Chapter 7 Fuzzy Logic Controller

Transcription

1 Chapter 7 Fuzzy Logic Controller 7.1 Objective The objective of this section is to present the output of the system considered with a fuzzy logic controller to tune the firing angle of the SCRs present in the TCSC considered. The objective is to limit the usage of membership functions in the fuzzy block. This is because more the number of membership functions higher will be the time taken for simulation and more will be the memory storage capacity. The objective of this research is to design controllers which are fast in action and requires less memory space. The membership functions used are limited to 7 and the rules are 49. The input to the fuzzy logic controller is the measured impedance and the reference impedance the output is the controlling signal to the firing angle of the SCR in TCSC. 7.2 Basics of Fuzzy logic controller Fuzzy logic has evolved as one of the emerging information processing technologies, especially from the last few years. A rapid growth has been witnessed in the number and variety of applications of fuzzy logic. Fuzzy Logic was initiated in 1965 by Lotfi A. Zadeh, professor for computer science at the University of California in Berkeley. Basically, Fuzzy Logic (FL) is a multi valued logic, which allows intermediate values to be defined between conventional evaluations like true/false, yes/no, high/low etc. Fuzzy logic implements human experiences and preferences via membership functions and fuzy rules. Fuzzy membership functions can have different shapes depending on the designers preference and/or experience. The fuzzy rules, which describe relationships at a high level (in a linguistic sense), are typically written as antecedent consequent pairs of IF-THEN statements. Basically, there are four approaches to the developing fuzzy rules (1) Extract from expert experience and control engineering knowledge, (2) Observe the behavior of human operators, PET Research centre, PESCE, Mandya 173

2 (3) Use a fuzzy model of a process, and (4) Learn relationships through experience or simulation with a learning process. These approaches do not have to be mutually exclusive. Due to the use of linguistic variables and fuzzy rules, the system can be made understandable to a nonexpert operator. In this way, fuzzy logic can be used as a general methodology to incorporate knowledge, heuristics or theory into controllers and decision-makers Fuzzification Fuzzification is a process whereby the input variables are mapped into fuzzy variables. The fuzzy input variables considered in this thesis are line flow before compensation (P line ) and change in line flow after series compensation ( P line ). The membership functions of input variables are shown. To relieve congestion, the location for placement of TCSC is considered as a major issue. Hence, TCSC can be placed where the low power loss occurs in the line. Therefore the change in power loss ( P loss ) is taken as an output variable. The membership function of output variable is shown and also the fuzzy variables for the test case has been considered Fuzzy Sets A fuzzy set is a set of ordered pairs with each containing an element and the degree of membership for that element. A higher membership value indicates that an element more closely matches the characteristic feature of the set. Zadeh proposed the concept of a fuzzy set. Fuzzy sets are functions that map a value that might be a member of the set to a number between zero and one indicating its actual degree of membership. A degree of zero means that the value is not in the set, and a degree of one means that the value is completely representative of the set. This produces a curve across the members of the set. The basis of the technology is a fuzzy set that is an extension of the classical set. In traditional set theory, membership of an object belonging to a set can only be one of the two values: 0 or 1. An object either completely belongs to a set or does not at all. No partial membership is allowed. PET Research centre, PESCE, Mandya 174

3 A fuzzy set is an extension of a crisp set. Crisp sets allow only full membership or no membership at all, whereas fuzzy sets allow partial membership. A membership function is essentially a curve that defines how each point in the input space is mapped to a membership value (or degree of membership) between 0 and 1. Fig 7.1 Crisp Membership Function Fig. 7.2 An Example of a Fuzzy Membership Function Various types of membership functions are used, including triangular, trapezoidal, generalized bell shaped, Gaussian curves, polynomial curves, and sigmoid functions. PET Research centre, PESCE, Mandya 175

4 An MF can have different shapes, as shown in Fig 7.3. The general classification of MFs are as follows Fig. 7.3 Different Types of Membership Functions Fuzzy set operations are analogous to crisp set operations. The important thing in defining fuzzy set logical operators is that if we keep fuzzy values to the extremes 1 (True) or 0 (False), the standard logical operations should hold. In order to define fuzzy set logical operators, let us first consider crisp set operators. The most elementary crisp PET Research centre, PESCE, Mandya 176

