–17. It is well known that such delays could significantly affect the performance of the control system; for example, large time delays may cause instability of the system, i.e. the frequency deviation will deviate far away from zero 18.
Recently, several researchers have considered time delays in design of load frequency controllers in deregulation and market environments 19. Among those studies, some of them did not consider the delay during the design procedure, e.g. see 20, and some others proposed a robust controller in which the time delay is dealt as a part of model uncertainties 21. Further to these results, a graphical method to compute the stability region in the parameter space of a PI controller for an LFC problem in a single-area power system with time delay is proposed in 22. In 23, the delay-dependent stability of the LFC scheme considering constant and time-varying delays is investigated. The impact of time delays on stability of PI-based LFC systems is presented in 24. In 25, a delay-distribution-dependent H? LFC scheme is designed for power systems with probabilistic communication delays.
1.4 Objective of the thesis
The objective of this thesis is to design robust predictive load frequency control for power systems with uncertain parameters and time delays in communication networks. The communication of delay restrictions between different control region and the relationship between delay restrictions and control gains are studied in detail. Moreover, usage of delay restrictions as a new performance index to steer controller design is discussed, as well as regulation of the controller for a trade-off between delay tolerance and dynamic reaction, an opting the upper bound of sampling period of a discrete realization of the controller and the upper bound of the fault counter of communication channel. Simulation studies are carried out all equipped with PID-type controllers for single area and three areas LFC schemes, to verify the effectiveness of the proposed method. The simulations were performed in the environment of MATLAB/SIMULINK.
1.5 Organization of the thesis
The report is organized as follows:
Chapter 1: In this Chapter, the concept of LFC, introduction and importance are discussed.
Chapter 2: In this Chapter, concept of load frequency control, different types of frequency control methods and its significance are discussed.
Chapter 3: In this Chapter, the concept of single area and multi area load frequency control is explained.
Chapter 4: In this Chapter, the concept of an integral and differentiator load frequency control are discussed.
Chapter 5: In this Chapter, the simulation results are analyzed by simulating the proposed model in the environment of MATLAB/SIMULINK.
Chapter 6: This Chapter gives conclusions and scope for further work in the area of multi area concept.
LOAD FREQUENCY CONTROL
This chapter in brief portrays electrical power system networks and their use in transferring electrical energy from its source of generation to the centers of consumer load. The state-space approach determines an optimal controller using linear optimal control theory. The parameters calculated by the optimal controller are strongly dependent, on the coefficients of the quadratic performance index. The main problem with the linear optimal control approach is the fact that no practical guide lines exist on the selection of the coefficients of the performance index.
The load Frequency Control problem of an interconnected power system is a well defined problem. The system is divided into groups of generators which are interconnected by tie–lines. Each group of generators is called an area, and each area must be able to meet its own load changes and any import or export targets set by the controllers in advance. Each area has its own response characteristics which relate the area frequency and total generation for load changes on the specific area. This curve is the regulation curve of an area and represents the area gain (in MW /Hz). The area gain or regulation is a direct measure of the effects of all the governors on the prime movers within the area, and plays an important part in the steady-state and dynamic performance of the system.
The normal procedure used in the design of L.F.C. functions is to construct a linear system model with fixed parameters. This is obtained by linearizing the system around an operating point; however, this approach is not strictly correct as the system response characteristics tend to be non-linear. Power system parameters, are a function of the operating point. Hence, as the operating conditions change, the calculated operating point will no longer be optimal. To keep the system performance near to optimum, a way of tracking the operating conditions of the system is required which will enable the continuous updating of the system parameters. The control signal can then be computed based on an optimal approach using the newly updated parameters.
Load changes m the system are random in magnitude and time. To control the transient response effectively and to take into account the sensitivity problem of L.F.C. in interconnected power systems the use of self-tuning controllers is considered. It seems more appropriate to consider the system as a stochastic system and to improve control performance, design an adaptive stochastic controller than the fixed schemes used previously.
2.2 CONTROL OBJECTIVES
The main objectives of the Automatic Generation Control (A.G.C.) function are
To match the generated power to the load demanded by the consumer.
Adjust the system frequency to the reference set frequency.
Control the power export to other areas in the interconnected case to keep to scheduled interchange agreements.
To control each individual area to share the generation in the most economic way.
The first three objectives are under the control of the L.F.C. and the last involves the use of economic dispatch. The system dynamics must be considered and the complete regulator built up from this start position. Generally there have been two main types of solutions proposed to the problem of L.F.C. the more conventional solutions which have been used in practice for many years and the more modern ones have only been proposed without any practical implementations.
As mentioned earlier the conventional controllers use the well known and understood proportional plus integral type control schemes. These are based on the theory associated with servomechanisms which was developed in the 1950’s. These schemes use what is normally termed’ unconstrained economic dispatch, that is the control is permitted to carry out any control action it deems necessary without any regard to the economics of this control action.
2.3 DEFINITION OF GOOD CONTROL
The future developments for A.G.C. are based on the definition of how a good L.F.C. function and economic dispatch are capable of acting. This definition is not a well defined quantity, but there are a number of properties which must be considered as compulsory for the completion of a good frequency controller.
It should be robust m its communication with other control function and numerically constant, it have got to avoid data complexity, and it must be decentralized based on area basis.
It has got to take the system constriction into account, and place extra emphasis on, system security, power rate limits and daily dynamic constraints.
It must be able to show some economic improvement in operation, seen directly through a suitable interface with economic dispatch, seen indirectly by the reduction of the spinning reserve and regulation plant margins.
It must be able to cope with system transients, by having large stability margins, by issuing smooth control commands which improves the economy, by decreasing the wear and tear on the generation units, the interface between L.F.C. and E.D. must be correctly solved.
2.4 EARLY LOAD FREQUENCY CONTROL
The L.F .C. problem has been a major area for research and investigation for both Power System Engineers and academics for several years. However, many of the proposed solution have relied on poor modeling and simulation techniques along with reduced order system models leading to uncharacteristic responses of the plant. The control problem has become more significant as the size of the systems has grown and the greater extents to which systems have become interconnected. The growth in such systems has required larger and more powerful computers to both monitor and in many cases control the system as a whole.
