Posts Tagged ‘Theory’

Chapter-7: Conclusion and Future Scope

7.1. Conclusion and Discussion

This research work mainly represents the modeling and simulations of some Intelligent controllers for an AGC in single power systems. In this thesis, the computer simulations results based on system non-linear differential equations have been carried out for different load changes. The differential equations have been solved by using MATLAB programming environment. In this research we basically use fuzzy logic control. We have reduced the settling time and minimized the error (∆f). We have made the settling time 4.5 second which was above 6 second in other previous records.

Our research of frequency control is suitable for universal load system. By this method, the system will be stable within 4.5 seconds which is very effective.

 7.2. Scope of Future Work

In this research, all modeling and simulation of the proposed scheme was performed by MATLAB program. These will also be performed by using the latest simulation techniques MATLAB Fuzzylogy system.

In this research, some intelligent Controllers for an AGC in single power system have successfully achieved zero steady state error, but this research has some future scopes described as follows:

a)      The research work can be further extending considering boiler dynamics.

b)      Also deferent type of intelligent controller can be tested to get the better performance.

7.3. References

1)      M.G.Rabbani, J.B.X Devotta, S. E langovan, “A fuzzy set theory based control of superconductive magnetic energy storage unit improve power system dynamic performance” Electric power system Research 40 (1997), 107-114

2)      Y.L Karnavas, D.P Papadopoulos “AGC for autonomous power system using combined intelligent techniques” Electric Power System Research 62 (2002), 225-239.

3)      prof. Wah-Chun Chan, Yuan-Yih Hsu, “Automatic Generation Control of Interconnected Power system Using variable -structure Controllers”, IEEE Proceeding Vol. 128 , pt,  C,  No. 5 September 1981

4)      P.M Anderson and A.A Fuad, “Power system control and stability” Iowa State University press, Ames, lowa, 1977

5)      Hadi Saadat, “Power System Analysis”, Tata McGraw-hill Publishing Company Limited, New Delhi.

6)      O.I. Elgerd, “Electric Energy Systems Theory” McGraw-Hill Book company New York.

7)      S.c Tripathy, T.S Bhatti, C.S. Jha   O. P. Malik, G. S. Hope, “Smpled Data Automatic Generation Control Analysis with Reheat Steam Turbine and Governor Dead-Band Effects”, IEEE Transaction on Power Apparatus and System, Vol. PAS-103, No.4 May,1984

8)      Gilberto CD Sousa, Bose, “A Fuzzy Set Theory Based Control of a Phase-Controlled Converter DC Machine Drive”, IEEE Transactions on Industry Applications, Vol. 30, No.1, January 1994

9)      D.Driankov, et al, “An Introduction to Fuzzy Control” Springer-Verlag Berlin- Heidelberg, New York, 1993

10)  P.N. Paraskeveopoulos, “ Digital Control System” prentice Hall Europe 1996

11)  Nzsser Jaleeli, Louis  S. Vanslyck, Donald N. Ewart, Lester H. Fink, Arthur G. Hoffmann, “Understanding Automatic Generator Control” IEEE Transactions on power system, Vol.7 No. 3 August, 1992.

12)  W.C. Chan, Y.Y. Hsu, “Automatic Generation Control of Interconnected Systems Using variable structure controllers” IEE proceedings, vol. 128, Pt.C.No.pp. 269-279, September, 1981.

13)  C.T. pan, C.M. Lian, “An Adaptive Controller for Power System Load-Frequency Control”, IEEE Transactions on Power System, Vol. 4, No. 1 February, 1988

14)  M.H. Ali, “A Fuzzy  logic controlled Braking Resistor for Power System Trnsietn Enhancement”, in partial fulfillment of the Ph.D. degree in Electrical & Electronic Engineering, Kitami Institute of Technology, Japan.

15)  Zhog He Shaohua Tan and Chang- Chieh Hang, “Control of dynamical processes using an on –line rule adaptive fuzzy control system”, Elsevier science publishers B.V. All rights reserved fuzzy Sets and Systems 54(1993), 11-22

16)  Dr.S.P.Ghoshal , “Multi area frequency and tie- line p, December overflow     control with fuzzy logic based  integral gain scheduling” IE(I) jouranal-El, Vol 84 December 2003

17)  J.B.X. Devotta, M.G.  Rabbni S.Elagovan “Effects of SMES Unit on AGC dynamics” Internationl conference on Energy management and power Delivery 1998 EMPD, 98 Singapore.

