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Methods of scheduling a hydro-thermal power system for optimum economy Smith, Bryan Robert 1961

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METHODS OF SCHEDULING A HYDRO-THERMAL POWER SYSTEM FOR OPTIMUM ECONOMY by  BRYAN ROBERT SMITH B.A. Sc., University of British Columbia,  1958  A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR T H E DEGREE OF MASTER  OF  APPLIED  SCIENCE  in the Department of Electrical Engineering  THE  UNIVERSITY  OF  BRITISH  April, 1961  COLUMBIA  METHODS OP SCHEDULING A  HYDRO-THERMAL  POWER SYSTEM FOR OPTIMUM ECONOMY  by BRYAN ROBERT SMITH B » A , S c o , U n i v e r s i t y of B r i t i s h Columbia, 1958 A THESIS SUBMITTED IN PARTIAL FULFILMENT OP THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE  In the Department of Electrical  Engineering  We accept t h i s t h e s i s as conforming to the standards r e q u i r e d from candidates f o r the degree of Master of A p p l i e d Science  Members of the Department of E l e c t r i c a l E n g i n e e r i n g  THE UNIVERSITY OP BRITISH COLUMBIA i  A p r i l 1961  In p r e s e n t i n g the  t h i s thesis i n p a r t i a l f u l f i l m e n t of  requirements f o r an advanced degree a t t h e U n i v e r s i t y  British  Columbia, I agree t h a t the  a v a i l a b l e f o r reference  and  study.  of  L i b r a r y s h a l l make i t f r e e l y I f u r t h e r agree t h a t  permission  f o r e x t e n s i v e copying o f t h i s t h e s i s f o r s c h o l a r l y purposes may g r a n t e d by  the  Head o f my  It i s understood t h a t f i n a n c i a l gain  Department o r by h i s  s h a l l not  The U n i v e r s i t y o f B r i t i s h Vancouver 8, Canada.  j^/lcyy. /  representatives.  c o p y i n g or p u b l i c a t i o n o f t h i s t h e s i s f o r be  a l l o w e d w i t h o u t my  Department  Date  be  Columbia,  /  written  permission.  ABSTRACT  In t h i s t h e s i s an attempt has been made to improve upon and to compare  e x i s t i n g methods of scheduling a hydro-thermal  power system., Two methods i n p a r t i c u l a r , the c o o r d i n a t i o n - e q u a t i o n s method and the incremental dynamic programming method, were investigated i n d e t a i l .  Numerical c a l c u l a t i ons ba sed on these  two methods were c a r r i e d out f o r a two-plant system on an Alwac I I I - E d i g i t a l computer, a medium-speed l i m i t e d amount of random-access memory.  computer with a  The economical s o l u t i o n  of these equations f o r a l a r g e r system would r e q u i r e a f a s t e r computer with a l a r g e random-access memory and p r e f e r a b l y automatic f l o a t i n g p o i n t a r i t h m e t i c . The r e s u l t s obtained i n d i c a t e that economic scheduling becomes important only f o r l a r g e power systems.  I t a l s o tppears  t h a t the dynamic programming method i s the more s u i t a b l e f o r d i g i t a l computer  solution.  iii TABLE OP CONTENTS Page Abstract  » • • • • • • . « . < . . . . . « . . . . . « . . « «  L i s t of I l l u s t r a t i o n s  11  . . . . . . . . . . . .  v  L i s t of Tables . .  vii  Acknowledgement  vin  Chapter Chapter  . . . . . . . . . . . . . . . . . . . . .  I. Introduction I  I  .  «  .  .  «  «  «  . . . . . . . . . . . . . . . . «  .  .  «  «  «  o  o  .  .  .  .  «  .  «  .  1 5  «  2.1  The System to Be Studied . . . . . . . .  5  2.2  Thermal P l a n t  5  2.3  Hydro P l a n t  . . . . . . . . . . . . .  9  Chapter I I I . S i m p l i f i e d Scheduling Equations 3.1  3.2 Chapter  11  F i x e d Head Case  11  3.1.1  C o o r d i n a t i o n Equations Method  . .  12  3.1.2  Equal Incremental Cost Method  . .  20  V a r i a b l e Head Case -.  24  IV. S o l u t i o n by Incremental Dynamic Programming Programming  .  35  .  35  4.1  D i s c u s s i o n of Dynamic Techniques . . . . .  4.2  Method of S o l u t i o n .  