A fuzzy knowledge-based system for bus load ...

CONTENTS

Foreword Preface L. Zadeh Introduction R.J. Marks II Suggested Additional Reading list

vii xvii xix xxi

Section I: Technology Chapter 1: Overviews

1

Selected Papers: Introduction to Fuzzy Models 1.1 "Editorial: Fuzzy Models - What are they, and why?" J.C. Bezdek

3

Consumer Products 1.2 "Application of Neural Networks and Fuzzy Logic to Consumer Products" Hideyuki Takagi

8

Vehicular Technology 1.3 "Fuzzy Logic Technology & the Intelligent Highway System (IHS)" Bruno Bosacchi and Ichiro Masaki

13

Control 1.4 "Twenty Years of Fuzzy Control: Experiences Gained and Lessons Learnt" E. H. Mamdani

19

Computer Vision 1.5 "Fuzzy Set Theoretic Approach to Computer Vision: An Overview" Raghu Krishnapuram and James M. Keller

25

Image Processing and Recognition 1.6 "Fuzzy Sets in Image Processing and Recognition" S.K. Pal

33

IX

Section II: Applications Chapter 2: Vehicular Technology

41

Selected Papers: 2.1 "Trainable Fuzzy and Neural-Fuzzy Systems for Idle-Speed Control" L. A. Feldkamp and G. V. Puskorius

43

2.2 "Evaluation of Fuzzy and Neural Vehicle Control" Jos Nijhuis, Stefan Neuber, Jurgen Heller, Jochen Sponnemann

50

2.3 "Follow-up Characteristics of a Small Automatic Guided Vehicle System with Fuzzy Control "Nobuhiko Yuasa, Mitsuo Shioya and Gunji Kimura

56

2.4 "Self Organizing Fuzzy Logic Control of a Level Control Rig" Nader Vijeh

62

2.5 "Fuzzy Logic Anti-Lock Brake System for a Limited Range Coefficient of Friction Surface" D.P. Madau, F. Yuan, L.I. Davis, Jr., and L.A. Feldkamp

68

2.6 "Intelligent Cruise Control with Fuzzy Logic" Rolf Muller and Gerhard Nocker

74

2.7 "Design of a Rule-Based Fuzzy Controller for the Pitch Axis of an Unmanned Research Vehicle" Deepak Sabharwal and Kuldip S. Rattan 81 2.8 "Fuzzy Expert System for Automatic Transmission Control" H.G. Weil, G. Probst and F. Graf

88

2.9 "Application of Fuzzy Logic to Shift Scheduling Method for Automatic Transmission" S. Sakaguchi, I. Sakai and T. Haga

94

2.10 "Adaptive Traffic Signal Control Using Fuzzy Logic" Stephen Chiu and Sujeet Chand

101

2.11 "A Control System of Carburization Using Fuzzy-PID Combined Controller" Liya Hou and Zhengqing Wang

107

2.12 "Fuzzy Control for Active Suspension Design" Edge C. Yeh and Yon J. Tsao

109

Chapter 3: Robotics

115

Selected Papers: 3.1 "Fuzzy Controlled Gait Synthesis for a Biped Walking Machine" Luis Magdalena and Feliz Monasterio

117

3.2 "A Fuzzy Logic Force Controller for a Stepper Motor Robot" J.G. Hollinger, R.A. Bergstrom and J.S. Bay

123

x

3.3 "Hierarchical Intelligent Control for Robotic Motion by Using Fuzzy, Artificial Intelligence, and Neural Network" Toshio Fukuda and Takanori Shibata

129

3.4 "Hierarchical Control for Autonomous Mobile Robots with Behavior-Decision Fuzzy Algorithm" Yoichiro Maeda, Minoru Tanabe, Morikazu Yuta and Tomohiro Takagi