2.5 CENTRALIZED LOAD FREQUENCY CONTROL
Much of the early research started using the classical control theory approach but further research continued due to this relatively simplistic approach in using simple modeling techniques and only two interconnected areas. Many investigations used models which represented non reheated steam plants and did not represent any –nonlinearities such as generator rate constraints, limitations on generator output, or the difference between regulating and non-regulating plant. Some of the ideas used involved the use of large central controllers with system variables being sent to the main control from many out laying stations. This was somewhat impractical due to the large amounts of data which need processing and the in some cases the use of system variables which were not readily measurable.
2.6 NON-CENTRALIZED CONTROL
The problems associated with centralized control schemes lead to the investigation of decentralized control of large dynamic systems. Many of these solutions were heuristic in design but were successful and worked well in practice. The emphasis of the control schemes became involved not only with the frequency deviation but also with the interaction with other control functions. These control functions tended to be the longer time scale operations to enhance the systems economic operation and security of operation.
2.7 QUALITY OF CONTROL ACTION
The quality of system frequency control depends on the quality of service that the utility is expected or required to provide for its customers. This quality of service is most clearly seen in the effect that it has on electric clocks, which will deviate from the actual time if the supply frequency deviates from the actual frequency. To keep the frequency error at zero would be clearly almost impossible, but a zero average time error over a 24 hour period may be acceptable. The quality of control must also take into account the maximum and minimum frequency bounds that a utility would require to stay within, whether it be a legally enforced limit or one defined by the utility.The L.F .C. schemes were originally designed to operate with three separate modes of operation. These were defined to be basic control, emergency action and corrective control.
2.7.1 Basic Control
This mode of basic control was designed to maintain the frequency error within a specified error boundary and any power exchanges over inter area tie-lines to scheduled values. The control was also required to maintain the distribution of generated power between all the generators nominated as regulatory units in an area constant. These generators which are defined to be available for regulation are allocated by the economic dispatch function and hence, the interface between the frequency control and the economic function must be well defined. The L.F.C. objective is to control the generator output by altering the set points of the regulating units in response to changes in system frequency and tie-line interchange values. This power balance is achieved by using units selected for regulation and tracking the set points of the turbine governor. This action is in addition to their primary local control loops, which take action when the locally measured frequency deviates from the reference value. Thus L.F.C. is sometime termed secondary control action.
Loading among the regulating units is determined using economic considerations. The use of static and dynamic programs to calculate the economic minimum operating point of each unit gives each turbine generator a base point value and a future output target. This optimization is termed economic dispatch, and has the task of supplying an area consumer load in the most economic manner possible, aiming to find the minimum operating costs for a given set of constraints. The operation and use of an economic control function is often thought of as a tertiary control, as it is one layer above the L.F .C. and governor. Due to the time scales involved, L.F.C. is regarded as a dynamic operation where as the optimization of the generating cost is regarded as a static operation. The economic dispatching and L.F.C. are both controlled from a longer range control function termed Unit Commitment. This control function is designed to calculate the time of start-up and shut-down of all the plant synchronized to the system and ensure there is sufficient generation available to fully supply the consumer demand. These three control functions are operated very closely and use the same system data; hence integration of these operations is essential for the smooth running of the overall control function.
2.7.2 Emergency Mode
The emergency mode of L.F .C is entered if there is a large disturbance on the system and the question of system security may become important. In this case the surrounding areas are required to give support to the problem area, as well as the area itself attempting to support the demand. Initially the support from surrounding areas may be able to assist the troubled area, by altering the tie-line interchange, but eventually spinning reserve may need to be allocated or the re-scheduling of the areas own generators will be required. In extreme circumstances load shedding may be required.
2.7.3 Corrective Control Action
The third standard control mode is that of corrective action. This control action is required during periods of inadvertent power interchange between neighboring areas, due to non-scheduled power exports or imports of power. The corrective action required is obtained by changing the reference set point of the tie-line interchange and the area frequency. The value of the offset depends on the time predicted for the corrective action and the value of the inadvertent power interchange. The L.F.C. and economic dispatch still ·operate in their normal modes as previously, but their calculations are based on a new set of reference points. The aim of this corrective action is to return the quantities to their scheduled values.
2.7.4 Decentralized Control
Generally the L.F.C. applied to many systems is of the decentralized type, which is carried out in each individual area. Some techniques use the tie-line bias technique, where the control is biased on the tie-line flows rather than the frequency error. This enables the controllers to operate separately in each area and still interact with the other interconnected areas, keeping the economic benefits of the single area calculation. Early developments of the classical control theory showed their ability to solve the L.F .C. problem, and further developments allowed refinements from the earlier work. Work on the control of transient oscillations and reaction to small disturbances, of the L.F.C. and its sensitivity to system changes enabled a more practical solution to the problem to be achieved. The integration of L.F .C. and economic dispatch was also investigated.
2.7.5 Operating Characteristics
The early L.F .C. proposals all had two common themes. Firstly the only mode which was covered in any great detail was the normal operating mode for the steady-state operation. The problems arising from the other modes mentioned earlier have not been provided with any clear solutions. The second common theme is the design of centralized controllers for the whole of an interconnected network. Several papers have made attempts to use a single system model to simulate various networks. This approach has been used even for the inter-area control problem and also investigation of decentralized control algorithms where separate system models would seem to be more appropriate. A few attempts have been made to apply art area dynamic model to the process of L.F.C. However, the use of a single overall system model to the inter area problem has lead to problems in the computational requirement. The number of variables required for the multiple area calculations is large, and often the data must be gathered from a wide range of transducers and geographic areas. This mass of data must then be solved using a single pass type calculation in real time, which is not possible due to the constraints placed on the data gathering system by the external plant. This problems lead to the development of decentralized design techniques. These techniques use the idea of dividing the multiple area power system into several decomposed subsystems. Each subsystem may then be controlled using a decentralized L.F .G. technique. This system requires the use of decentralized model and area controller.
The practical use of the centralized or decentralized controllers does not appear to be possible at the present time due to the implementation problems of optimization and system data collection. The theoretical problems encountered are due to the fact that optimal control theory has difficulties coping with the large size and complexity of power systems as well as the data structures required for a well programmed L.F.C. scheme.
