Reduced Size Rule Set Based Fuzzy Logic Dual Input Power System Stabilizer

Transcription

1 772 NATIONAL POWER SYSTEMS CONFERENCE, NPSC 2002 Reduced Size Rule Set Based Fuzzy Logic Dual Input Power System Stabilizer Avdhesh Sharma and MLKothari Abstract-- The paper deals with design of fuzzy logic dual input power system stabilizers (FL-DIPSS) considering seven and five Gaussian shaped membership functions (MFs) A step by step procedure for designing the different type of reduced size rule set based FL-DIPSS has been presented The performance of the SMIB system with reduced side rule set based FL-DIPSS is compared with that of full size rule set based FL-DIPSS Investigations reveal that the system with FL- DIPSS based on reduced size rule set of seven Gaussian shaped MFs, provides some what superior performance in comparison to others The robustness of the reduced size rule s et based FL- DIPSS for wide variations in line reactance and loading conditions is studied in detail Index Terms-- Dual input Power System Stabilizer, Fuzzy Logic System, Intelligent controllers, SMIB System I INTRODUCTION Fuzzy Logic Controllers (FLC) are suitable for systems that are structurally difficult to model due to naturally existing non-linearties and other model complexities This is because, unlike a conventional controller such as PID controller, rigorous mathematical model is not ruired to design a fuzzy logic controller FLC can also be implemented easily Power system is a highly nonlinear system and it is difficult to obtain exact mathematical model of the system In view of this during the last one decade, an attempt has been made to design and apply fuzzy logic controllers for effectively damping power system oscillations In contrast to a conventional PSS which is designed in the fruency domain, a fuzzy logic PSS is designed in the time domain A fuzzy controller determines the operating condition from the measured values and selects the appropriate control actions using the rule base created from the expert knowledge Depending on the system state, the controller operates in the range between no control action and full control action in an extremely nonlinear manner The fuzzy controller in itself has no dynamic component, ie, it can immediately perform the desired control action IEEE digital excitation committee [3] has presented PSS2B model of the dual input PSS (DIPSS) and its advantages are also discussed in detail Unlike the conventional PSS, the main advantage with DIPSS is that it does not excite any torsional mode However it has a fixed structure and provides the optimal performance at the operating condition for which it is designed and provides sub-optimal performance at Avdhesh Sharma is with Department of Electrical Engineering, MBM Engineering College, JNVUniversity, JODHPUR , INDIA ( avdhesh_2000@yahoocom) MLKothari is with Department of Electrical Engineering, Indian Institute of Technology, Delhi; Hauz Khas, New Delhi , INDIA ( mohankothari@hotmailcom) other operating condition To obtain better performance, fuzzy logic based dual input PSS is proposed in this paper The main objectives of the work presented in this paper are: 1 To present a systematic approach for designing a fuzzy logic dual input power system stabilizer (FL-DIPSS) 2 To analyze the dynamic performance of the system with FL-DIPSS and hence to compare it with that obtained using a conventional DIPSS (CDIPSS) 3 To design FL-DIPSS based on reduced size rule set and hence to analyze the performance of the system with such a FL-DIPSS 4 To study the dynamic performance of the system with FL-DIPSS considering wide variations in loading condition and line reactance X e II SYSTEM INVESTIGATED A single machine infinite bus (SMIB) system with synchronous generator provided with IEEE Type-ST1 static excitation system is considered III BASIC CONFIGURATION OF A FUZZY LOGIC CONTROLLER (FLC) The basic configuration of a FLC is presented in ref1 It comprises of four principal components: a fuzzification interface, a knowledge base, decision making logic and a defuzzification interface A DESIGN PARAMETERS OF A FLC The principal design parameters for a FLC are the following: 1 fuzzification strategies and the interpretation of a fuzzification operator (fuzzifier), 2 data base: a universes of discourse, b fuzzy partition of the input output spaces, c choice of the membership function of a primary fuzzy set; 3 rule base: a choice of process state (input) variables and control (output) variables of fuzzy control rules, b source and derivation of fuzzy control rules 4 decision making logic: a definition of fuzzy implication, b interpretation of the sentence connective and, c interpretation of the sentence connective also, d definition of a compositional operator, e inference mechanism; 5 defuzzification strategies and the interpretation of defuzzification operator (defuzzifier)

2 INDIAN INSTITUTE OF TECHNOLOGY, KHARAGPUR , DECEMBER 27-29, µ B Membership Function of A Primary Fuzzy Set A functional definition expresses the membership function of a fuzzy set in functional form, typically a Gaussianshaped, triangle-shaped, and trapezoid-shaped functions, etc Such functions are used in FLC because they lead themselves to manipulation through the use of fuzzy arithmetic The functional definition can readily be adapted to a change in the normalization of a universe Gaussian Membership Functions: Fig1 show an example of a functional definition of Gaussian membership functions with two parameters, expressed as: µ i ( x ) 2 ( x c i ) = exp 2 2 σ i ; i = 1,,n (1) where, n is the number of MFs, (c i,σ i ) is the parameter set of a Gaussian function NB NM NS ZO PS PM PB Fig1: Gaussian membership functions The linguistic labels of the primary fuzzy set are; negative big (NB), negative medium (NM), negative small (NS), zero (ZO), positive small (PS), positive medium (PM) and positive big (PB) C Knowledge Base The knowledge base of a fuzzy logic controller comprises of the following two major components 1 Database (a) Identification of Process Variables and their universes of discourse It includes the definition of scale mapping (b) Deduction of the fuzzy labels are used to classify measured values of each process variables and to define the fuzzy sets corresponding to these fuzzy labels over their respective universe of discourse 2 Rule Base A fuzzy system is characterized by a set of linguistic statements based on the expert knowledge The expert knowledge is usually in the form of if-then rules, which are easily implemented by fuzzy conditional statements in fuzzy logic The collection of fuzzy control rules that are expressed as fuzzy conditional statements form the rule base or the rule set of a FLC Fuzzy control rules are more conveniently formulated in linguistic rather than numerical terms The proper choice of process state variables and control variables is essential to characterization of the operation of a fuzzy system Furthermore, the selection of linguistic variables has a substantial effect on the performance of a FLC The experience and engineering knowledge play an important role during this selection stage Typically, the linguistic variables in the FLC are the state, state error, state error derivative, state error integral, etc Table 1 shows a typical decision (rule base) for a FL-DIPSS It has ω, and & ω as input signals and the stabilizing signal TABLE 1 DECISION TABLE WITH SEVEN MEMBERSHIP FUNCTIONS FOR EACH OF THE TWO INPUT SIGNALS ( ie, ω, & ω ) AND V S D Mamdani Fuzzy Inference System & ω ω A variety of Fuzzy Inference Systems are in vogue A commonly used Mamdani Fuzzy Inference system briefly explained considering following two fuzzy control rules; Rule1: IF x is A 1 AND y is B 1 THEN z is C 1 (2) Rule2: IF x is A 2 