18)  M.H.Ali, et al,” Braking Resistor Switching By Genetic algoritam optimized Fuzzy logic controller in Muli-machine power system” Transation of IEE,Japan, Vol 123 –B No.315-323, 2003.

Chapter 6: FUZZY LOGIC CONTROLLER

6.1. Introduction

Fuzzy logic is a powerful problem-solving methodology with a myriad of application in embedded control and information processing. Fuzzy provides a remarkable a sense, fuzzy logic resembles human decision making with its ability to work from approximate data and find precise solution. Unlinked classical logic which requires a deep understanding of a system, exact equations, and precise numeric values, fuzzy logic incorporation an alternative way of thinking, which allows modeling complex systems using a higher level of abstraction organizing from our knowledge and experience. Fuzzy logic allows expressing this knowledge with subjective concepts such as very hot, medium cold, and a long time which are mapped into exact numeric ranges. Fuzzy logic is recently finding wide in various applications which cover a variety of practical systems, such as the control of cement kilns, train operation, parking control of car, heat exchangers, and in many other system, such as home appliances, video cameras, elevators, aero space, etc [10, 14]. Nearly every application can potentially realize some of the benefits of fuzzy logic, such as performance simplicity lower cost, and productivity [9, 13]. Fuzzy logic was first introduced by Zadeh in 1965, whereas the first fuzzy logic controller was implemented by Mamdani in 1974 [10, 14].

In this work, different types of fuzzy based controllers are designed for an automatic generation control in single power system.

6.2. Basic Control of Fuzzy Logic

A fuzzy logic, unlike the crispy logic in Boolean theory that uses two logic (0 to 1) is a branch of logic that admits infinity logic level (from 0 to 1) to solve a problem that has uncertainties or imprecise situations. A variable in fuzzy logic has sets of values which are characterized by linguistic expressing, such a SMALL, MEDIUM, LARGE, etc. linguistic expressions are represented numerically by fuzzy sets. Even Fuzzy set is characterized by a membership function, which varies from 0 to1 .Although fuzzy theory deals with imprecise information; it is based on sound quantitative mathematical theory [8, 14].

Again fuzzy control is a process that is based on fuzzy logic and is normally characterized by if-THEN rules. A fuzzy control algorithm for a process control system embeds the intuition and experience of operator, designer and researcher. The control does not need accurate mathematical model of a plant, and therefore, it suits well to a process where the model is unknown or ill defined. The fuzzy control also works well for complex nonlinear multi dimensional system, system with parameter variation problem, or where the sensor signals are not precise. The fuzzy control is basically nonlinear and adaptive in nature, giving robust performance under parameter variation and load disturbance effect [8, 14].

To gain an in depth on fuzzy logic, the following terms the need to be studied.

  1. Degree of Membership (µ): It is a number between (0 to 1) that expresses the confidence that a given element belongs to a fuzzy set.
  2. Fuzzy Set (fuzzy subset): This is defined as a set consisting of elements having degree of membership varying between 0 (member) to 1 (full member). It is usually characterized by a membership function, and associated with linguistic valuably characterized by a membership function, and associated with linguistic values, such as SMALL, MEDIUM, and LARGE etc.
  3. Membership Function: It is a function that defines a fuzzy subset, by associating every element in the set with a number between 0 and 1.
  4. Linguistic Variables: Any variable (such as temperature, speed, etc) whose values are defined by language, such as LAGE, SMALL, etc. is called a linguistic variable or fuzzy variable.
  5. Universe of Discourse: It is the range of values associated with a fuzzy variable.

6.3. Advantages of Fuzzy Logic Controller (FLC)

When compared to classical control theory a fuzzy logic approach to control offers the following advantages [14].

  1. It can be used in systems which cannot be easily modeled mathematically in particular, systems with non linear responses are difficult to analyze may respond to a fuzzy control approach.
  2. It is inherently robust since it does not require precise, noise free inputs. The output control is a smooth control function despite a wide range of input variations.
  3. Since the fuzzy logic controller (FLC) processes user defines rules governing the target control system, it can be modified easily to improve or drastically alter system performance.
  4. Continuous variable may be represented by linguistic terms that are easier to understand, making the controller easier to implement and modify. For example, instead of using numeric values, temperature may be represented as “cold, cool warm, or hot”.
  5. Complex processes can often be controlled by relatively few logic rules, allowing a more understandable controller design and faster computation for real time applications.