135

3.5 "Fuzzy Navigation of a Mobile Robot" Kai-Tai Song and Jen-Chau Tai

141

3.6 "Blending Reactivity and Goal-Directedness in a Fuzzy Controller" Alessandro Saffiotti, Enrique H. Ruspini and Kurt Konolige

148

3.7 "Fuzzy Logic Based Robotic Arm Control" Robert N. Lea, Jeffrey Hoblit, and Yashvant Jani

154

3.8 "Robotic Deburring Based on Fuzzy Force Control" M.H. Liu

160

3.9 "Manipulator for Man-Robot Cooperation (Control Method of Manipulator/Vehicle System with Fuzzy Inference)" Yoshio Fujisawa, Toshio Fukuda, Kazuhiro Kosuge, Fumihito Arai, Eiji Muro, Haruo Hoshino, Takashi Miyazaki, Kazuhiko Ohtsubo and Kazuo Uehara

168

Chapter 4: Motors, Servos, and Drives

175

Selected Papers: 4.1 "Adaptive Fuzzy Control of High Performance Motion Systems" E. Curruto, A. Consoli, A. Raciti and A. Testa

177

4.2 "Fuzzy Algorithm for Commutation of Permanent Magnet AC Servo Motors without Absolute Rotor Position Sensors" Dong-II Kim, Jin-Won Lee and Sungkwun Kim

184

4.3 "Fuzzy Logic-Based Control of Flux and Torque in AC-Drives" Wilfried Hofmann and Michael Krause

190

4.4 "Adaptive Fuzzy Techniques for Slip-Recovery Drive Control" L.E. Borges da Silva, G. Lambert-Torres, V. Ferreira da Silva and K. Nakashima

196

4.5 "Fuzzy Controller for Inverter Fed Induction Machines" Sayeed A. Mir, Donals S. Zinger and Malik E. Elbuluk

204

4.6 "A Fuzzy Current Controller for Field-Oriented Controlled Induction Machine by Fuzzy Rule" Seong-Sik Min, Kyu-Chan Lee, Jhong-Whan Song and Kyu-Bock Cho

212

XI

Chapter 5: Power Systems

219

Selected Papers: 5.1 "A Fuzzy Knowledge-Based System for Bus Load Forecasting" H.G.Lambert-Torres, L.E. Borges da Silva, B. Valiquette, H. Greiss and D. Mukhedkar

221

5.2 "A Symptom-Driven Fuzzy System for Isolating Faults" Jiann-Liang Chen, Ronlon Tsai and Huan-Wen Tzeng

229

5.3 "Comparison of Fuzzy Logic Based and Rule Based Power System Stabilizer" Juan Shi, L. H. Herron and A. Kalam

233

5.4 "Analysis of Power System Dynamic Stability via Fuzzy Concepts" Pei-Hwa Huang

239

Chapter 6: Industry Applications

245

Selected Papers: 6.1 "Application of Neuro-Fuzzy Hybrid Control System to Tank Level Control" Tetsuji Tani, Shunji Murakoshi, Tsutomu Sato, Motohide Umano and Kazuo Tanaka

247

6.2 "Minimization of Combined Sewer Overflows Using Fuzzy Logic Control" Sheng-Lu Hou and N. Lawrence Ricker

253

6.3 "Identification and Analysis of Fuzzy Model for Air Pollution - An Approach to Self-Learning Control of CO Concentration" Kazuo Tanaka, Manabu Sano and H. Watanabe

261

6.4 "Self-Learning Fuzzy Modeling of Semiconductor Processing Equipment" Raymond L. Chen and Costas J. Spanos

267

6.5 "Range Tests Made Fuzzy: An Alternate Perspective on the Built-in-Test of Real Time Embedded Systems" Jennifer D. Brown and George J. Klir

274

6.6. "Fuzzy Seam-Tracking Controller" Yoshito Sameda

280

6.7 "Neural Network Based Decision Model Used for Design of Rural Natural Gas Systems" W. Pedryczk, J. Davidson and I. Goulter