36  4.3  D e r i v a t i o n of C r i t e r i o n F u n c t i o n . . . .  39  4.4  Choice of a New P o l i c y  41  4.5  S o l u t i o n by Incremental Dynamic Programming .....<> . . . . . . .  4.6  Computer S o l u t i o n  . . . . . . . . . . .  .  42 46  iv  Page 61  Chapter V 5.1  D i s c u s s i o n of Previous S o l u t i o n  61  5.2  General S o l u t i o n of S c h e d u l i n g E q u a t i o n  .  . . . . . . . . . . .  67  5.2.1  Introduction  67  5.2.2  The C o o r d i n a t i o n E q u a t i o n s Method .  67  5.2.3  The I n c r e m e n t a l Dynamic Programming Approach . . . . . . .  69  Conclusions . . . . . . . .  71  References  73  Nomenclature  .  . . . . . .  77  Appendix A  78  Appendix B  ' 89  Appendix C  .  Appendix D  . . . . . . .  . . . . . . . .  93 101  V  LIST OP  ILLUSTRATIONS i  Figure 1.  The  Page P o r t i o n of the BCPC System used i n the  Scheduling 2.  . . . . . .  Problem .  Input-Output C h a r a c t e r i s t i c of One i n the Georgia Thermal P l a n t  3. 4.  Generator  . <>  8  V a r i a t i o n of Storage and R e s e r v o i r Area Y i t h T o t a l Head f o r the Ash River R e s e r v o i r Input-Output C h a r a c t e r i s t i c s of the Ash Hydro P l a n t  8  River 10  5.  Flow Diagram f o r the Computer S o l u t i o n of the Coordinations Eqtiations f o r the F i x e d Plead Case  6.  V a r i a t i o n of P^ with A f o r C o n s t a n t .  7.  V a r i a t i o n of PJJ with IT  10.  . .  f o r F i x e d Load L e v e l s  17 18  . . . . . . .  8. V a r i a t i o n of Q with IT f o r F i x e d Load L e v e l s 9.  6  . . . .  18  . ...  18  .  A T y p i c a l Schedule f o r the F i x e d Head Case (7=6)  .  19  . .  19  V a r i a t i o n of the T o t a l Volume of Water Used with IT f o r the L o c a l P a t t e r n  shown i n F i g . 9 .  .  11.  V a r i a t i o n of Pp with A f o r Constant  12.  V a r i a t i o n of P^ with IT f o r F i x e d Load L e v e l s  . . . .  21  13.  V a r i a t i o n of Q with IT f o r F i x e d Load L e v e l s  . . . .  21  14.  Flow Diagram f o r the Computer S o l u t i o n of V a r i a b l e Head Problem V a r i a t i o n of Discharge with Time f o r the Schedule shown i n  15. 16. 17.  V a r i a t i o n of Head with Time f o r the shown i n Table IV. A T y p i c a l Load Demand and  the .  .  28  Schedule 30  the R e s u l t i n g Hydro  P l a n t Schedule f o r a Value of ^ = 18.  21  5.0  . . . . . . .  Computer Flow Diagram f o r Dynamic Programming S o l u t i o n of Short-Range Scheduling Problem . . . . .  32 47,48  vi Page  Figure 19. A P l o t of the L i m i t s V.  .  and V.  1 min  1 max corresponding to a T o t a l Discharge of 3270298 3 3 yds at a maximum r a t e of 60 yds / s e c . The i n i t i a l t r i a l p o l i c y V.° and the f i n a l optimum f p o l i c y V^ are a l s o shown ..• 1  49  20. A T y p i c a l Load Demand and the R e s u l t i n g Hydro Schedule Obtained by Incremental Dynamic Programming . . . . . .  52  21. The M o d i f i e d V e r s i o n of the BCPC System Used i n the Power Plow A n a l y s i s  81  22. Convergence of P and Q to the C o r r e c t Value £L~b Nod© 07  86  23. D i s t r i b u t i o n of R i v e r Plows i n t o Comox Lake. C l a s s Length = 200 . . . . . . . . . . . .  98  •  *  0  «  0  0  o  o  0  «  0  «  »  »  e  e  24. D i s t r i b u t i o n of R i v e r Plows when Normalized a Log B a s i s . C l a s s Length = 0.1 . . . . . . 25. The Regression Line f o r the J u n e - J u l y R i v e r Flows and the Corresponding 95 and 80$ Confidence L i m i t s . . . . . . . . . . . o  n  «  98  »  99  vii  LIST OF TABLES Table  Page  I . S o l u t i o n of the C o o r d i n a t i o n Equations f o r the Load Schedule shown i n F i g . 9 . . . .  