 

6.4. Design of Fuzzy Logic Controller (FLC)

Fuzzy control is special form of knowledge-based control. In designing a fuzzy control system, the precise mathematical model of target plant is not needed. Only the relevant experiences and heuristics concerning the pant are utilized to from a set of fuzzy control rules. These are linguistic in nature and often use the simple cause- effect relationship to link a fuzzy partitioning of certain state-space of the plant with a fuzzy petitioning of the control action. The final control signal is generated by an appropriate defuzzifying process [15].

The reference signal and plant output, which are crisp values but non-fuzzy variables, must be fuzzified by a fuzzification procedure. Similarly, the fact that the controlled plant can not directly respond to FL controls accounts for the reason why the FL control signal generated by the fuzzy algorithm must be defuzzification before applied to control the actual plant. A rule base consists of a set of fuzzy rules. The data base contains the membership functions of the fuzzy subsets. A fuzzy rule may contain fuzzy variables and fuzzy subsets characterized by membership functions and conditional statement. The fuzzy control algorithm

Fig: 6.1 Block diagram of a typical close-loop fuzzy control system

6.4.1. Fuzzification:

The fuzzification procedure consists of finding appropriate membership function to describe crisp data. The membership functions for the fuzzy variables may have several shapes. The most popular choices for the shape of the membership function include, triangular trapezoidal, and bell-shaped functions. Among them, triangular membership function is the most economic one because of its minimal use of memory & efficiency, in terms of real time requirements, by the inference engine. To design the fuzzy controller in the work, the triangular membership functions for an input & output variables. The memberships functions are system dependent. The membership functions used for the fuzzy logic controller (FLC) design in this work. Usually, membership functions are determined by trail & error. In this work, the membership functions have been determined by trail & error method. The precise numerical values are obtained by measurements that are converted to membership values of the various linguistic variables. For the FLC controller the inputs are defined as:

Input 1: error =∆f=fnom– ft=et

Input 2: change in error=∆f2-∆f1=cet.

Usually, two input variables (error of the variable of interest for the control and change of error) are used fore fuzzy logic controller (FLC) design [18]. However, in this work, two input and single output are used for FLC design. The use of two input and single output variable makes the design of the controller very straightforward [9-13].

6.4.2. Fuzzy Rule Base:

The rule base is the heart of a fuzzy controller, since the control strategy used to control the closed-loop system is stored as a collection of control rules. The heuristic rules of the knowledge base are used to determine the fuzzy controller action.

Controller has 2 inputs et, and cet,and one output ut. Then a typical control rule has the form.

If et, is A and cet, is B then and uis C. …………………………(6.1)

Where A, B and C are linguistic terms, such as very low, very high and medium, etc. the control rule (6.1) is composed of two parts: ‘if’ part and the ‘then’ part. The ‘if’ part is the input to the controller and the ‘then’ part is the output of the controller. The ‘if ’part is called the premise (or antecedent or condition) and the ‘then’ part is called the consequence (or action).

Though it is possible to derive a membership value for this variable in many possible ways, one of the rules that has been chosen is,

µ(et,cet.)=min[µ(et.),µ(cet.)] ……………………………(6.2)

The fuzzy rules are system dependent. The most usual source fore constructing linguistic control rules is human experts. However, it is often the case that no expert is available.

Therefore trial and error method is usually used to find fuzzy control rules.

6.4.3. Fuzzy Inference

The basic operation of the inference engine is that it infers i.e. it is deduces (from evidence or data) a logical conclusion. Let us consider the following example describe by the logical rule known as modus ponens:

Premise 1: If and animal is a cat, then it has four legs.

Premise 2: My pet is a cat.

Conclusion: My pet has four legs.

Here, premise 1 is the rule base, 2 are the fact (or the data) and the conclusion is the consequence. Actually, the inference engine is a program which uses the rule base and the input data of the controller to draw the conclusion, very much in the manner shown by the above the modus ponens rule [9]. The conclusion of the inference engine is the fuzzy output of the controller, which subsequently becomes the input to the defuzzification interface.