2.8 IMPLEMENTATION OF FREQUENCY CONTROL
The L.F.C. function is required to carry out the calculation on which the control correction signals are based for the future alteration of the power output levels, of specified generation units on the power system. Based on the mode of operation of the L.F.C. computation set by the dispatch operator, the program will operate to control a selection of system frequency, interchange powers, system time and interchange energy as closely as possible to scheduled values. The main computation for the L.F.C. function is the calculation of the Area Control- Error, (A.C.E.). This calculation is normally carried out on the known regulating characteristics of the power system together with deviations in system frequency and interchange power levels. These deviations are computed on the basis of telemeter values and target values input from the system. The basic objective of the L.F.C. control function is to control the system frequency within the statutory bounds, in the region of minimum operating cost, and hence there is no requirement to correct for random fluctuations in load provided that the system remains within the specified bounds. Unnecessary control action is undesirable due to the continuous operation and unnecessary use of the speed-changing (governor) equipment of is each generation unit. So that, the control action not excessive, the A.C.E. is filtered over several cycles of telemeter data. It is now recognized .that improved system performance can be achieved, particularly in the case of slower acting thermal units by varying the gains adaptively to suit different classes of disturbance on the system, and different system gains in .different operating points.
2.9 CALCULATION OF THE AREA CONTROL ERROR
The task of the Load Frequency Control is to regulate the power output of the electrical generators within a given area, in response to changes in the system frequency, tie-line loading or the relation of these to each other. This will enable the scheduled system frequency to be maintained and/or the agreed power interchange with other areas to be kept within the pre.-defined limits.
So that the control of both, the system frequency and the tie-line power interchange, may be controlled as required, the control action is designed to alter the set point of each of the generators regulating on the system. Further corrective action may be required if the frequency, or clock error deviates from a certain band. Each change of the generator set points is a step change in the system power, which itself may introduce an unwanted frequency transient before the control action has chance to carry out its intended action. So that these transients may be avoided, the generators units must be ramped to their new required outputs over a reasonable time interval, and be constrained by their ramp limitations. The automatic computer control schemes that have been proposed to allow the change in generator set point to control the system frequency error and the tie-line power exchange to be smoothly controlled.
To enable the control scheme to operate as .required, the control is based on the calculation of the A.C.E. The A.C.E. is calculated based on the equation
A.C.E. =? Ptie+ K ?f (2.1)
? Ptie is the error between the scheduled and actual tie-line power interchange, in MW. ?f is the error between the reference frequency and the actual system frequency. K is the system stiffness, or gain (MW /Hz).
The control scheme must calculate an A.C.E. before it is due to alter the generator power set points for each unit. At each time interval, the A.C.E. represents the change in generation that must be allocated to various units by the alteration of the individual set points of the regulating generators, taking into account the dynamic limitations of the individual units.
2.10 LOAD FREQUENCY CONTROL
The purpose of load frequency control loop is to achieve power balance between generation and demand by load tracking using speed governors. Under steady state condition, the generator supplies power to the load. The system will be under equilibrium i.e., the generator delivers constant power and the turbine provides constant accelerating torque. The equilibrium will be affected when the system is subjected to the load change.
When there is an increase in load, the turbine generator set will decelerate. This deceleration leads to the change in frequency i.e. decrease in frequency. Similarly, during removal of load, the turbine generator set will accelerate. This leads to the incremental increase in frequency. Either increase in load or decrease in load, the frequency is subjected to change. This makes frequency to be an indirect indicator to do the power balance.
Basically two types of inner control loops are there in LFC as shown in Fig.2.1. They are primary control and secondary control loop.
The primary control loop adjusts the turbine power based on frequency change using control valve. This loop works much faster. The response time will be in seconds. The loop does only coarse on frequency.
The secondary control loop eliminates the small error in frequency. It works after the control action of primary loop completes. It does the control action through speed changer until the frequency error becomes zero. It also controls the net power interchange between the power pool members.
Fig.2.1: Schematic diagram of load frequency control
In this Chapter, the concept of load frequency control, objectives of the control methods and different type of control classifications are presented.
MULTI-AREA POWER SYSTEM MODEL
In a two area interconnected power system, where the two areas are connected through tie lines, the control area are supplied by each area and the power flow is allowed by the tie lines among the areas. Whereas, the output frequencies of all the areas are affected due to a small change in load in any of the areas so as the tie line power flow are affected. So the transient situation information’s of all other areas are needed by the control system of each area to restore the pre defined values of tie line powers and area frequency. Each output frequency finds the information about its own area and the tie line power deviation finds the information about the other areas. For example in a two area power system, the information can be written as Bi?fi+?Ptie. B = frequency bias, f = predefined frequency and Ptie is the power in tie line. This is the Area Control Error (ACE) which is the input to the controller.
Thus the load frequency control of a multi area power system generally incorporates proper control system, by which the area frequencies could brought back to its predefined value or very nearer to its predefined value so as the tie line power, when the is sudden change in load occurs.
3.2 INTERCONNECTED POWER SYSTEMS
According to practical point of view, the load frequency control problem of interconnected power system is much more important than the isolated (single area) power systems. Whereas the theory and knowledge of an isolated power system is equally important for understanding the overall view of interconnected power system. Generally now days all power systems are tied with their neighboring areas and the Load Frequency Control Problem become a joint undertaking. Some basic operating principle of an interconnected power system is written below:
The loads should strive to be carried by their own control areas under normal operating conditions, except the scheduled portion of the loads of other members, as mutually agreed upon.
Each area must have to agree upon adopting, regulating, control strategies and equipment which are beneficial for both normal and abnormal conditions.
3.2.1 Advantages of Interconnection
Effect of size: This one is one of the most important advantages for the whole interconnected power system. When a load block is added, at the initial time, the required energy is temporarily borrowed from the system kinetic energy. Generally the availability of energy is more for larger systems. So there is comparatively less static frequency drop. Whereas, for a single area power system the frequency drop may be a bit higher for same amount in load change.
Need of reduced reserve capacity: As the peak demands do not have any certain time, they may occur at any random time of the day in many areas, for a large power system the ratio between load peak and load average is smaller as compared to smaller systems. Therefore it is obvious that all interconnected power system areas may benefit from a decreased need of capacity reserved by the scheduled arrangement of interchanging energy.
3.3 REASONS FOR THE LIMITS ON FREQUENCY
The speed of the ac motors depends on the frequency of the supply power. There are situations where speed consistency is expected to be of high order.
The electric clocks are driven by the synchronous motors. The accuracy of the clocks is not only dependent on the frequency but also is an integral of the frequency error.