16  I I . Comparison of Schedules Obtained by the Equal Incremental Cost Method and the C o o r d i n a t i o n E q u a t i ons Method . . . . . . . . . . . . . . . . . I I I . Comparison of the Cost of Operation of the C o o r d i n a t i o n Method and Equal Incremental Cost Method of Scheduling  22  23  IV. S o l u t i o n of the V a r i a b l e Head C o o r d i n a t i o n Equations f o r a Constant System Load of 120 MW ( — 3*5}. » • « « « • • • . V. S o l u t i o n of the C o o r d i n a t i o n Equations, C o r r e s ponding t o the Load Demand shown i n P i g . 17, f o r the V a r i a b l e Head Case  29  33  V I . P o s s i b l e States of Hydro P l a n t and R e s u l t i n g Discharges . . . . . . . . . . . . . . .  43  V I I . Allowable States and Corresponding Weighted V I I I . The Hourly System Load Studied along w i t h the I n i t i a l Discharges Q-°. The Hourly Values of V. . , V. and PTTare a l s o shown l mm' x max H i max IX. Optimum Hydro Schedule f o r the System Load  53  shown i n F i g . 20  54  X. T a b u l a t i o n of Table V I I f o r the F i n a l I t e r a t i o n . . 55,56 X I . Cost of Operation and Net Savings a t the End of each I t e r a t i o n . . . . . . . . . . . . . . . . . X I I . Dependence of the Speed of Convergence and of the Cost of Operation on AQ X I I I . Hourly Worths of Water  57 58 63  XIV. Optimum Schedule f o r the System Load shown i n F i g . 20 as Obtained from the C o o r d i n a t i o n XV. Scheduled V o l t a g e s and Powers on a Per U n i t B a s i s of 138 KV and 500 MVA XVI. F i n a l R e s u l t s of the F i r s t Power Flow A n a l y s i s XVII. The Regression and C o r r e l a t i o n C o e f f i c i e n t s f o r the R i v e r Flow a t Comox Lake  82 . .  83 97  ACKNOWLEDGEMENT  This i n v e s t i g a t i o n was supported  by the N a t i o n a l Research  C o u n c i l of Canada. The  author wishes t o express h i s s i n c e r e g r a t i t u d e to  Dr. Prank Noakes f o r h i s guidance and advice, and to Dr. John P. Szablya, s u p e r v i s o r , f o r h i s guidance and advice during the course  of t h i s r e s e a r c h work.  The Engineer  author wishes to thank Mr. A.¥. Lash,  of the B r i t i s h Columbia Power Commission, who provided  the data necessary f o r t h i s The  Consulting  author  research.  i s g r a t e f u l t o the personnel  centre f o r t h e i r h e l p f u l suggestions  of the computing  on,the s u b j e c t of  computer programming. Thanks are due t o the author's  colleagues i n the power  group of the Department of E l e c t r i c a l Engineering f o r much e n l i g h t e n i n g and s t i m u l a t i n g d i s c u s s i o n .  CHAPTER I INTRODUCTION One of the b i g problems i n a l a r g e hydro-thermal system i s the scheduling of the power output of the v a r i o u s hydro and thermal p l a n t s so t h a t the use of the a v a i l a b l e water resources i s optimizedo  The e f f i c i e n t c o n t r o l of a  complex  system l e a d s to many problems such ass (1) F l o o d  control,  (2) P r e d i c t i o n of v a r i a b l e and unknown p r e c i p i t a t i o n , (3) U n c e r t a i n t y of f u t u r e stream flows and system l o a d s , (4) Guarantee of f i r m loads during dry s p e l l s , (5) P i s h  restrictions,  ( 6 ) Shipping r e s t r i c t i o n s , e t c . D e c i s i o n s have to be made- r e l a t i n g to monthly, weekly and d a i l y r e s e r v o i r drawdowns.  These d e c i s i o n s must conform to  a l l the above r e s t r i c t i o n s while g i v i n g economic  and reasonable  operation. That supplementary power must be used to supply the p o r t i o n of the l o a d which the hydro s t a t i o n s cannot meet i s obvious, ¥hatis  not so obvious i s how much supplementary power t o supply -  and d u r i n g what p e r i o d s i t should be s u p p l i e d .  Since hydro  g e n e r a t i o n depends not only on d i s c h a r g e but also on head, i t i s l o g i c a l t h a t the supplementary power sources should be scheduled i n such a manner that the r e s e r v o i r heads w i l l be kept as h i g h as p o s s i b l e f o r as long as p o s s i b l e .  