285

6.8 "Application of Fuzzy Control System to Hot Strip Mill" Naoki Sato, Noriyuki Kamada, Shuji Naito, Takashi Fukushima and Makoto Fujino

293

6.9 "A Fuzzy Classification Technique for Predictive Assessment of Chip Breakability for Use in Intelligent Machining Systems" J. Fei and I.S. Jawahir

298

Xll

6.10 "A Rule-Based Fuzzy Logic Controller for a PWM Inverter in Photo-Voltaic Energy Conversion Scheme" Rohin M. Hilloowala and Adel M. Sharaf

304

6.11 "Fuzzy Control of Wire Feed Rate in Robot Welding" Satoshi Yamane, Guillermo Alzamora and Takefumi Kubota

312

Chapter 7: Electronics

317

Selected Papers: 7.1 "Autonous Navigation of a Mobile Robot Using Custom-Designed Qualitative Reasoning VLSI Chips and Boards" Francois G. Pin, Hiroyuki Wantanabe, Jim Symon and Robert S. Pattay 319 7.2 "Fuzzy Control of an Industrial Robot in Transputer Environment" Jarmo Franssila and Heikki N. Koivo

325

7.3 "Fuzzy Logic Approach to Placement Problem" Rung-Bin Lin and Eugene Shragowitz

331

7.4 "A Fuzzy Algorithm for Multiprocessor Bus Arbitration" Robert T. Tran, Timothy R. Slator and Anaikuppam R. Marudarajan

337

Chapter 8: Sensors

343

Selected Papers: 8.1 "Improving Dynamic Performance of Temperature Sensors with Fuzzy Control Technique" Wang Lei and Volker Hans

345

8.2 "Multi-Sensor Integration System Based on Fuzzy Inference and Neural Network for Industrial Application" Toshio Fukuda, Koji Shimojima, Fumihito Arai and Hideo Matsuura

347

8.3 "A Fuzzy Logic Approach for Handling Imprecise Measurements in Robotic Assembly" H.B. Gurocak and A. de Sam Lazaro

353

Chapter 9: Aerospace

361

Selected Papers: 9.1 "Space Shuttle Attitude Control by Reinforcement Learning and Fuzzy Logic" Hamid R. Berenji, Robert N. Lea, Yashvant Jani, Pratap Khedkar, Anil Malkani and Jeffrey Hoblit

363

9.2 "Intelligent Control of a Flying Vehicle Using Fuzzy Associative Memory System" T. Yamaguchi, K. Goto, T. Takagi, K. Doya and T. Mita

369

X1U

9.3 "Development and Simulation of an F/A-18 Fuzzy Logic Automatic Carrier Landing System" Marc Steinberg

380

Chapter 10: Communications

387

Selected Papers: 10.1 "An RLS Fuzzy Adaptive Filter with Application to Nonlinear Channel Equalization" Li-Xin Wang and Jerry M. Mendel

389

10.2 "Model-Reference Neural Color Correction for HDTV Systems Based on Fuzzy Information Criteria" Po-Rong Chang and C. C. Tai

395

10.3 "Classified Vector Quantization Using Fuzzy Theory" Ferran Marques and C.C. Jay Kuo

401

Chapter 11: Bioengineering

409

Selected Papers: 11.1 "Fuzzy Control of Blood Pressure During Anesthesia with Isoflurane" R. Meier, J. Nieuwland, S. Hacisalihzade, D. Steck and A. Zbinden

411

11.2 "Real-Time Fuzzy Control of Mean Arterial Pressure in Post-surgical Patients in an Intensive Care Unit" Hao Ying, Michael McEachern, Donald W. Eddleman and Louis C. Sheppard

418

11.3 "Fuzzy ARTMAP Neural Network Compared to Linear Discriminant Analysis Prediction of the Length of Hospital Stay in Patients with Pneumonia" Philip H. Goodman, Vassilis G. Kaburlasos and Dwight D. Egbert