5 set operations are union, intersection, and complement, which essentially correspond to OR, AND, and NOT operators, respectively. In FL, the truth of any statement is a matter of degree. In order to define FL operators, we have to find the corresponding operators that preserve the results of using AND, OR, and NOT operators. The answer is min, max, and complement operations. Most applications use min for fuzzy intersection, max for fuzzy union, and complement for complementation. We have to remember that operators used in FL, such as union, intersection, and complement, reduce to their crisp logic counterparts when the membership functions are restricted to 0 or 1. The graphical representation of union, intersection and compliment is shown in figure 7.4. Fig. 7.4 The Graphical Representation of Union, Intersection and Complement PET Research centre, PESCE, Mandya 177

6 7.2.3 Graphical Examples of Containment, Union, Intersection and Complement Union (Disjunction).The union of two fuzzy sets A and B is a fuzzy set C, written as C=A B or C=A OR B, whose MF is related to those of A and B by: µ C (x) = max(µ A (x),µ B (x)) Intersection (Conjunction). The intersection of two fuzzy sets A and B is a fuzzy set C, written as C = A B or C =A AND B. whose MF is related to those of A and B by: µ C (x) = min(µ A (x),µ B (x)) Complement (Negation).The complement of fuzzy set A, denoted by Ā or NOT A is defined as µ Ā (x)= 1-µ A (x) Fuzzy inference systems consist of if then rules that specify a relationship between the input and output fuzzy sets. Fuzzy relations present a degree of presence or absence of association or interaction between the elements of two or more sets [80]. 7.3 Fuzzy Inference System A fuzzy inference system (FIS) essentially defines a nonlinear mapping of the input data vector into a scalar output, using fuzzy rules. The mapping process involves input/output membership functions, FL operators, fuzzy if then rules, aggregation of output sets, and defuzzification. CRISP INPUT FUZZIFIER INFERENCE DEFUZZIFICATION CRISP OUTPUT RULE BASE Fig 7.5 Block Diagram of a Fuzzy Inference System. An FIS with multiple outputs can be considered as a collection of independent multi input, single-output systems. A general model of a fuzzy inference system (FIS) is shown in above figure 7.5.The FLS maps crisp inputs into crisp outputs. It can be seen PET Research centre, PESCE, Mandya 178

7 from the figure that the FIS contains four components: the fuzzifier, inference, rule base, and defuzzifier. Rule Base: The rule base contains linguistic rules that are provided by experts. It is also possible to extract rules from numeric data. Once the rules have been established, the FIS can be viewed as a system that maps an input vector to an output vector. The fuzzifier maps input numbers into corresponding fuzzy memberships. This is required in order to activate rules that are in terms of linguistic variables. The fuzzifier takes input values and determines the degree to which they belong to each of the fuzzy sets via membership functions. The inference defines mapping from input fuzzy sets into output fuzzy sets. It determines the degree to which the antecedent is satisfied for each rule. If the antecedent of a given rule has more than one clause, fuzzy operators are applied to obtain one number that represents the result of the antecedent for that rule. It is possible that one or more rules may fire at the same time. Outputs for all rules are then aggregated. During aggregation, fuzzy sets that represent the output of each rule are combined into a single fuzzy set. Fuzzy rules are fired in parallel, which is one of the important aspects of an FIS. In an FIS, the order in which rules are fired does not affect the output. The defuzzifier maps output fuzzy sets into a crisp number. Given a fuzzy set that encompasses a range of output values, the defuzzifier returns one number, thereby moving from a fuzzy set to a crisp number. Several methods for defuzzification are used in practice, including the centroid, maximum, mean of maxima, height, and modified height defuzzifier. The most popular defuzzification method is the centroid, which calculates and returns the center of gravity of the aggregated fuzzy set. FISs employ rules. However, unlike rules in conventional expert systems, a fuzzy rule localizes a region of space along the function surface instead of isolating a point on the surface. For a given input, more than one rule may fire. Also, in an FIS, multiple regions are combined in the output space to produce a composite region. PET Research centre, PESCE, Mandya 179