If the frequency is 50 c/s and the system frequency falls below 47.5 c/s or goes up above 52.5 c/s, then the turbine blades will get damaged and it is required to prevent the stalling of the generator.
The frequency below the system frequency operation of the power transformer is not desirable. For constant system voltage if the frequency is below the desired level then the normal flux in the core increases. This sustained under frequency operation of the power transformer results in low efficiency and over-heating of the transformer windings.
The most serious effect of subnormal frequency operation is observed in the case of Thermal Power Plants. Due to the subnormal frequency operation the blast of the ID and FD fans in the power stations get reduced and thereby reduce the generation power in the thermal plants. This phenomenon has got a cumulative effect and in turn is able to make complete shutdown of the power plant if proper steps of load shedding technique is not engaged. It is pertinent to mention that, in load shedding technique a sizable chunk of load from the power system is disconnected from the generating units so as to restore the frequency to the desired level.
3.4 MATHEMATICAL MODELLING OF VARIOUS COMPONENTS
If the system is connected to a number of different loads in a power system then the system frequency and speed change with the governor characteristics as the load changes. If it is not required to keep the frequency constant in a system then the operator is not required to change the setting of the generator. But if constant frequency is required the operator can adjust the speed of the turbine by changing the governor characteristic as and when required. If a change in load is taken care by two generating stations running at parallel then the complexity of the system increases. The possibility of sharing the load by two machines is as follow:
Suppose there are two generating stations that are connected to each other by tie line. If the change of load is either at A or at B and the generation of A is alone asked to regulate so as to have constant frequency then this kind of regulation is called Flat Frequency Regulation.
The other possibility of sharing the load the load is that both A and B would regulate their generations to maintain the constant frequency. This is called parallel frequency regulation.
The other possibility is that the change of the frequency of an exacting area is taken care of by means of the generator of that area, in this manner the tie-line loading remains the same. This method is known as flat tie-line loading control.
In Selective Frequency control each system in a group is takes care of the load changes on its own system and does not aid the other systems in the group for changes outside its own limits.
In Tie-line Load-bias control all the power systems in the interconnection aid in regulating frequency regardless of where the frequency change originates. The equipment consists of a master load frequency controller and a tie line recorder measuring the power input on the tie as for the selective frequency control.
The error signal i.e. ?f and ?Ptie are amplified, mixed and transformed to real power command signal ?PV which is sent to the prime mover to call for an increase in the torque. The prime mover shall bring about a change in the generator output by an amount ?PG which will change the values of ?f and ?Ptie within the specified tolerance. The first step to the analysis of the control system is the mathematical modeling of the system’s various components and control system techniques.
3.5 MATHEMATICAL MODELLING OF GENERATOR
Applying the swing equation of a synchronous machine to small perturbation, we have:
2H?d2??dt2=?Pm-?PeOr in terms of small deviation in speed
Fig.3.1: Mathematical modeling block diagram for generator
3.6 MATHEMATICAL MODELLING OF LOAD
The load on the power system consists of a variety of electrical drives. The equipments used for lighting purposes are basically resistive in nature and the rotating devices are basically a composite of the resistive and inductive components. The speed-load characteristic of the composite load is given by:
Where ?PL is the non-frequency- sensitive load change,
D?? is the frequency sensitive load change, D is expressed as percent change in load by percent change in frequency.
Fig.3.2: Mathematical modeling Block Diagram for Load
3.7 MATHEMATICAL MODELLING FOR PRIME MOVER
The source of power generation is commonly known as the prime mover. It may be hydraulic turbines at waterfalls, steam turbines whose energy comes from burning of the coal, gas and other fuels. The model for the turbine relates the changes in mechanical power output ?Pm to the changes in the steam valve position ?PV.
Where ?t, the turbine constant is, in the range of 0.2 to 2.0 seconds.
3.8 MATHEMATICAL MODELLING FOR GOVERNOR
Fig.3.3: Graphical representation of speed regulation by governor
When the electrical load is unexpectedly increased, subsequently the electrical power exceeds the mechanical power input. As a result of this the shortage of power in the load side is extracted from the rotating turbine energy. Due to this cause the turbine kinetic energy i.e. the energy stored by the machine is reduced and the governor sends a signal to supply more volumes of water or steam or gas adds to the prime-mover speed, so as to compensate speed insufficiency.
The curve slope represents the speed regulation R. The typical speed regulation of Governors is 5 to 6 percentage between no load and full load.
Or in s- domain
The command ?Pg is transformed through hydraulic amplifier to the steam valve position command ?PV. We assume a linear relationship and consider simple time constant ?g we have the following s-domain relation:
Combining all the block diagrams from earlier block diagrams for a single system we get the following:
Fig.3.4: Mathematical Modeling of Block Diagram of single system consisting of Generator, Load, Prime Mover and Governor
3.9 AUTOMATIC GENERATION CONTROL
Owing to the importance of the distribution of the electrical power, the organizations are responsible for providing it with great reliability, availability and efficiency. The designed and operated power system must deal with changes in the load, with system disturbances; provide acceptable high level of power quality and maintaining both voltage and frequency with intolerance restrictions. Due to any disturbance, the actual working position of a power system changes as of its pre-specified position. Consequently the difference occurs about the operating position such as actual system frequency, exchange of scheduled power to the added areas which is undesirable. Problems have been tackled by the various researchers in diverse moment in time through the regulator of AGC, the controller design of excitation and performance in control relating to variation/uncertainties of parameter and different t load characteristics.
If the load on the system is increased suddenly then the speed of the turbine drops earlier than the governor can regulate the input of the steam to the fresh load. Since the change of speed value diminishes the inaccuracy signal becomes lesser and the position of the governor and not of the fly balls gets nearer to the required position to continue the constant speed. The approach to re-establish the speed or frequency to its so-called an integrator value is insert on the way. The integrator will unit shall monitor over a period of time of average error and the offset will be overcome. Thus as the load of the system changes constantly the generation is in tune routinely to bring back so-called value of frequency. The method is referred as automatic generation control. Multi area power system consisting of numerous pools, the AGC role of is to divide the load among the system, stations and generators so as to accomplish utmost cost-cutting measure and logically standardized frequency.