Usually, two types of fuzzy inference are available in the literature. One is Takagi-Sugeno-Kang (TSK) fuzzy inference and other is Mamdani type fuzzy inference. For the inference mechanism of the fuzzy logic controller in this work, Mamdanis method [8] has been utilized. Compared to other methods, the advantages of Mamdant’s are:

  1. Calculation time is very short;
  2. Inference mechanism is very simple;
  3. Parameters can be changed easily.

6.4.4. Defuzzification

In this last operation, the fuzzy conclusion of the inference engine is defuzzified, i.e. it is converted into a crisp signal .this last signal is the final product of the FLC which is of course, the crisp control single to the process. There are several methods for defuzzification  available in the fuzzy interaction, such as Center -of -Area  or  Center -of –Gravity defuzzification Center -of –Sums defuzzification Center -of –Largest-Area, defuzzification  first- of- Maxima defuzzification Middle-of- Maxim defuzzification, Height defuzzification etc. the Center -of -Area  or  Center -of –Gravity method which is implemented in this work to determine the output crisp value. The well-known center of gravity defuzzification method is given by the following expression:

……………………………………….. (6.3)

Where, Z is the crispy output function and symbols have already been defined in the previous section. The membership function, knowledge base and method of defuzzification essentially determine the controller performance [1].

6.5. The Proposed Fuzzy Logic Control

Fig (6.2): fuzzy controller for the SMES unit

The proposed controller along with SMES unite is shown in fig (6.2), the ∆f are the input to the corresponding fuzzy controllers. The output of the Fuzzy Frequency Controller (FFC) is KI. ∆f is changing with changing KI.

Unlikely, the conventional controller, in the proposed method  the change in  ∆f the signal are also consider as explained below. In general, the input variables considered in the fuzzy rule base are:

E(k)=R(k)-C(k)

CE(k)=E(k)-E(k-1)

Where E(k) is the loop error (present deviation), CE(k) is the change in loop error, R(k) is the reference signal, C(k) is the present signal, and k is the sampling interval.

The structure of a general rule can be given as:

IF E(K) is X AND CE(K) is Y THEN U(K) is Z

The variable can be expressed as per unit quantities as follows:

e(p.u) = E(k)/GE

ce(p.u) = E(k)/GCE

Where, GE and GCE are the respective gain factor of the controllers. Fig (6.4) shows the membership function of e(p.u), ce(p.u) and their respective output variable.

Note that the fuzzy the subsets for output variable have an asymmetrical shape causing more crowding near the origin. This allows precision control near the steady state operating point. Also large number of subsets is selected to obtain accurate control.

Table 1 gives the rule base matrix for frequency control can be summarized as follows:

  1. Sample the reference frequency f* and the actual frequency fact
  2. Compute error (e) and change of error (ce) in their respective p.u. values are as follows:

                                                       e(k)=(f*(k)-f(k))/GE

                                                      ce=(e(k)-e(k-1))/GCE

  1. Identify the interval indices for e (p.u) and ce(p.u) respectably, by the comparison method.
  2. Compute the degree of membership of e (p.u) and ce (p.u) for the relevant fuzzy subsets.
  3. Identify the four valid rules in table 1 and calculate the degree of membership µRi MIN operation.

Fig (6.4): Membership function of fuzzy frequency control (FFC)

            6. Integrating gain KI for each rule from table

           7. Calculate the crisp value of KI higher defuzzification  method as follow:

………………………. (6.4)

Table 1: Rule base of frequency control

The fuzzy sets of the each linguistic variable adopted in this work are: NVB: Negative very Big, NB: Negative Big; NM: Negative Medium NS: Negative small; NVS: Negative Very small Z: Zero; PVS: Positive Very small; PS: Positive small; PM: Positive Medium PB: Positive Big. PVB: Positive Very Big. Each fuzzy set has a triangular shape and is determined by three parameters consequently, if a linguistic variable is supported by N fuzzy sets, then the total number of parameters needed is N+1, considering that the zero.