AND y is B 2 THEN z is C 2 (3) where, x and y denote input signals and z denotes the output Here fuzzy min operation, fuzzy min implication and max aggregation method have been used [4] E Defuzzification NB NM NS ZO PS PM PB NB NB NB NB NM NM NS ZO NM NB NB NM NM NS ZO PS NS NB NM NS NS ZO PS PM ZO NB NM NS ZO PS PM PB PS NM NS ZO PS PS PM PB PM NS ZO PS PM PM PB PB PB ZO PS PM PM PB PB PB To obtain a deterministic control action, a defuzzification strategy is ruired Basically, defuzzification is a mapping from a space of fuzzy control actions defined over an output universe of discourse into a space of non-fuzzy (crisp) control actions A defuzzification strategy is aimed at producing a non-fuzzy control action that represents best the possibility distribution of an inferred fuzzy control action The centre of area (COA) defuzzification technique is most prevalent and physically appealing of all the defuzzification method It is given by the algebraic expression: s z = (4) µ (z) dz s µ c (z)z c dz Where s denotes the support of µ c (z)

3 774 NATIONAL POWER SYSTEMS CONFERENCE, NPSC 2002 V ref v s IV ALGORITHM FOR DESIGNING FUZZY LOGIC DUAL INPUT POWER SYSTEM STABILIZER A step by step procedure for designing a FL-DIPSS is presented by considering seven Gaussian membership functions [1,2] The labels of the seven linguistic variables (Fig1) are Negative Big (NB), Negative Medium (NM), Negative Small (NS), Zero (ZO) Positive Small (PS), Positive Medium (PM), and Positive Big (PB) An universe of discourse, -3 to 3 is chosen Centre of area (COA) defuzzification technique is used Step1: Input Signals to Fuzzy Logic DIPSS : The transfer function model of the IEEE type PSS2B-dual input PSS is given in Ref 3 The input signals to this PSS are the speed deviation ω and terminal power deviation P e The type PSS2B dual input power system stabilizer may be considered as comprising two cascade connected blocks, ie (i) the processing block with P e and ω as input signals, while ω as the output signal and (ii) phase compensator or conventional PSS block, with ω as the input signal and stabilizing signal as output V t AVR Fuzzy Logic Controller ω & ω FL-DIPSS ω Ρ e Fig2: Schematic block diagram of a synchronous generator, excitation system and FL-DIPSS For the present investigations generator uivalent speed deviation ω and its derivative & ω are chosen as the input signals to the FL-DIPSS (Fig2) In practice, only uivalent speed deviation ω is readily derived from signal processing bock The can be derived from the & ω ω computed at two successive sampling instants, ie, [ ω (kt ) ω [(k 1)T]] ω& (kt ) = (5) T Step2: Selection of normalization factor: A scaling is done in order that the range of scaled values of input variables is spread over the complete universe of discourse (UOD) In order to obtain the scaling factors for the input signals ie, ω and, the dynamic performance of the system is & ω Terminal Voltage Transducer Exciter d dt obtained for different operating conditions considering 5% step increase in mechanical torque The maximum values Generator Signal Processing block To Power System obtained for the input signals ω and & ω are and respectively Thus, scaling factor for ω, 3 K = = 3750 ω 4 8 x 10 Scaling factor & ω for signal, 3 K ω& = = (6) (7) These values are kept fixed for all the investigations presented in this paper Step 3: Fuzzification: Fuzzification is the process of making a crisp quantity fuzzy Step 4: Fuzzy Rule Base: Fuzzy rule base is formed using the decision table and for seven membership functions, 49 rules are formed A typical rule has the following structure: Rule: IF ω is PM AND & ω is NS THEN Vs is PS (10) Step 4: Computation of fuzzy control signal: The control signal in the fuzzy form is obtained by applying Mamdanifuzzy inference system Since controller has two inputs, 4 rules shall be fired and corresponding to every rule