424

11.4 "Fuzzy Classification of Heart Rate Trends and Artifacts" Dean F. Sittig, Kei-Hoi Cheung and Lewis Berman

430

Chapter 12: Image Processing and Recognition

441

Selected Papers: 12.1 "Application of the Extended Fuzzy Pointing Set to Coin Grading" P.A. Laplante, D. Sinha and C.R. Giardina

443

12.2 "Region Extraction for Real Image Based on Fuzzy Reasoning" Kohi Miyajima and Toshio Norita

447

12.3 "A Fuzzy Approach to Scene Understanding" Weijing Zhang and Michio Sugeno

455

xiv

Chapter 13: Pattern Recognition

461

Selected Papers: 13.1 "Recognition of Facial Expressions Using Conceptual Fuzzy Sets" Hirohide Ushida, Tomohiro Takagi and Torn Yamagugchi

463

13.2 "Qualitative/Fuzzy Approach to Document Recognition" Hiroko Fujihara and Elmamoun Babiker

469

13.3 "Fuzzy Artificial Network and its Application to a Command Spelling Corrector" N. Imasaki, T. Yamaguchi, D. Montgomery and T. Endo

476

13.4 "A New Similarity Measurement Method for Fuzzy-Attribute Graph Matching and its Application to Handwritten Character Recognition" G.M.T. Man and J.C.H. Poon

482

13.5 "Automatic Target Recognition Fuzzy System for Thermal Infrared Images" Christiaan Perneel, Michel de Mathelin and Marc Acheroy

486

13.6 "A Vowel Recognition Using Adjusted Fuzzy Membership Functions" Sung-Soon Choi and Kyung-Whan Oh

493

Chapter 14: Management

501

Selected Papers: 14.1 "Dynamics and Fuzzy Control of a Group" Kenji Kurosu, Tadayoshi Furuya, Masaaki Nakamura, Hiroshi Utsunomiya and Mitsuru Soeda

503

14.2 "The Fuzziness Index for Examining Human Statistical Decision-Making" Sumiko Takayanagi and Norman Cliff

509

14.3 "Linking the Fuzzy Set Theory to Organizational Routines: A Study in Personnel Evaluation in a Large Company" Alessandro Cannavacciuolo, Guido Capaldo, Aldo Venire and Giuseppe Zollo

515

Chapter 15: General and Multi-Discipline

521

Selected Papers: 15.1 "An Electronic Video Camera Image Stabilizer Operated on Fuzzy Theory" Yo Egusa, Hiroshi Akahori, Atsushi Morimura and Noboru Wakami'

523

15.2 "Fuzzy Logic Based Banknote Transfer Control" Masayasu Sato, Tohru Kitagawa, Takehito Sekiguchi, Keisuke Watanabe and Masao Goto

531

15.3 "Electrophotography Process Control Method Based on Neural Network and Fuzzy Theory" Tetsuya Morita, Mitsuhisa Kanaya, Tatsuya Inagaki, Hisao Murayama and Shinji Kato

537

XV

15.4 "Fuzzy Logic Implementation of Intent Amplification in Virtual Reality" John Dockery and David Littman

543

15.5 "On-Line Analysis of Music Conductor's Two-Dimensional Motion" Zeungnam Bien and Jong-Sung Kim

549

15.6 "Multilevel Database Security Using Information Clouding" Sujeet Shenoi

556

15.7 "Active Control of Broadband Noise Using Fuzzy Logic" Oscar Kipersztok

562

Author Index

569

Subject Index

571

Editor's Biography

575

XVI

A FUZZY KNOWLEDGE-BASED SYSTEM FOR BUS LOAD FORECASTING G. Lambert-Torres1

L.E.Borges da Silva'