I f t h i s procedure i s  f o l l o w e d the energy y i e l d e d by a l l f u t u r e stream flows w i l l be i n c r e a s e d , thus l e a d i n g to a net increase i n hydro energy. On the other hand should the r e s e r v o i r s be kept too f u l l may  spillage  occur, i n which case the energy l o s t through s p i l l a g e may more  2 than o f f s e t the energy gained by maximizing it  the heads.  Thus,  can be seen that the e f f i c i e n t use of storage water can become  q u i t e a complex problem. Several methods have been suggested f o r the long-range 19 20 23 o p t i m i z a t i o n o f hydro-thermal solve the long-range problem d e s c r i b e the system  systems,  '  '  In order to  i t appears that i t i s necessarjr to  stream flows by the use of p r o b a b i l i t y i  d i s t r i b u t i o n s and to f i n d a method of s o l u t i o n which w i l l use these p r o b a b i l i t y d i s t r i b u t i o n s to f i n d the most economic mode of o p e r a t i o n , 20 L i t t l e used such an approach to solve t h i s problem, 21 24 His methods have been subjected to s e v e r a l c r i t i c i s m s ' but a r e c e n t l y p u b l i s h e d paper  i n d i c a t e s that these c r i t i c i s m s can  21 be overcome.  In some cases, i t seems c e r t a i n t h a t r u l e curve  o p e r a t i o n as d e s c r i b e d i n reference 23 i s s t i l l  the best method  f o r long-range s c h e d u l i n g . Many of the problems mentioned above apply mainly to the long-range  scheduling problem.  The p r e d i c t i o n of stream flows  i s g e n e r a l l y not too important when d e a l i n g with the short-range problem.  Except f o r f l a s h y streams, water to be used tomorrow  i s upstream today.  B i g r i v e r s r i s e and f a l l r e l a t i v e l y slowly  and i f there are p l a n t s with l a r g e r e s e r v o i r s , then these p l a n t s d i c t a t e the flows. The d i f f i c u l t y  of f i n d i n g the best short-range o p e r a t i o n  depends s t r o n g l y on the system to be considered. the best o p e r a t i o n may be obvious.  ^  In some cases  For examples, i t may be that  peaking with steam power i s too expensive.  A given hydro  plant  may be the best f o r t a k i n g the peak loads and so i t i s used f o r  3 peaking as much as p o s s i b l e .  The remainder  of the hydro p l a n t s  could then be scheduled i n order of e f f i c i e n c y . problems may  Transmission  make i t impossible to take more than a c e r t a i n  l e v e l of power from a d e s i r a b l e source, t h e r e f o r e , as much as p o s s i b l e of the output of t h i s source i s set at t h i s  level  0  R e s t r i c t i o n s such as f l o o d c o n t r o l s and n a v i g a t i o n c o n t r o l s may  l i m i t the discharges to w i t h i n a r e l a t i v e l y small range.  Thus, good o p e r a t i o n may fairly  be set by o p e r a t i n g extremes which are  obvious to someone f a m i l i a r with the  system.  This t h e s i s w i l l be l i m i t e d to the short-range problem. There appeared  to be no p o s s i b l e s o l u t i o n of the general scheduling 24  equations d e r i v e d i n a p r e v i o u s t h e s i s  , even f o r a smaller  system, with the f a c i l i t i e s  The  available.  s o l u t i o n of these  equations would be best c a r r i e d out on a s o p h i s t i c a t e d or  analog d i g i t a l computer.  at  the U n i v e r s i t y of B r i t i s h Columbia,  the problem  can be s o l v e d .  On the d i g i t a l  computer a v a i l a b l e  a d i s c r e t e v e r s i o n of  Under these circumstances i t i s  b e t t e r to use the s i m p l i f i e d scheduling equations. computations,  analog  Considerable  based on the s i m p l i f i e d equations, have been c a r r i e d  out f o r both f i x e d head and v a r i a b l e head p l a n t s . from the B r i t i s h Columbia  T y p i c a l data  Power Commission (BCPC) i s used.  The  equations used are p e r f e c t l y general and a p p l i c a b l e to an N source system; however, i t was case would be s t u d i e d .  