3.9.1 Overview of AGC Schemes
Flywheel governor of synchronous machine may be the initial idea implemented for AGC of a power system. But this approach was found inadequate. Therefore, a secondary was incorporated to the governor with the help of a signal directly proportional to the frequency deviation with its integral. This technique composes the classical approach to the AGC of power systems. Secondary AGC schemes are developed to control the area control error (ACE). Perhaps and Cohn 5-10 was the first to introduce the scheme for the control of bulk power transfer in interconnected power systems based on tie-line bias control strategy, particularly deciding the frequency bias setting and techniques timing error and inadvertent interchange correction for large multi area power system. In 11, he has presented a comprehensive study on extensive growth and expansion of interconnected electric power systems. Concordia and Kirchmayer 12, 13 have analyzed the AGC problem of two area hydrothermal power systems. D. M. Patel et al. 15 dealt with AGC in power system operation with reference to the tie-line control and the requirement of reactive power and voltage regulation under normal operating conditions in the model. A model was proposed to show the interaction between the Automatic voltage regulator (AVR) and the Load frequency control (LFC) loops. The coupling effects of the AVR and LFC loops were studied by including the excitation system in system dynamic model 16. T. W. Reddoch et al. 17 have presented a state variable model for AGC of a linear interconnected power system.
Other advanced techniques in control, such as optimal and adaptive control, are also formulated instate space. More recent trends in the science have been towards intelligent control systems that tend to use both the ideas of conventional control as well as methods such as fuzzy logic, Petri nets, search and genetic algorithms and neural networks. Following these developments in control systems, many AGC schemes have been proposed in the literature as their applications to power systems. Bhatt et al. 18 proposed a traditional AGC loop with modifications incorporated for simulating AGC in restructured power system model and the concept of distribution participation matrix for power system model. This model is used to simulate the bilateral transactions contract in the multi-area models.
3.9.2 Classical Control Based AGC Schemes:
In classical control a basic concept is to describe close loop properties in terms of easily measurable open loop properties. A few examples of classical control scheme include Nyquist, Bode, and Root Locus plots which are drawn based on open loop transfer function. A major limitation of classical control methods was the use of single-input, single-output (SISO) methods. Also the use of transfer functions and frequency domain limited to linear time invariant systems. In literature, a limited work has been reported concerning AGC of interconnected power systems using classical control theory 22-34. Also the load frequency control system is investigated using root locus techniques by J. E. Van Ness 22 and W. R. Barcelo 23. O. I. Elgerd and C. E. Fosha had presented a work on AGC concerned with the classical approach to determine the optimum integrator gains for ACEs 1833. Willems28 has proposed the classical approach to determine optimum parameter values of conventional load-frequency regulation of interconnected power systems. T. Hiyama 30 has proposed a method for designing a discrete-time load frequency controller based on conventional tie-line bias control strategy of a two area reheat thermal system considering generation rate constraint. Based on classical control scheme M. L. Kothari et. al. have discussed some aspects of sampled AGC in 31. Later, in 32 they have studied the AGC problem of an interconnected power system in continuous and discrete-mode using classical control theory.
3.9.3 Modern control concept based AGC schemes:
In present times the demand for electrical energy and load fluctuations is increasing so modern power systems are multi input and multi output (MIMO). The classical control schemes used for single input, single output (SISO) systems are incapable of handling modern power systems. The AGC regulator design techniques using recent be in command of theory enable the power engineers to design optimal control schemes with respect to a given performance criteria. In literature, volumes of research articles are reported using various aspects of modern control concepts for modern power systems.
22.214.171.124 Optimal Control AGC Schemes
The optimal control theory has provided new schemes to solve the problems of multivariable control problem of modern power systems in a simplified form. In this control scheme the state space representation of the model is considered and an objective functions to be minimized. In a two area interconnected power system consisting of two identical power plants of non-reheat thermal turbines Elgerd and Fosha have exhibited their pioneering work on optimal AGC regulator design using modern control concept. Also, two area interconnected power system investigated by Tacker et al. K. Yamashita and T. Taniguchi have analyzed the AGC problem for interconnecting systems considered from the point of view of optimal control theory. M. L. Kothari and J. Nanda introduce other algorithm for AGC regulators of an interconnected hydro-thermal power system using a performance index that circumvents the need for a load demand estimator. K. P. S. Parmar et al. Developed dynamical response of the AGC problem in an interconnected reheat type power system under consideration with a practical viewpoint by designing the optimal full state feedback controller. Ibraheem and P. Kumar have dealt a computational approach for the solution of the Matrix Riccati (MR) equation. A control strategy was proposed by Mariano et al. Later, they considered the stabilization and performance of the AGC regulator by using the theory of the optimal control.
126.96.36.199 Sub Optimal Control AGC Schemes
The optimal controller design requires the measurement of all the state variables for their feedback which is a serious limitation because measurement and access of all the state variables is not possible all the times. Therefore, the idea of sub-optimal AGC regulator design was introduced to overcome the limitations of former scheme. A remarkable work was presented by V. R. Moorthi and R. P. Aggarawal on sub-optimal and near optimal control of AGC system. Many aspects of sub-optimal AGC regulator designs for power systems have been considered in the publications. The authors have proposed a suboptimal controller design technique such that the proportional part of the regulator is a linear function of a smaller number of states of the system plus integral function of the area control error (ACE). Planned suboptimal control law was obtained by Eigen values grouping technique and has shown the feasibility of the design technique of the sub-optimal regulator for load frequency control of power system consisting of non-reheat thermal turbines. S.S. Choi discussed a design of an LFC using the feedback of only the directly measurable system state variables. P. Kumar et. al. designed AGC regulators for hydrothermal power system based on decentralized control strategies. O. P. Malik et. al. presented a sub-optimal load-frequency control for hydrothermal power systems. A sub-optimal control method using the area decomposition technique to the multi-area power system LFC is presented by Yoshibumi. Sub-optimal AGC schemes have also been reported using the concept reduced order modeling of the system. Elangovan et al. proposed a method by which a sub optimal control policy of a given linear system is derived using its simplified model whose order is less than that of the original system.
188.8.131.52 Centralized AGC scheme
Originally, the Load Frequency Control problem was based on the strategy of centralized control. On the origin of disturbances classes, the control strategy has been proposed. Elgerd and Fosha suggested a loop gain and feedback to eradicate the commotion, and by means of a state variable model and the state regulator problem of optimal control theory new feedback control law is developed.