H.Greiss3

B.Valiquette 2

D.Mukhedkar 2

1. Escola Federal de Engenharia de Itajubs - Brazil 2. Ecole Polytechnique de Montreal - Canada 3. Harris Computer Systems - Canada ABSTRACT This paper describes an alternative approach to short-term load forecasting. The approach m e r g e s t r a d i t i o n a l m a t h e m a t i c a l techniques and fuzzy concepts in a knowledge base. The rules of this knowledge base are devised using the historical data of the bus and are represented by f u z z y c o n d i t i o n a l s t a t e m e n t s An example using real data from an actual power system is presented. INTRODUCTION In Operation Control Centers, many programs and computational routines [l] are available to operators to provide the best possible solutions for power system operation problems. Today many advisor programs using IA techniques are available to aid in operators' decidion-making [ 2 1 . Short-term load forecasting is one such program available to operators. Therearemanyreasons to study short-term load forecasting and they can be classified according to their focus on either objectives or time. The economical operation of power plants, and solution of the distribution systems, are examples of load forecasting focusing on objectives [31. Day-to-day operation, scheduling of fuel supplies, and planning operations, are examples which focus on time. This paper describes an alternative approach to short-term load forecasting to assist in the operation and planning of distribution systems. The proposed alternative approach is a knowledge based system composed of three knowledge bases. The first knowledge base includes historical data of the load, the second includes data of recent performance of the load. These knowledge bases are composed of fuzzy conditional statements The which are obtained using a technique described in [ 4 1 . third knowledge base enables the inclusion of operator knowledge of the load from the next hour to the next 24 hours This base includes fuzzy values in the description of the rules. The operator knowledge inclusion is very important for

0-7803-0236-2192 $3.00 Q 1992 IEEE

1211

special loads or for special characteristics due to external factors. The load forecasted value is calculated by an inference engine using fuzzy logic. Following an outline of load f o r e c a s t i n g c l a s s i f i c a t i o n , t h e proposed alternative approach is described and demonstrated using Hydro-Quebec Power System data (Canada). CLASSIFICATION OF LOAD FORECASTING This section presents a classification of load forecasting focusing on either objectives or time. A brief description of each focus is presented below. Load Forecasting According to Objectives There are two different objectives for studying load forecasting economical operation of power plants, and economical solution of the distribution systems. The first objective is to study the global performance of the load without taking into account the individual performance of each feeder. The second objective takes into account the characteristics of each load. The first objective is applied to power system's operation and planning, while the second is applied to the distribution system's operation and planning. Load Forecasting According to Time Currently, there are three possible classifications for forecasting focusing on time: short-term (the next half-hour to twenty four hour ahead), medium-term (the next day to the next year), and long-term (beyond the next year). Criteria's for selecting classification are different. As an example,the most important factors for short-term load forecasting include: the day of the week, temperature, humidity, seasonal effects and so on. Long-term load forecasting factors include economic aspects, political aspects, industrial development degree of a region etc. Thus, the objectives for each classification range from: short-term forecasting which concerns itself with the day-to-day operation and scheduling of the power system, medium-term forecasting whith deals with the scheduling of fuel supplies and maintenance operations and long-term forecasting which involves planning operations. DESIGN OF THE ALTERNATIVE APPROACH There are many methods to calculate load forecasting. all cases, a complete database is suitable as it allows major degree of accuracy.

In the

Database The database must contain historical information about load and weather parameters. Dry bulb temperature, wet bulb temperature, relative humidity, wind direction and wind speed are the most common weather parameters. For special load, additional factors such as sun inclination and cloudiness may also bear an influence. The data structure discriminates all data: its value (v),