8 7.4 Defuzzification A fuzzy inference system maps an input vector to a crisp output value. In order to obtain a crisp output, we need a defuzzification process. The input to the defuzzification process is a fuzzy set (the aggregated output fuzzy set), and the output of the defuzzification process is a single number. Many defuzzification techniques have been proposed in the literature. The most commonly used method is the centroid. Other methods include the maximum, the means of maxima, height, and modified height method. The five methods may be described as follows: (a) Centroid Defuzzification Method: In this method, the defuzzifier determines the center of gravity (centroid) and uses that value as the output of the FLS. The centroid defuzzification method finds the balance point of the solution fuzzy region by calculating the weighted mean of the output fuzzy region. It is the most widely used technique because, when it is used, the defuzzified values tend to move smoothly around the output fuzzy region. The technique is unique, however, and not easy to implement computationally. (b) Maximum-Decomposition Method: In this method, the defuzzifier examines the aggregated fuzzy set and chooses that output y for which is the maximum method has some properties that are applicable to a narrower class of problems. The output value for this method is sensitive to a single rule that dominates the fuzzy rule set. In addition, the output value tends to jump from one frame to the next as the shape of the fuzzy region changes. (c) Center of Maxima: In a multimode fuzzy region, the center-of-maxima technique finds the highest plateau and then the next highest plateau. The midpoint between the centers of these plateaus is selected. (d) Height Defuzzification: In this method, the defuzzifier first evaluates at and then computes the output of the FLS, where denotes the center of gravity of fuzzy sets. This technique is easy to use because the centers of gravity of commonly used membership functions are known ahead of time. Regardless of whether minimum or product inference is used, the fuzzy inference process essentially defines the mapping of the given vector of crisp values to an output crisp value using fuzzy rules stored in the knowledge base. PET Research centre, PESCE, Mandya 180

9 The fuzzy inference process just discussed is known as Mamdani s fuzzy inference method. Sugeno (1977) suggested a fuzzy inference method that is similar tomamdani s. In Sugeno s method, the first two parts, namely, mapping inputs tofuzzy membership functions and applying same fuzzy operators. In this research work centroid defuzzification method has been used and is explained above. A fuzzy control system is a control system based on fuzzy logic a mathematical system that analyzes analog input values in terms of logical variables that take on continuous values between 0 and 1, in contrast to classical or digital logic, which operates on discrete values of either 1 or 0 (true or false, respectively). Fig. 7.6 Block diagram of fuzzy logic controller Fuzzy sets can be applied to many applications of power systems. This section discusses the fuzzy method used for proper selection of TCSC impedance to enhance stability of the power system. The Fuzzy used in this research uses two inputs the reference impedance and the measured impedance and produces an output proportional to the difference between the impedances, that is the impedance error (Zreference Zmeasured). The output produces a triggering pulse which tunes the firing angle of the SCR present in the TCSC which in turn modifies the impedance of the line. The rules for the Fuzzy controller are formed taking 7 different values of the impedance Totally 49 set of rules are designed. An intelligent controller Fuzzy logic controller is used in this research work. The following block diagram shows the actual model of Fuzzy logic controller used in this research work. PET Research centre, PESCE, Mandya 181

10 7.5 Simulink Model of System Under Consideration with Expanded Blocks Fig 7.7 SIMULINK model of the system considered Fig 7.8 Control system model of the system with fuzzy logic controller. PET Research centre, PESCE, Mandya 182

11 Fig 7.9 Simulink model of the impedance calculation Fig 7.10 Firing circuit model used in the system PET Research centre, PESCE, Mandya 183

12 Fig 7.11 TCSC used in the model Fig 7.12 TCSC used in the model PET Research centre, PESCE, Mandya 184

13 Fig 7.13 Scopes used for plotting the waveforms in the SIMULINK model PET Research centre, PESCE, Mandya 185

14 Fig 7.14 (a) Data entered for Fig 7.14 (b) Data entered for programmable voltage source 1. programmable voltage source 2 Fig 7.14(c) parameter entered for Z reference Fig 7.14(d) parameter entered for transmission line PET Research centre, PESCE, Mandya 186

15 Fig 7.14 (e) Entry for measuring voltage and current Fuzzy Logic Matlab Details Double click open input variable icon to open the Membership Function Editor. Double click on the system diagram to open the Rule Editor The name of the system is displayed here. It can be changed using one of the Save as.. Menu option Double click on the icon for the output variable to open the Membership Function Editor. These pop-up menus are used to adjust the fuzzy inference functions, such as the defuzzification method. The edit field is used to name and edit the names of the input and output variable. This status line describes the most recent operation. Fig 7.15(a) FIS Editor in general PET Research centre, PESCE, Mandya 187