184.108.40.206 Decentralized AGC Scheme
The main limitation of the works presented on AGC considering centralized control strategy is the need to exchange information among control areas spread over distantly connected geographical territories along with their increased computational and storage complexities. A wide range of research papers on decentralized AGC control strategy for large scale power systems with continuous and discrete time system models have appeared in the literature. Velusami and Chidambaram proposed a decentralized (PI) biased dual mode controllers for two areas interconnected thermal power system. Decentralized AGC regulator has been proposed by Elemetwally and Rao assuming a constrained structure with a minimum error excitation concept.
220.127.116.11 Adaptive and Self-tuning AGC scheme
The operating point of modern systems may not remain same so the performance of the controller may not be optimal. Pan and Liaw presented an adaptive controller for the LFC problem of power systems where the controller uses a PI adaptation. Rao and Ahson examined the use of a two-level control scheme for a two-area power system with nonlinear interactions between the areas. Rubaai and Udo proposed a multilevel adaptive load frequency controller which is based on the self-tuning regulator. Yamashita and Miyagi devised a method for designing a multi variable self-tuning regulator for an LFC system with the inclusion of the interaction of voltage on load demand.
18.104.22.168 Intelligent AGC Schemes
Abdel-Magid and Dawoud used genetic algorithms to handle the LFC problem. Rerkpreed apong et al. proposed a robust load frequency controller using genetic algorithms and linear matrix inequalities. Birch et al. applied a robust control technique to a nonlinear LFC problem. Demiroren et al. used an artificial neural network (ANN) controller for the LFC problem. Talaq and Al-Basri used an adaptive fuzzy gain scheduling technique to deal with the LFC problem. Other works on the application of intelligent control techniques to the load frequency control problem were reported in the literature. J. Talaq et. al. has proposed an adaptive fuzzy gain scheduling scheme for conventional PI and optimal AGC regulators.
22.214.171.124 Other controllers for AGC
Most of the AGC schemes presented so far have been formulated considering linear power system models. However, like other physical systems, power systems are also highly non-linear in nature. An overview of applications of robust control techniques in power systems is illustrated by L. Fan. This review has considered a variety of robust control techniques such as non-linearity H?, linear matrix inequalities, MR equation approaches, Kharitonov’s theorem, structured singular value theory, linear quadratic Gaussian, quantitative feedback theory and pole placement technique have been used, and an investigation is carried out for power system reliability against the uncertainties. A combination of matching conditions and Lyapunov stability theory has been adopted to implement a robust stabilizing controller of interconnected power systems with uncertain parameters. The Q-parameterization method is used to design robust AGC regulators while the set of all robust controllers of the power system was characterized by a parameter free ‘Q’ matrix. Exploration carried by Tripathi et al revealed that improved dynamic performance of the system could be achieved by simultaneous control of steam turbine and energy storage device.
3.9.4 AGC in a Single Area
With the Load Frequency Control primary loop, a alteration in the system load will result in a steady state frequency deviation, depending on the governor speed regulation. In order to reduce the frequency deviation to zero we must provide a reset action by introducing an integral controller to act on the load reference setting to change the speed set point. The integral controller increases the system type by 1 which forces the final frequency deviation to zero. The integral controller gain must be adjusted for a satisfactory transient response.
Fig.3.5: Mathematical modeling of AGC for an isolated power system
The closed loop transfer function of the control system is given by:
3.9.5 AGC in the Multi Area System
In many cases a group of generators are closely coupled internally and swing in unison. Furthermore, the generator turbines tend to have the same response characteristics. Such a group of generators are said to be coherent. It is possible to let the LFC loop represent the whole system and the group is called the control group. For a two area system, during normal operation the real power transferred over the tie line is given by
Where X12= X1+Xtie+X2 and ?12=?1-?2
For a small deviation in the tie-line flow
The tie-line power deviation then takes on the form
Fig.3.6: Two area system with primary loop LFC
Modern Control design is especially based on the multivariable state vector system. In this design algorithm we make use of the state variable parameters that can be obtained from the system. For the systems where all the state variables are not available a state estimator is designed.
3.10 MULTI-AREA POWER SYSTEM MODEL
Fig. 3.7 is a control block diagram for the ith area of a multi area power system. Although a power system is nonlinear and dynamic, the linearized model is permissible in the load frequency control problem because only small changes in load are expected during its normal operation 21.
Fig. 3.7: Block diagram of the ith area of a multi-area power system
The dynamic equations of the ith area of a multi-area power system are as follows:
In this chapter, the concept, an advantage and importance of an interconnected power system is analyzed. The mathematical model of various components are presented and analyzed. The concept of AGC scheme and its different control based schemes are discussed.
LFC PID CONTROLLER
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4.2 LFC PID CONTROLLER
The largest part used control strategy in industry is P-I-D controller. It is used for various control problems such as automated systems or plants. A PID-Controller includes three different fundamentals, which is why it is at times called a three term controller. Proportional control, Integral control and Derivative control is the expansion of PID.
To meet up different design specifications for the system, PID control is able to put into operation. These can include the settling and rise time plus the accuracy and overshoot of the system step response. The three stipulations should be measured separately to realize the function of a feedback controller for PID.
Proportional Control: to supply the driving input to the process, the Proportional control is a chaste gain tuning acting on the error gesture. The system speed can be adjusted using the P-term from the PID controller.
Integral Control: by preamble of an integrator, an Integral control is employed. To get the control system desired accuracy, an Integral control is used
Derivative Control: to enhance the damping in the system, Derivative deed is usually introduced. The derivative term also magnifies the presented noise which can cause evils together with instability.
Primarily we can well thought-out with a single generator supply of an isolated power system.
Fig. 4.1: Linear model of a single area power system
The PID controller tuning is to improve the performance of power system load frequency control. The design control law u = -K(s) ?f, where K(s) has the form
In general, PID controller is put into practice to decrease the effect of noise. So, for this case K(s) can be written as
Where N is termed as the filter constant
Where ‘T’ is an incredibly little sampling rate
In view of the fact that the power system load frequency control deems a little load change, it can be symbolized by the single area model.
In this Chapter, the benefits of control method are discussed and also the PID LFC is introduced and explained.
The single area and three-area interconnected power system is analyzed to illustrate the effectiveness of the proposed control scheme. In order to validate the proposed topology, simulation is carried out using the Matlab/Simulink.