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hour(h), day(d), year(y1 and season(s1. The general forms for the data are presented below. Real Power: P (vrhtd,w,y,S ) Dry Bulb Temperature: D(V,h,d,WrY,s) Wet Bulb Temperature: M(V,h'd'W,YlS) Relative Humidity: H(v,h,d,w,y,s ) Wind Direction: WD (v h d r w,y s 1 Wind Speed: WS(v,h,d,w,y,s) Real power values, temperature values, relative humidity and wind speed are stored in [MW], [ O C I , [%I and [km/hl, reg pectively. For the wind direction values, the following convention is employed: Nort is 1, and each additional 22.50 in a clock-wise sense is a further 1. For a calm d a y O i s used S o for example, the direction WSW is 12 and NE is 3 . Twenty four hour notation is used for hourly values. The day values are numbered as 1 for Monday through 7 for Sunday. Holidays receive the number 6 (Saturday). The week values are numbered by season, the most recent receiving the number 1. This database is built to manipulate the current year through three years previous. The yearly values range from 1 for the current year to 4 for three years ago. The seasonalvaluesare as follows: winter - 1, spring - 2, summer - 3 , and fall - 4. For a maximum of two missing values (v) of one data type, a single average between the previous and next values is calculated. When more than two values are missing, the data is forgotten. Special days (such as: snow storm) are stored in the data base with a flag. These values are not used for current calculations, except when this particular weather condition is forecasted. Alternative Approach The proposed alternative approach is based on a knowledge base formed by three bases. The first knowledge base contains information about the standard load B(t) where the historical data set is represented.The second knowledge base contains information about the deviation load R(t) where the recent load performance is represented. The third knowledge base contains rules-of-thumb about special characteristics of the load. The first two knowledge bases expresses information by fuzzy conditional statements, whilst the third base is composed of information given by operator experience. A fuzzy mechanism is able to make inferences using the actual data,to provide the load forecast. First Knowledge Base: Standard Load The standard load represents the historical data set. An hourly load shape must be represented using weather conditions. The relationship between hourly load and weather conditions must be established directly from a mathematical function : B(t) = f (P,K,M,H,WD,WS) or through a standard shape from each of the elements.

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The first konwledge base is composed of a standard shape for load and weather conditions.The standard shape is created using a weighted average with a decreasing degree of member ship for each year of the database. If the load evolution has not been large over the last years, the decrement is small. If the load evolution has been large, however, the decrement is consequential. Thus, the membership p1 (year), and p2 (year) represent a large and a not so large evolution of the load, respectively. pl(year) = ~ ~ 1 ~ 1 ~ , ~ ~ ~1 0 ~ 5 ~ , ~2(year)= {(111),(2(0.8),(310.6) (410.4) I Equation (1) shows a standard shape value calculation for time t, of the season s.

Where @ (. . . ) represents each one of the database elements namely, real power, dry bulb temperature, wet bulb temperat2 re, etc. The values kl and k2 depend on the day (working days or weekend) of the specific week, year, and season. The value k3 depends on the specific year and season. After calculations of all the times-of-day t, an equation of the shape must be calculated. An algorithm to express a shape by fuzzy conditional statements has been proposed in [4]. These statements present a linear membership function as a primise and a straight line as a consequence. Second Konwledge Base: Deviation Load The second knowledge base represents the deviation load, and contains information from the most recent four weeks, and represents recent load performance. This data is merged in a multiple linear model through the method of least squares. Equation (2) shows the general form for this model, using differences between actual, and standard weather information. 2 Rl(t) = ao+al. (Da(t)-Ds(t))+a2. (Da(t)-Ds(t)) +a3. (Da(t) 3 2 -Ds (t)1 +a4. (Ma(t)-Ms (t)+a5. (Ma(t)-M5 (t)1 2 +a6. (Ha(t)-Hs(t))+a7. (Ha(t)-Hs(t) 1 +a8. (Da(t)-Da(t-l))+ag.(Ma(t)-Ma(t-l)) +al0. (Ha(t)-Ha(t-l))+all.(Wa(t)-Wa(t-l)1

(2)

Where Qa(t) and Q2(t) represent actual, and standardvalues of the elements and of time t, respectively. 1214