16 This is the Variable palette area. Click on a variable here to make it current and edit its membership function. This graph field display all the membership Functions of the current variable. Click on a line to select it and you can change any of its attributes, including name,type and numerical parameter. Drag your mouse to move or change the shape of the selected membership function. These text fields display the name and type of the current variable. This edit field lets you set the display range of the current variable. This edit field lets you set the display range of the current plot. This status line describes most recent operation. This edit field lets you change the name of current membership function. This pop-up menu lets you change the type of the current membership function. This edit field lets you change the numerical parameters for the current membership function. Fig 7.15(b) Membership Function Editor Fig 7.15(c) FIS editor used in this research PET Research centre, PESCE, Mandya 188

17 Fig 7.15 (d) Membership function editor used for input Fig 7.15 (e) Membership function editor for the output PET Research centre, PESCE, Mandya 189

18 Fig 7.15 (f) Rule editor with explanation Fig 7.15 (g) Rule editor used in this research PET Research centre, PESCE, Mandya 190

19 Each column of plots (yellow) shows how the input variable is used in the rules. The input values are shown here at the top This column of the plots (blue) shows how the output variable is used in the rules. Each rows of plots represents one rule(here there are 3). Click on a rule to display it in the status bar. Slide this line to change your input values, and generate a new output response. This edit field allows you to set the input explicitly. This line provides a defuzzified value. The button right plot shows how the output of each rule is combined to make an aggregate output and then defuzzified. This status line describes the most recent operation. Fig 7.15 (h) Rule viewer with explanation To implement fuzzy logic technique to a real application requires the following three steps: 1. Fuzzification convert classical data or crisp data into fuzzy data or Membership Functions (MFs) 2. Fuzzy Inference Process combine membership functions with the control rules to derive the fuzzy output 3. Defuzzification use different methods to calculate each associated output and put them into a table: the lookup table. Pick up the output from the lookup table based on the current input during an application A fuzzy inference system maps an input vector to a PET Research centre, PESCE, Mandya 191

20 crisp output value. In order to obtain a crisp output, we need a defuzzification process. The input to the defuzzification process is a fuzzy set (the aggregated output fuzzy set), and the output of the defuzzification process is a single number. Many defuzzification techniques have been proposed in the literature. The most commonly used method is the centroid. In the second step, to begin the fuzzy inference process, one need combine the Membership Functions with the control rules to derive the control output, and arrange those outputs into a table called the lookup table. The control rule is the core of the fuzzy inference process, and those rules are directly related to a human being s intuition and feeling. For example, still in the air conditioner control system, if the temperature is too high, the heater should be turned off, or the heat driving motor should be slowed down, which is a human being s intuition or common sense. Different methods such as Center of Gravity (COG) or Mean of Maximum (MOM) are utilized to calculate the associated control output, and each control output should be arranged into a table called lookup table. Other methods include the maximum, the means of maxima, height, and modified height method. During an actual application, a control output should be selected from the lookup table developed from the last step based on the current input. Furthermore, that control output should be converted from the linguistic variable back to the crisp variable and output to the control operator. This process is called defuzzification The fuzzy set is a powerful tool and allows us to represent objects or members in a vague or ambiguous way. The fuzzy set also provides a way that is similar to a human being s concepts and thought process. However, just the fuzzy set itself cannot lead to any useful and practical products until the fuzzy inference process is applied. To implement fuzzy inference to a real product or to solve an actual Fundamentals of Fuzzy Logic Control problem, three consecutive steps are needed, which are: Fuzzification, fuzzy inference and defuzzification. Fuzzification is the first step to apply a fuzzy inference system. Most variables existing in the real world are crisp or classical variables. One needs to convert those crisp variables (both input and output) to fuzzy variables, and then apply fuzzy inference to process those data to obtain the desired output. Finally, in most cases, those fuzzy outputs need to be converted back to crisp variables to complete the desired control objectives. PET Research centre, PESCE, Mandya 192