6.6. Simulation Results and Discussion:

In Order to Demonstrate the beneficial damping effect of the proposed fuzzy set theory based-gain scheduling of automatic generation controller in a single area power system, computer simulations results based on system non-liner differential equations have been carried out for different load changes. The differential equations have been solved by using MATLAB environment. Figure 6.5, 6.6 and Fig 6.7 depict the simulation results with & without considering the load changes of ∆PL =0.01, 0.015, and 0.02 P.U M W respectively.

The system frequency is not found satisfactory with fixed integral gain, KI. With the addition of proposed schemes, the damping is improved significantly. The generator frequency is almost diminished with the proposed mode of control. It is clearly shown that considering governor is greatly minimized error. Moreover, this eventually reduces the settling time of the speed for both cases, which in turn brings the FGPI controller in more advantageous position for subsequent use. The frequency deviation is diminished with the Fuzzy controller. The system frequency deviation is almost minimized. The settling time becomes smaller.

Fig: 6.5 Frequency deviation and variation of KI of a typical single area power system with the step load ΔPL =0.01(p.u)

Fig: 6.6 Frequency deviation and variation of KI of a typical single area power system with the step load ΔPL =0.015(p.u)

Fig: 6.7 Frequency deviation and variation of KI of a typical single area power system with the step load ΔPL =0.02(p.u)

6.7. Effect of Variable Load

Fig: 6.8. Frequency deviation for single area power system with variable load change

Table: 2 Table for frequency deviation for single area power system with variable load change. In the table 2 that get from the fig 6.8., Load change from .005(pu) to 0.25(pu), then maximum deviation, settling time, overshoot change. Above table settling time is nearly 4.5(s).

6.8. Conclusion

Fuzzy logic is getting increasing acceptance in control application over the past few years. In this work the fuzzy logic control (FLC) is used for the designed of some controllers for an Automatic Generator Control (AGC) of single and single-area power system. Again in order to see how well, robust and effect designed of the fuzzy controllers are and its performance compare to that of the control. This chapter presents overview of fuzzy controller, design of fuzzy logic control.

Chapter- 5: Integral Gain Control of Automatic Generation Control (AGC)

5.1. Introduction

The first and overriding requirement is the selection of parameter which will result in a power stable system .Having secured a stable system; our next objective is to adjust the parameter until we have a best optimum response [6]. One way to improve the stability of power system optimizes the basic factor controller parameter with change of frequency. This factor is integrating gain [7].

5.2. Automatic Generation Control (AGC)

If the load on the system is increased, the turbine speed drops before the governor can adjust the input of the stream to the new load .As the change in the value of speed diminishes, the error signal becomes smaller and the position of the governor fly balls gets closer to the point required to maintain a constant speed. However, the constant speed will not be the point, and there will be an offset. One way to restore the speed or frequency to its nominal value is add an integrator. The integral unit monitors the average error over a period of time and will overcome the offset. Because of its ability to return a system to its point, integral action is also known as the reset action. Thus as the system load changes continuously, the generation is adjusted automatically to restore the frequency to the nominal value. This scheme is known an the automatic generation control (AGC) [13].

5.3. Integral Control

By using the control strategy shown in figure 5.1 maintains an overall system that will meet performance. The speed change is commanded by a signal stained by first amplifying and then integrating the frequency error [2].

  ………………………….…………(5-1)

Note the negative polarity of the integral controller. This polarity must be chosen so as to cause a positive frequency error to given rise to a negative, or decrease command. The signal fed into the integrator is referred to as area control error (ACE).

ACE = ∆f ………..……………………………………………… (5-2)

The integral control will give rise to zero static frequency error following a step load change for the physical reason. As long an error remains the integrator output will increase, causing the speed changer to move. The integrator output, and thus the speed changer position, attains a constant value only when the frequency error has been reduced to zero [5].

 

5.3.1. Integral Gain Value KI of Automatic Generation Control

The gain constant KI control the rate of integration, and thus the speed of response of the loop. The integration is actually performed in electronic integrators of the same type as found in analog computers. Here follows an analysis of the proposed system, subject to a step load change. To avoid cumbersome numerical analysis, we shall as before neglect the time constants TT and TG .In addition we also make the assumption that the speed changer action is instantaneous. This is not perfect correct, since the device is electromechanical and will therefore have a nonzero response time. These approximations will make possible relatively simple analysis without distorting essential features of the response. It is also worth mentioning that the errors we thus introduce our analysis affect only the transient, not the static, response.