~

3

~

0

~

In equation(2), the square of the difference between the dry bulb temperatures (due to the strong correlation between the load this temperature) was considered. The increase of parcel number is possible (for example, other square differences), but it is not recommended since problems of multivari ables increase rapidly in complexity, in this case. Without creating a better representation of the deviation load. Third K n o w l e d g e B a s e : Operator Experience In special cases, load forecast can be manipulated by a set of rules according to certain characteristics. Two sets of rules are presented, namely those of daily routine loadand thermic inertia. A daily routine load has regular attributes for specific parts of the day. For example, is there is a load point at 9 : 0 0 a.m. and the actual load at 8:OO a.m. is greater than the forecasted load calculated for the same weather conditions, then part of the load point has already been consumed. Likewise, if the load at 8 : O O a.m. is less than the forecasted load, then the load point maypossiblyincrease This special characteristic is applicable in load forecasting for predicting a maximum of 2 hours ahead. Rule: If the percentage error between the actual load and the forecasted load is big then add 5 0 % of this error, times the membership value for one hour ahead forecasting, and add 25% of this error, times the membership value for two hours ahead forecasting. 'Big' is a fuzzyvalue for adeviation The values for 'big', 50% and 25% were obtained from many forecasts and were consid ered acceptable for all simulated residential buses. When the error is negative, the extra values can be subtracted. Thermic inertia is the difficulty in changing from abetter to a a worse temperature. For example, if the average winter temperature for the previous day was -5OoC, and today is -2OoC, the load will increase due to the temperature difference plus a certain thermic inertia. If the next day, the temperature remains at -2OoC, this inertial component will decrease. In Canada for example, this factor isveryimportant due to the excessive temperature ranges. This special charac teristics is mainly applicable for load forecasting 24 hours in advance. Rule: If the 24 hour temperature variation is big then add 10% to forecasted load. The values used in this rule must be calculated for each bus of the system using the historical data. In our case, these values ('big temperature variation' and 10%) reference the example in the next section. M e c h a n i s m of Inference Following the calculation of the knowledge bases, the inference engine is activated to determine the load forecast value. The mechanisms of the inference engine are described below. The inference is made in several parts, namely,the initial estimate, the second estimate and the final estimate. The initial estimate uses only the first knowledge base and the

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actual values of load and weather conditions. It updates the standard load values. Equation (3) presents the initial estimate. Pa(t) L'(t+A) = B(t+A). for A = 1,2, ... (3) B (t) The second estimate is a rated average between the initial estimate and the second knowledge base (deviation load). The ratios of this average are calculated for each bus of the system and can be modified according tothekindof forecasting and the time-of-day. Equation (4) presents this expression. (4) L"(t+A) = a1 .L' (t+A)+ %.R(t+A) The final estimate is obtained after a treatment of the value L"(t+A) by the third knowledge base. If any rule of this knowledge base is applicable then the final estimate is like a second estimate. ILUSTRATIVE EXAMPLE This example uses data from the Hydro-Quebec Power Systen Canada. The numerical data extends from April 1984 to July 1988. It contains information related to real power flows in lines 1288 and 1289, as well as actual weather conditions.The weather conditions include parameters such as dry bulb temps rature, wet bulb temperature, relative humidity, wind speed and direction, all of which were measured hourly. Since the two lines worked in parallel to supply the Plouffe bus, load performance is a function of the addition of these lines. The example forecasts load for the winter month of January 1988. Application of the algorithm to compute the model of the shape proposed in [4], for the real power, results in: R1: If t is pl(t) then Pl(t) = 0.85t + 116.25 R2: If t is %(t) then P2(t) = -1.70t+ 165.33 R3: If t is u3(t) then P3(t) = 4.556+ 71.60 R4: If t is p4(t) then P4(t) = -1.30t+ 153.87

Where: u,(t) = -0.0419t p2(t) = 0.1079t p3(t) = -0.0184t p4 (t) = 0.0254t

+

+

-

0.3881 0.6190 0.4495 0.4725

for 3 5 for 5 < f o r 1 5-< for 16