5.2 PROPOSED TOPOLOGY
The validity and performance of the proposed LMI-based robust predictive LFC (LMI-PLFC), and non-predictive LFC (LMI-NPLFC) schemes are investigated for one-are and multi area power systems. The results have been compared with the LMI based LFC (LMI-LFC) approach presented in 35, the popular PI-based LFC (PI-LFC) reported in 24, and decentralized PI-LFC (DPI-LFC) in 21. It is remarkable that all systems suffer from the delays, and step load demand of ?Pd = 0.01 is applied to all units. Different cases of uncertain and time varying systems for one-area LFC, and decentralized control structure for a multi-area LFC system are considered to show the superior performance of the proposed LFC approach.
Fig.5.1: Dynamic model of the ith control area in a multi-area LFC system
5.3 SIMULATION RESULT ANALYSIS
The simulation is performed based on a one-area LFC system studied in 35. The responses of the LFC system equipped with PI controller 24, load frequency control method based on linear matrix inequalities presented in 35, and the proposed LMI-based robust predictive and non-predictive controllers are shown in Fig.5.1. It demonstrates good time responses, lower fluctuations, and more robustness of the proposed LMI-based LFC approaches.
5.3.1 Uncertain One-Area LFC System
Fig.5.2: Time responses of different controllers for uncertain one-area LFC system
The simulation is performed based on a one-area LFC system studied in 35. The responses of the LFC system equipped with PI controller 24, load frequency control method based on linear matrix inequalities presented in 35, and the proposed LMI-based robust predictive and non-predictive controllers are shown in Fig.5.2. It demonstrates good time responses, lower fluctuations, and more robustness of the proposed LMI-based LFC approaches. It is remarkable that the communication delay of the system is assumed to be three seconds, (d = 3 sec).
5.3.2 Time-Varying One-Area LFC System
As it is mentioned, the communication fault in power system may cause data drop and can be modeled as an equivalent time-varying system. The single-area LFC system in 35 is considered again with a constant time delay, d = 3 sec. A sudden change in dynamic model of the system is considered at t = 30 second. As shown in Fig.5.3, the varying model of system affected the responses of all LFC schemes. The approaches in 35 and 24 become unstable, whereas the proposed LMI-PLFC and LMINPLFC approaches have stable and fast transient responses. Due to the predictive nature of the LMI-PLFC, it shows better performance compared with the non-predictive LFC approach.
Fig. 5.3: Time responses of different controllers for varying one-area LFC system
5.3.3 Decentralized Predictive LFC Scheme for Three-Area Power System
Consider a three-area power system reported in 35, where the first area is modeled by two generators and the other areas have single generator equivalents. For performance studies, the decentralized LMIPLFC and LMI-NPLFC are compared with the decentralized LMI-based LFC approach in 35, and the decentralized PI controller (DPI-LFC) in 21. The time responses of frequency deviation, ?f, and ?E for all areas are depicted in Fig.5.4 and Fig.5.5. The applied control inputs at each area are also shown in Fig. 5.6. The results show two aspects of advantages for the proposed LMI-PLFC scheme in comparison with the other decentralized LFC approaches. First, its responses in Fig.5.4 have very good time specifications such as lower settling time and overshoot. Second, this good performance is obtained with better control efforts as shown in Fig. 5.5.
Fig.5.4: ?f responses of all areas in a multi-area LFC scheme with different methods
Fig.5.5: ?E responses of all areas in a multi-area LFC scheme with different methods
Fig.5.6: Applied control of all different methods
In this chapter, the proposed method is explained with simulation results. The proposed method validated through simulation and the simulation results for the different cases is presented.
CONCLUSIONS AND FUTURE SCOPE
One-area and multi-area power systems with communication delays are considered to evaluate the performance of the proposed robust predictive LFC approach. For a single-area LFC system, the better performance of the proposed control strategy compared with the two existing LFC approaches. It should be pointed out that considering the communication delay with uncertainty and time-varying parameters in the LFC system are key features different from some existing works. Moreover, for a multi-area LFC system, the proposed decentralized predictive LFC strategy is preferable than the existing decentralized LFC approaches. Considering decentralized control structure for a multi-area LFC system with time-delays, uncertainty, and time varying parameters is the other new feature which is out of the scope of those previous works.
The proposed method will also validate through experimental studies. The practical implementation of the designed controller depends on the accuracy of local states of each area. Those states can be obtained from the measurements of monitoring system or using the state estimation methods. Although the detailed methods of the state estimations are not focused in this thesis, the errors from the measurements and state estimations have been considered as a future work of this thesis in controller design to guarantee the robustness and effectiveness of the proposed controller.
H. Bevrani, Robust Power System Frequency Control. New York, NY, USA: Springer, 2009.
N. Chuang, “Robust H? load-frequency control in interconnected power systems,” IET Control Theory Appl., vol. 10, no. 1, pp. 67–75, 2016.
S. Snmez and S. Ayasun, “Stability region in the parameter space of PI controller for a single-area load frequency control system with time delay,” IEEE Trans. Power Syst., vol. 31, no. 1, pp. 829–830, Jan. 2016.
C.-K. Zhang, L. Jiang, Q.Wu,Y.He, and M.Wu, “Delay-dependent robust load frequency control for time delay power systems,” IEEE Trans. Power Syst., vol. 28, no. 3, pp. 2192–2201, Aug. 2013.
A. Khodabakhshian and M. Edrisi, “A new robust PID load frequency controller,” Control Eng. Pract., vol. 16, no. 9, pp. 1069–1080, 2008.
W. Tan, “Unified tuning of PID load frequency controller for power systems via IMC,” IEEE Trans., Power Syst., vol. 25, no. 1, pp. 341–350, Feb. 2010.
R. Vijaya Santhi, K. Sudha, and S. Prameela Devi, “Robust load frequency control of multi-area interconnected system including SMES units using type-2 fuzzy controller,” in Proc. 2013 IEEE Int. Conf. Fuzzy Syst., 2013, pp. 1–7.
C. Boonchuay, “Improving regulation service based on adaptive load frequency control in LMP energy market,” IEEE Trans. Power Syst., vol. 29, no. 2, pp. 988–989, Mar. 2014.
S. Nag and N. Philip, “Application of neural networks to automatic load frequency control,” in Proc. Int. Conf. Control, Instrum., Energy Commun., 2014, pp. 345–350.