By derivations equation (5-1), we get equation,

…………………………..…………………….. (5-3)

5.3.2. Effect of Constant Gain KI

The value of gain KI is constant then ∆f create more overshot from the x –axis

And get more time to create ∆f =0. When ∆f is allow age zero then curve flow through the x –axis. We see that settling time is more than 6s.

When KI=1, 0.7, 0.5 effect of the ∆f curve see in figure below:

Fig: 5.2 frequency deviation for single area power system with step load change ∆PL=0.01(p.u) MW

In the above picture we see that when KI =1 then overshoot 0.02 (pu) and settling time is more than 10s. When KI =0.7 then overshoot 0.012 (pu) and setting time is more than 9s. And when KI =0.5 then overshoot 0.005 (pu) and setting time is more than 7s.

5.3.3. Effect of Variable Gain KI

Frequency deviation for single area power system with step load change ∆PL=0.01(p.u) MW:

Fig: 5.3 frequency deviations for single area power system with variable gain controlled


5.3.4. Compare Between Constant and Variable Gain

When KI=1, 0.5 and vary (1-0.25) then effect of the ∆f curve see in figure below:

Fig: 5.4 frequency deviation for single area power system with step load change ∆PL=0.01(p.u) MW

In the above picture we see that when KI =1 and 0.5, then overshoot is more than 0.009 (p.u) and setting time is more than 7s. We also see when KI is variable gain vary (1-.25) then overshoot is more than 0.002 (pu) and setting time is more than 4.5s.

5.4. Matlab Program:

Variable Gain Control of Automatic Generation Control (AGC):

Chapter-4: MODELING OF A SINGLE AREA POWER SYSTEM

4.1. Introduction

The single area power system model without the LMES unit is the same as that described in for convenience of the reader; both the original model and the changes due to addition of the LMES unit are described here. It is assumed that initially the system is in a steady state, characterized by a constant frequency fO, a constant prime mover value setting xOE, and constant generator output pOG. Initially the speed changer command signal is zero. In response to the step load change, the deviations of these parameters from their nominal steady-state values are ∆f, ∆XE, ∆PG and ∆PC respectively.

4.2.1. Mathematically Representation Single Area Power Pystem

The complete block diagram of the frequency control of an isolated power system is shown figure:

Fig: 4.1. Load Frequency control of an isolated power system

The governor adjusted XE, the value setting of the turbine depending on the frequency deviation signal ∆f and the speed changer command signal ∆PC. R denotes the speed regulation due to governor action. Thus the gain of the frequency feedback loop is 1/R. If KG and TG are the gain and time constant of the speed governing mechanism, then the transfer function of the speed governor is,

Thus the deviation in turbine value setting is given by,

………………………………………(4-1)

The prime mover considered here is a reheat type steam turbine, with one high pressure stage and one low pressure stage. There is reheating arrangement is between the two stages.

Suppose the high-pressure stage is rated at a fraction Kr of the total power developed by the turbine. Let KT and TT be the gain and time constant respectively of the individual turbine and Tr , the time delay due to the passage of steam in the reheater. So the power developed in the H. P. stage is,

…………………………….(4-2)

And that developed in the L. P. stage is,

The total power developed by the turbine is obtained by adding the power developed in the two stages.

∆PG = ∆PG, HP + ∆PG, LP

……………………(4-3)

When a sudden change in loading occurs, the governor turbine control system does not react instantaneously and the immediate requirement of extra power demand has to come from the inertia of the generator rotor. Consequently the generator decelerates and the frequency falls. As the frequency deviation is fed into the governor control system, the governor comes into action after its characteristic time delay and increases the output. At any point of time in the process the difference between load and generated power is made up in two ways.

  1. By decreasing the area kinetic energy Wkin at the rate 
  2. By a reduced load consumption caused by the reduction in frequency. This reduction is found empirically and is denoted by D, which is equal to   MW/HZ.

Thus the power balance equation can be written as,

……………………………….(4-4)

The kinetic energy of the rotor varies with the square of the speed or frequency of the area. Therefore,

……………………………………………(4-5)

Where f is the instantaneous frequency and WOkin is the initial kinetic energy of the area.