21 Generally, fuzzification involves two processes: derive the membership functions for input and output variables and represent them with linguistic variables. This process is equivalent to converting or mapping classical set to fuzzy set to varying degrees. In practice, membership functions can have multiple different types, such as the triangular waveform, trapezoidal waveform, Gaussian waveform, bell-shaped waveform, sigmoidal waveform and S-curve waveform. The exact type depends on the actual applications. For those systems that need significant dynamic variation in a short period of time, a triangular or trapezoidal waveform should be utilized. For those system that need very high control accuracy, a Gaussian or S-curve waveform should be selected. The figure 7.7 shows the system considered with TCSC and a fuzzy logic controller. The input to the fuzzy controller block is the Zerror (Zreference Z measured). The Fuzzy logic controller based on the rules provided gives the output which is the angle alpha, the angle of firing instant to the SCR present in the TCSC. Based on this firing angle the TCSC impedance changes which are the measured impedance. This measured impedance is compared with the reference impedance and the error that is the Zerror is again compared with Fuzzy logic controller and the process continues until the Zerror is minimized. The fuzzy control system design is based on empirical methods, basically a methodical approach to trial and error. The general process is as follows: Document the system s operational specifications and inputs and outputs. Document the fuzzy sets for inputs. Document the rule set. Run through test suite to validate system, adjust details as required. Complete document and release to production. In this work for both inputs (Zreference and Zmeasured, Zerror) and output alpha six subsets have been used. They are PVS positive very small PMS- positive medium small PS- positive small PB- positive big PMB- positive medium big PVB-Positive very big PET Research centre, PESCE, Mandya 193

22 Fuzzy Control Rules: Fuzzy control rule can be considered as the knowledge of an expert in any related field of application. The fuzzy rule is represented by a sequence of the form IFTHEN, leading to algorithms describing what action or output should be taken in terms of the currently observed information, which includes both input and feedback if a closed-loop control system is applied. The law to design or build a set of fuzzy rules is based on a PET Research centre, PESCE, Mandya 194

23 human being s knowledge or experience, which is dependent on each different actual application. A fuzzy IF-THEN rule associates a condition described using linguistic variables and fuzzy sets to an output or a conclusion. The IF part is mainly used to capture knowledge by using the elastic conditions, and the THEN part can be utilized to give the conclusion or output in linguistic variable form. This IF-THEN rule is widely used by the fuzzy inference system to compute the degree to which the input data matches the condition of a rule. Two types of fuzzy control rules are widely utilized for most real applications. One is fuzzy mapping rules and the other is called fuzzy implication rules. In the beginning the measured impedance, reference impedance and error impedance will be converting to fuzzy variables. After this fuzzification, fuzzy inputs enter to inference mechanism level and with considering membership function and rules; outputs are sent to defuzzification to calculate the final output. After evaluating inputs and applying them to the rule base, a control signal will be generated by the fuzzy logic controller. The output variable of the inference system are linguistic variables. They will be evaluated for the derivation of the output signal. This process is the defuzzification. The defuzzification is achieved using the centre of gravity (COG) method and output of the fuzzy coordinated controller is Centre of gravity (set of real numbers). COG(A) = Where Xmin = 1 X max = 128 A(X) = alpha X = membership function In this method AND relationship between mappings of two variables are considered. Analysis of the result: The reference impedance is set as 0 secs to 2.5 secs 128 ohms 2.5 secs to 5 secs 121 ohms. Sending end voltage = KV Receiving end voltage=439.4 KV Steady state is not reached within 5 seconds. PET Research centre, PESCE, Mandya 195

24 Fig 7.16 Waveforms for Fuzzy logic controller X axis time in seconds, Y axis-top, power in MW, bottom firing angle of TCSC in degrees 7.6 Conclusion The following conclusions can be drawn from the above analysis. The advantage of the fuzzy logic controller is that the variations of the alpha, the firing angle to SCRs in the TCSC present can be varied depending on the number of membership functions and the control rules. In this research 7 membership functions and 49 set of rules have been considered. The time of evaluation is restricted to 5 seconds. As the rest of the analysis in this research is carried out for this time duration only. Even though Fuzzy has many advantages, with the restrictions on to the number of membership functions and the rules in this work, it can be observed from the waveform that there is a lot of fluctuations in the output waveform of power output and the impedance also varies a lot. Corresponding to this there are variations in the firing angle alpha also. With increase in membership functions and rules for the same time duration or increasing the time will help to get smooth variations of the output. PET Research centre, PESCE, Mandya 196