C. Boonchuay, “Improving regulation service based on adaptive load frequency control in LMP energy market,” IEEE Trans. Power Syst., vol. 29, no. 2, pp. 988–989, Mar. 2014.
L. Dong and Y. Zhang, “On design of a robust load frequency controller for interconnected power systems,” in Proc. Amer. Control Conf., 2010, 2010, pp. 1731–1736.
H. Bevrani, Y. Mitani, and K. Tsuji, “Robust decentralized load-frequency control using an iterative linear matrix inequalities algorithm,” IEE Proc.-Gener., Transm. Distrib., vol. 151, no. 3, pp. 347–354, 2004.
A. Bensenouci and A. A. Ghany, “Mixed H?/H2 with pole-placement design of robust LMI-based output feedback controllers for multi-area load frequency control,” in Proc. EUROCON 2007—Int. Conf. Comput. Tool, 2007, pp. 1561–1566.
M. Rahmani and N. Sadati, “Hierarchical optimal robust load-frequency control for power systems,” IET Gener., Transm. Distrib., vol. 6, no. 4, pp. 303–312, Apr. 2012.
W.Yao, L. Jiang, Q.Wu, J.Wen, and S. Cheng, “Delay-dependent stability analysis of the power system with a wide-area damping controller embedded,” IEEE Trans., Power Syst., vol. 26, no. 1, pp. 233–240, Feb. 2011.
S. Wen, X. Yu, Z. Zeng, and J. Wang, “Event-triggering load frequency control for multi-area power systems with communication delays,” IEEE Trans. Ind. Electron., vol. 63, no. 2, pp. 1308–1317, 2016.
K. Vrdoljak, I. Petrovic, and N. Peric, “Discrete-time sliding mode control of load frequency in power systems with input delay,” in Proc. 12th Int. Power Electron. Motion Control Conf., 2006, pp. 567–572.
M. Liu, L. Yang, D. Gan, D. Wang, F. Gao, and Y. Chen, “The stability of AGC systems with commensurate delays,” Eur. Trans. Elect. Power, vol. 17, no. 6, pp. 615–627, 2007.
L. Yongjuan, M. Yang, Y. Yang, and W. Limin, “The study of sliding mode load frequency control for single area time delay power system,” in Proc. 2015 27th Chin. Control Decis. Conf., 2015, pp. 602–607.
X. Xie, Y. Xin, J. Xiao, J. Wu, and Y. Han, “WAMS applications in Chinese power systems,” IEEE Power Energy Mag., vol. 4, no. 1, pp. 54–63, Jan./Feb. 2006.
H. Bevrani and T. Hiyama, “Robust decentralized PI based LFC design for time delay power systems,” Energy Convers. Manage. vol. 49, no. 2, pp. 193–204, 2008.
S. Sonmez and S. Ayasun, “Stability region in the parameter space of PI controller for a single-area load frequency control system with time delay,” IEEE Trans. Power Syst., vol. 31, no. 1, pp. 829–830, Jan. 2016.
L. Jiang,W.Yao, Q.Wu, J.Wen, and S. Cheng, “Delay-dependent stability for load frequency control with constant and time-varying delays,” IEEE Trans. Power Syst., vol. 27, no. 2, pp. 932–941, May 2012.
S. Sonmez, S. Ayasun, and C. Nwankpa, “An exact method for computing delay margin for stability of load frequency control systems with constant communication delays,” IEEE Trans. Power Syst., vol. 31, no. 1, pp. 370–377, Jan. 2016.
C. Peng and J. Zhang, “Delay-distribution-dependent load frequency control of power systems with probabilistic interval delays,” IEEE Trans. Power Syst., vol. 31, no. 4, pp. 3309–3317, Jul. 2016.
M. S. M. Cavalca, R. K. H. Galv˜ao, and T. Yoneyama, “Robust model predictive control using linear matrix inequalities for the treatment of asymmetric output constraints,” J. Control Sci. Eng., 2012. Online. Available: http://dx.doi.org/10.1155/2012/485784P. Ojaghi, N. Bigdeli, and M. Rahmani, “An LMI approach to robust model predictive control of nonlinear systems with state-dependent uncertainties,” J. Process Control, vol. 47, pp. 1–10, 2016.
J. Zhang, X. Zhao, Y. Zuo, and R. Zhang, “Linear programming-based robust model predictive control for positive systems,” IET Control Theory Appl., vol. 10, no. 15, pp. 1789–1797, 2016.
X. Liu, H. Nong, K. Xi, and X. Yao, “Robust distributed model predictive load frequency control of interconnected power system,” Math. Probl. Eng., vol. 2013, 2013, Art. no. 468168.
X. Liu, X. Kong, and K. Y. Lee, “Distributed model predictive control for load frequency control with dynamic fuzzy valve position modeling for hydro-thermal power system,” IET Control Theory Appl., vol. 10, no. 14, pp. 1653–1664, 2016.
A. M. Ersdal, L. Imsland, andK.Uhlen, “Model predictive load-frequency control,” IEEE Trans. Power Syst., vol. 31, no. 1, pp. 777–785, Jan. 2016.
L. Dong, Y. Zhang, and Z. Gao, “A robust decentralized load frequency controller for interconnected power systems,” ISA Trans., vol. 51, no. 3, pp. 410–419, 2012.
Y. Mi, Y. Fu, C. Wang, and P. Wang, “Decentralized sliding mode load frequency control for multi-area power systems,” IEEE Trans. Power Syst., vol. 28, no. 4, pp. 4301–4309, Nov. 2013.
M. Yang, L. Wenlin, and J. Yuanwei, “Decentralized load frequency variable structure control of multi-area interconnected power system,” in Proc.2011 Chin. Control Decis. Conf., May 2011, pp. 666–669.
X. Yu and K. Tomsovic, “Application of linear matrix inequalities for load frequency control with communication delays,” IEEE Trans. Power Syst., vol. 19, no. 3, pp. 1508–1515, Aug. 2004.
P. Kundur, N. J. Balu, and M. G. Lauby, Power System Stability and Control, vol. 7. New York, NY, USA: McGraw-Hill, 1994.
H. K. Khalil, Nonlinear Systems. Englewood Cliffs, NJ, USA: Prentice-Hall, 2002.
L. El Ghaoui and S.-I. Niculescu, Advances in Linear Matrix Inequality Methods in Control. Philadelphia, PA, USA: SIAM, 2000, vol. 2.