there will be an output Step5: Defuzzification: Applying centre of area (COA) defuzzification method, crisp value of v s is obtained Step 6: Selection of Denormalization Factor : The output denormalization maps the point wise value of the control output onto its physical domain The dynamic performance of the system is now obtained for the nominal operating condition with the fuzzy logic PSS Fig3 shows the dynamic response for ω following a 5% increase in mechanical torque from its nominal value ie T m = 005 pu The dynamic response for ω of the system with conventional optimum DIPSS is also shown in the diagram for the purpose of comparison[4] The optimum parameters of the conventional DIPSS are K s1 =194437, T 1 = sec, and T 2 = 005 sec It is clearly seen that the dynamic performance of the system with FL-DIPSS is superior identical to that obtained with conventional DIPSS V REDUCED SIZE RULE SET BASED FL-DIPSS CONSIDERING SEVEN GAUSSIAN SHAPED MFs Investigations reported on fuzzy logic based PSS in the literature, have considered a rule set comprising of maximum possible number of rules (49 rules with seven membership functions for each of the two input signals) An attempt has now been made to design and investigate the performance of FL-DIPSS based on reduced size rule sets Studies are carried out considering following rule sets 1 a rule set comprising 19 rules as shown in Table-2 2 a rule set comprising 29 rules as shown in Table-3 3 a rule set comprising 37 rules as shown in Table-4 For each of the above cases optimum denormalization factor and σ are computed using ISE technique (Table-5) Table-5 clearly shows that the denormalization factor and σ decrease with increase in number of rules The corresponding

4 INDIAN INSTITUTE OF TECHNOLOGY, KHARAGPUR , DECEMBER 27-29, values of J min is also tabulated in Table-5 It can be seen that the J min is more or less of the same order for all the cases considered This implies that the dynamic performances of the system with FL-DIPSS based on reduced size rule sets are comparable to that obtained with FL-DIPSS based on a full size rule set This fact is further corroborated by plotting dynamic responses considering?t m =005pu of the system with FL-DIPSS based on rule sets of different sizes (Fig 4) Fig 3: Dynamic responses for ω at nominal loading condition with (a) Conventional dual input PSS (K S1 = , T 1 * = sec, and T 2 = 005 sec) (b) FL-DIPSS TABLE 2 DECISION TABLE WITH REDUCED SIZE RULE SET (19 RULES) WITH SEVEN MEMBERSHIP FUNCTIONS FOR EACH OF TWO INPUT SIGNALS (ie, ω, ω ) AND STABILIZING SIGNAL Vs ω ω P e = 10 pu, Q e = pu V t = 09 pu and X e = 065 pu FL-DIPSS Conventional DIPSS & NB NM NS ZO PS PM PB NB NS ZO NM NS ZO PS NS NS ZO PS - ZO - - NS ZO PS - - PS - NS ZO PS PM NS ZO PS PB ZO PS TABLE 3 DECISION TABLE WITH REDUCED SIZE RULE SET (29 RULES) WITH SEVEN MEMBERSHIP FUNCTIONS FOR EACH OF TWO INPUT SIGNALS (ie, ω, ω ) AND STABILIZING SIGNAL Vs ω ω & NB NM NS ZO PS PM PB NB NM NS ZO NM NM NS ZO PS NS - - NS NS ZO PS PM ZO - NM NS ZO PS PM - PS NM NS ZO PS PS - - PM NS ZO PS PM PB ZO PS PM TABLE 4 DECISION TABLE WITH REDUCED SIZE RULE SET (37 RULES) WITH SEVEN MEMBERSHIP FUNCTIONS FOR EACH OF TWO INPUT SIGNALS (ie, ω, ω ) AND STABILIZING SIGNAL Vs ω & NB NM NS ZO PS PM PB ω NB NM NM NS ZO NM - - NM NM NS ZO PS NS - NM NS NS ZO PS PM ZO NB NM NS ZO PS PM PB PS NM NS ZO PS PS PM - PM NS ZO PS PM PM - - PB ZO PS PM PM P e = 09 pu, Q e = pu V t = 10 pu and X e = 065 pu (a) a rule set of 19 rules a) 29 rules b) 37 rules c) 49 rules From the above investigations it may be inferred that with judiciously designed FL-DIPSS based on seven Gaussian MFs and reduced size rule sets, one can obtain dynamic performance comparable to that a FL-DIPSS based on full size rule set Fig 4: Dynamic responses for ω with Gaussian shaped MFs based FLDIPSS with different set of rules