…………………………….(4-6)

Every term of the above equation is expressed in Megawatts. If we divide the equation by the area capacity PR, the power deviation terms and the term D will be given in per units.

Thus, equation becomes,

…………………………(4-7)

Or, ………………………………………(4-8)

Where, is the per unit inertia constant and has the dimension of seconds.

So equation (4-9) becomes,

……………………………………..(4-9)

To achieve a zero steady state error condition, an integral control has to be added to the system. This is achieved by making the speed changer signal ∆PC commanded by a signal obtained by first amplifying and then integrating the frequency error. Thus,

…………………………………………………..(4-10)

By derivative this equation we get,

………………………………………………….(4-11)

Where KI = 1/R is the integrator gain. The negative sign in equation (4-11) appears because a decrease in frequency gives rise to an increase command.

The two transfer function blocks for governor and turbine are in series, with a total gain of that portion of the block diagram equal to (KG. KT). It is convenient to set the two gains so that the combined gain KG. KT equals unity. Considering this setting we omit these two gain parameters from further analysis.

When the model of an LMES unit is added to the power system model, the input signal to the LMES unit is the frequency deviation of the power area and the power coming from the LMES unit is added in the summing junction of the signal area model.

The addition of the LMES power in the summing junction leads to the modification of equation (4-10) which becomes,

…………………………………(4-12)

Here ∆Pd , the per unit change in converter power is shown negative because the power into the inductor is assumed to be positive.

On manipulation (4-13) becomes,

………………………….(4-13)

Equation (4- 1), when written in state variable equation from becomes,

………………….(4-14)

As the turbine transfer function is not of first order, we define a dummy variable ∆DM as,

………………………………………(4-15)

This gives,

Or, ………………………..(4-16)

The dummy state variable brings equation (4-3) down to first order,

Or, ……………..(4-17)

The resulting systems of state variable equations defining the single area power system with an LMES unit are,

Setting K0 = 0 gives the system model without LMES unit.

4.2.2. The Uncontrolled Case of Single Area Power System

Uncontrolled case means ∆PC =0.So the integrator gain KI =0.That   means do not change the speed change portion.

 Speed regulation R=2.4;Hz/p.u MW

Power system gain KP=120

Power time constant TP=20s

Load increases ∆PL=.001(p.u) MW

Generator time constant TG=.08s

Turbine time constant TT=.3s

Integrating gain KT =0(uncontrolled case)

Fig: 4.2 frequency deviation following a step load change (uncontrolled case)

4.2.3. Controlled Case of Single Area Power System

It is necessary to achieve must better frequency constancy than is by the speed governor system itself, as demonstrated above. To accomplish this we must manipulate the speed changer in according with some suitable control strategy. It is necessary to settle for a set of control specifications the stringency of which will in the end determine the sophistication of the proposed control method.

We are presently discussing a single area power system and the control specifications for such a system are considerably simpler than those imposed upon a multiple area system.

Here follow some realistic specifications:

  1. The control loop must be characterized by a sufficient degree of stability.
  2. Following a step load change the frequency error should return to zero. This is referred to a specifications control. The magnitude of the transient frequency division should be minimizing.
  3. The integral of the frequency error should not exceed a certain maximum value.
  4. The individual generators of the control area should divide the load for optimum economy.

So, the equation of the frequency division of governor system is,

Chapter-3: MODELING OF ELECTRICAL POWER SYSTEM COMPONENTS

3.1. Introduction

 First consider the problem of controlling the real power output of electric generator within a prescribed area in response to changes in system frequency and the tie line loading or the relation of these to each other, so as to maintain the scheduled system frequency and the established interchange with other areas within pre-determined limits. This term automatic load frequency control (LFC) is also often used to identify this problem area [5, 6].

The operation objectives of the LFC are to maintain reasonably uniform frequency, to divide the load between generators, and to control the tie line interchanging schedules. The change in frequency and tie line power are sensed, which is measure of the change in rotor angle δ (i.e. the error ∆ δ) to be corrected. The error signal ∆f and ∆Ptie, are amplified, mixed and transformed into a real power command signal ∆PV, which is sent to the prime mover to call for an amount ∆PG which will change the values∆f and ∆Ptie within the specified tolerance.