5 776 NATIONAL POWER SYSTEMS CONFERENCE, NPSC 2002 TABLE 5 OPTIMUM DENORMALIZATION FACTOR, σ AND J min FOR RULE SETS OF DIFFERENT SIZES VI REDUCED SIZE RULE SET BASED FL-DIPSS CONSIDERING FIVE GAUSSIAN SHAPED MFs At this stage it is important to investigate whether it is permissible to reduce the size of the rule set for FL-DIPSS based on five Gaussian membership functions [4] An attempt has now been made to design FL-DIPSS based on reduced size rule set considering five Gaussian shaped membership functions Studies are carried out considering FL-DIPSS based on the following rule sets: 1 a rule set comprising 13 rules 2 a rule set comprising 19 rules 3 a rule set comprising 25 rules For each of the above cases optimum σ and denormalization factors are computed using ISE technique (Table-6) Table-6 clearly shows that the denormalization factor and σ decrease with increase in number of rules The corresponding value of J min is also tabulated in Table 6 It can be clearly seen that J min increases with reduction in number of rules from 25 Dynamic responses of the system are now obtained considering FL-DIPSS based on rule sets comprising 13, 19, and 25 rules (Fig5) Examination of Fig5 clearly shows that the dynamic responses of the system obtained with FL-DIPSS based on rule sets comprising 19 and 25 rules are virtually identical while that obtained with FL-DIPSS based on a rule set of 13 rules, is somewhat inferior Rule Set (number of rules) Denormalization Factor Fig5 Dynamic responses for ω with FL-DIPSS considering five Gaussian MFs with different set of rules From the above investigations it may be inferred that for designing FL-DIPSS based on five Gaussian shaped MFs, one can reduce the number of rules from 25 to 19 without compromising the dynamic performance σ J min x x x x10-5 P e = 09 pu, Q e = pu V t = 10 pu and X e = 065 pu (a) a rule set of 25 rules (b) 19 rules (c) 13 rules TABLE 6 OPTIMUM DENORMALIZATION FACTORS, σ AND J min FOR DIFFERENT SIZE RULE SETS Rule Set (Number of Rules) Denormalizat ion factor σ Jmin x x x10-5 VII ANALYSIS A Effect of Variation of Loading on Performance of FL- DIPSS At this stage, it is extremely important to assess the robustness of the FL-DIPSS to wide variations in loading condition and uivalent reactance X e The robustness of the following FL-DIPSS is examined 1 FL-DIPSS based on 7 Gaussian MFs with a rule set of 19 rules 2 FL-DIPSS based on 5 Gaussian MFs with a rule set of 25 rules 3 FL-DIPSS based on 5 Gaussian MFs with a rule set of 19 rules The above fuzzy logic dual input PSS are chosen for assessing their robustness since their performances were comparable at nominal operating conditions For simplicity of presentation the above FL-DIPSS shall henceforth be denoted by the nomenclature FLDIPSS-719, FLDIPSS-525 and FLDIPSS-519 respectively The dynamic performance of the system with FLDIPSS- 719, FLDIPSS-525, and FLDIPSS-519 are evaluated considering the following widely different loading conditions [4] For all these cases X e set ual to its nominal value (X e,= 065 pu) 1 P = 12 pu, Q = pu, and V t = 11 pu 2 P = 10 pu, Q = pu, and V t = 10 pu 3 P = 050 pu, Q = pu, and V t = 09 pu 4 P = 025 pu, Q = pu, and V t =09 pu The performance has also been evaluated considering wide variations in X e, ie, the operating condition characterized by 1 X e = 030 pu, Q e = pu, Vt = 08 pu, and P = 025 pu 2 X e = 040 pu, Q e = pu, Vt = 09 pu, and P = 025 pu 3 X e = 065 pu, Q e = pu, Vt = 10 pu, and P = 10 pu 4 X e = 085 pu, Q e = pu, Vt = 10 pu, and P = 10 pu The dynamic performances of the system with the above alternative structures of the FL-DIPSS are evaluated by plotting dynamic responses for ω considering a step increase in T m (ie, T m = 005 pu) Fig 6 shows the dynamic responses of the system for ω with FLDIPSS-719 for a wide variation in loading condition It is clearly seen that dynamic performance of the system with FLDIPSS-719 is quite robust to wide variation in loading condition Fig 7 shows the dynamic responses of the system for ω with

6 INDIAN INSTITUTE OF TECHNOLOGY, KHARAGPUR , DECEMBER 27-29, FLDIPSS-719 for wide variations in X e Examining Fig 7 it can be inferred that the FLDIPSS-719 is quite robust to wide variations in X e Fig 6: Dynamic responses for ω with FLDIPSS719 for different loading conditions P e = 12 pu, Q e = pu, V t = 11 pu P e = 10 pu, Q e = pu, V t = 10 pu P e = 050pu, Q e = pu,v t = 09 pu P e = 025 pu, Q e = pu, V t = 09 pu X e = 030 pu, Q e = pu, V t = 08 pu X e = 040 pu, Q e = pu, V t = 09 pu X e = 065 pu, Q e = pu,v t = 10 pu X e = 085 pu, Q e = pu, V t = 10 pu Fig7: Dynamic responses for ω with FLDIPSS719 for several values of X e Further, the comparison of responses, the dynamic performance of the system with FLDIPSS-719 is some what superior to those obtained with FLDIPSS-525 and FLDIPSS- 519 under wide variations in loading conditions The detailed investigations presented above reveal the following: 1 All the three FL-DIPSS are quite robust to the variation in loading condition and X e 2 The performance of FLDIPSS-719 is some what superior to those of FLDIPSS-519 and FLDIPSS-525 under wide variation in loading conditions In view of the above, it may be inferred that any one of the three structures may be chosen for practical implementation of FLDIPSS However, the FLDIPSS-719 (ie, FL-DIPSS based on reduced size rule set comprising 19 rules [Table 2]) may be preferred for realizing FLDIPSS, since its performance is somewhat better as compared to the other two Such a PSS is simple for practical implementation and fast in operation X CONCLUSIONS The following are the significant contributions of the research work presented in this chapter 1 A systematic approach for designing a Fuzzy Logic Dual Input Power System Stabilizer (FL-DIPSS) has been presented FL-DIPSS comprising different primary fuzzy sets, shapes of the membership functions and reduced size rule sets have been designed and their performances evaluated A systematic approach for tuning the parameters of FL-DIPSS using ISE technique has been presented 2 Studies also show that the proposed reduced size rule set based FL-DIPSS when appropriately designed exhibits robust dynamic performance comparable to those based on Full size rule set either with 7 or 5 primary fuzzy sets of Gaussian-shape 3 Investigations reveal that the dynamic performance of the system with FL-DIPSS is quite robust to wide variations in loading condition and line reactance X e XI ACKNOWLEDGMENT The authors gratefully acknowledge the financial support received AICTE (research project no R&D /2001-2/ ) which made it possible to conduct this research XII APPENDIX 1 The nominal system parameters and operating condition are: H =35 sec, T do =8 sec,x d =181 pu,x d =03 pu,xq=176 pu K A =400, T R =002 sec,t A =005 sec,t B =10 sec, T C = 80 sec P e =09 pu, Q e =03 pu, V t =10078 pu, X e = 065 pu XIII APPENDIX 2 The IEEE recommended settings of the processing block of the type PSS2B model of the dual input PSS are [3]: K s2 =099, K s3 =10, T 5 =0033 sec, T 6 =00 sec, T 7 =10 sec, T 8 = 05 sec, T 9 =01 sec, T 10 =00, n=1, m=5, T w1 = T w2 = T w3 = T w4 = 10 sec XIV REFERENCES [1] YYHsu, CHCheng, Design of Fuzzy Power system stabilizer for multimachine power system, IEEE proceedings, Vol 137 Part c, No 3 May (1990) [2] MAMHassan,OPMalik,GSHope, A Fuzzy logic based stabilzer for a synchronous machine IEEE Transactions on Energy Conversion, Vol 6, No3, September 1991 [3] IEEE Digital Excitaion System Sub-committee report, Computer models for representation of digital?based excitation systems, IEEE Transactions on Energy Conversion, Vol 11, No 3, September 1996, pp [4] Avdhesh Sharma, Artificial Neural Network and Fuzzy Logic System based Power System Stabilizers, PhD thesis, Indian Institute of Technology, Delhi, 2001