The first step in analysis and design of control is mathematical modeling of the system. The two most common methods are the transfer function method and the state variable approach. The state variable approach can be applied to portray linear as well as nonlinear systems. In order to use the transfer function and linear state equation, the system must first be linearized. Proper assumptions and approximations are made to linearize the mathematical equations describing the system, and a transfer function model is for the following components [5].

3.2.1 Generator Model

Applying the swing equation of a synchronous machine given by to small perturbation, we have,

Or, in terms of small deviation in speed,

With speed expressed in per unit, without explicit per unit rotation, it can be written as,

The speed deviation is directly proportional to the system frequency deviation,

The above relation is shown in block diagram form:

Fig: 3.1 Generator block diagram [4]

3.2.2. Load model

The load on a system consists of a variety of electrical devices. For resistive loads, such as sighting and heating loads, the electrical power is independent of frequency [11]. Motor loads are resistive to changes in frequency. How sensitive it is to frequency depends on the composite of the speed-load characteristics of all the devices. The speed-load characteristic of a composite load is approximated by,

Where ∆PL is the nonfrequency-sensitive load change and D∆f is the frequency sensitive load change. D is expressed as the percent change in load divided by percent change in frequency [4, 5]. For example, if the load is changed by 1.6 percent a 1 percent change in frequency shown in figure below:

Fig: 3.2 (a) Combined Generator and Load block diagram

Fig: 3.2(b) Equivalent block diagram

The above model can also be modified by the following block diagram in fig: 3.2(b)

Fig: 3.2. (c) Modified Block diagram of the Combined Generator and Load
Fig: 3.2. Block diagram of the Combined Generator and Load

Where,

, is power system proportional gain value.

, is power system time constant.

 

3.2.3. Governor Model

When the generator electrical load is suddenly increased, the electrical power exceeds the mechanical power input. This power defiance is the kinetic energy store in the rotating system the reduction in kinetic energy causes the turbine speed and consequently, the generator frequency to fall. The change in speed is sensed by the turbine governor which acts to adjust the turbine input valve to change the mechanical power output to bring the speed to a new steady-state. The earliest governors were the watt governors which sense the speed by means of rotating fly balls and provide mechanical motion in response to speed changes. However most modern governors use electronic means to sense speed changes [4, 5]. Figure 3.3 shows mathematically the essential element of a conventional watt governor, which consist of the following major parts.

Fig: 3.3. Speed Governing System

Speed Governor: The essential parts are centrifugal fly balls driven directly or through gearing by the turbine shaft. The mechanism provides up ward and downward vertical movement proportional to the change in speed.

Linkage Mechanism: These are links for transforming the fly balls movement to the turbine valve through a hydraulic amplifier and providing a feedback from the turbine valve movement.

Hydraulic Amplifier: Very linkage mechanical forces are needed to operate the system valve. Therefore, the governor movements are transformed into high power factor via several stages hydraulic amplifiers.

Speed Changer:  The speed changer consists of servomotor that can be operated manually or automatically for scheduling load nominal frequency. By adjusting this set point, a desired load dispatch can be scheduled at nominal frequency.

For stable operation, the governors are designed to permit the speed to drop as the load is increased. The steady-state characteristic [4] of such a governor is shown in fig 3.3.

The slopes of the curve represent the speed regulation R. Governors have speed regulation of 5-6 percent from zero to full load. The speed governor mechanism acts as a comparator whose output ∆P is the difference between references set power ∆Pref  and the power, as given from the governor speed characteristics,

Define in s domain,

This can also written as,

The command ∆Pg is transformed through the hydraulic to the steam valve position command ∆Pv. Assuming a linear relationship and considering a simple time constantan Tg, it can be expressed by the following s-domain relation.

Fig: 3.4 .block diagram representation of speed governing system turbine

3.2.4. Turbine Models:

 The change in valve position ∆XE causes an incremental increase in turbine power, ∆PT, via the electromechanical interaction the generator, will result in an increased generator power ∆PG.

The overall mechanism is relatively complicated, particularly if the generator voltage simultaneously undergoes while swings due to major network disturbances. The voltage level is constant and the torque variations are of small size, then an incremental analysis of the type we performed for the speed governor, above will give a relatively simple dynamic relationship between ∆XE and ∆PG.   It use any type turbine single gain factor KT and a single time constant TT and thus write.