One  decided that only the two  reason f o r t h i s d e c i s i o n was  Alwac I I I - E computer i s r e l a t i v e l y slow and i t l a c k s f e a t u r e s , such as random access to memory. of  these equations becomes r e l a t i v e l y  plant t h a t the  several  Thus, the  solution  slow.  More recent developments i n the f i e l d of power  system  a n a l y s i s have i n d i c a t e d the u s e f u l n e s s of dynamic programming procedures i n the s o l u t i o n s of power system problems„  A recent  paper d e s c r i b e s the use of t h i s method f o r the f i x e d head case*, This type of approach was for  30  a p p l i e d to the above two-plant problem  the v a r i a b l e head case.  Dynamic programming methods lend  themselves more e a s i l y to the i n c l u s i o n of the v a r i o u s r e strictions i n c l u d i n g v a r i a b l e t a i l w a t e r e l e v a t i o n and v a r i a b l e r e s e r v o i r surface area.  A l l r e s t r i c t i o n s were i n c l u d e d i n t h i s  s o l u t i o n except problems  of i n t e r r u p t a b l e power.  The p o s s i b i l i t y of u s i n g Monte Carlo procedures i n the s o l u t i o n of the short-range problem was  examined.  short-range problem does not lend i t s e l f of  solution.  In f a c t , i t i s d i f f i c u l t  a p p l i e d at a l l .  However, the  e a s i l y to t h i s method  to see how  i t can be  I t i s to be noted that t h i s type of procedure  can be used to advantage  i n the long-range problem to d e s c r i b e  and f o r e c a s t stream f l o w s . A study of t y p i c a l stream flows was t i o n with t h i s t h e s i s .  c a r r i e d out i n conjunc-  The r e s u l t s i n d i c a t e that d u r i n g c e r t a i n  p e r i o d s of the year i n B r i t i s h Columbia, the v a r i a t i o n s i n stream flows are so l a r g e that i t i s n e a r l y impossible t o p r e d i c t them a c c u r a t e l y .  To p r e d i c t stream flows a c c u r a t e l y , a d e t a i l e d  knowledge of r a i n f a l l  and snowfall f i g u r e s would be r e q u i r e d .  r e s u l t s of t h i s study are shown i n Appendix In  The  C.  t h i s t h e s i s i t i s assumed t h a t the amount of water  scheduled f o r use has been determined from the long-range schedule. The expected stream flows may  be determined from l a s t month *s  stream flows and from an examination of the l a t e s t figures„  precipitation  5  CHAPTER II 2.1  The System to be Studied The system to be studied consists of one hydro and one  thermal plant, (see Pig. l ) . BCPC system.  This i s a modified version of the  I t was necessary to modify the system for the  following reasons: (1) i t had to conform with a two-plant system, (2)  i t was necessary to put i t i n a form suitable for the existing programme for doing power flow studies.  The data for the plants and the transmission l i n e impedances were supplied by the BCPC.  The thermal plant constants were  obtained from data given for the Georgia plant, while the hydro plant constants correspond with data taken from the Ash River development.  To give the problem a reasonable basis, the  megawatt output of the hydro plant was scaled up so that i t s maximum output would correspond with that of the thermal plant. The loss constants used were obtained for the network shown i n Pig. 1.  The procedure used to f i n d the loss constants 39  is described i n a paper published by E.E. George.  A brief  description of the method used, and of the results obtained i s given i n Appendix A. 2.2  The Thermal Plant The data f o r the thermal plant was taken from information  obtained from the Georgia-• Tifermal: plant.  The linpU-t-OEitpti-t char-  a c t e r i s t i c s of v t h e , four units present are>\ "&©-* i d e n t i c a l . . This fact leads to some d i f f i c u l t i e s i n the solution of the scheduling equations; so f o r this reason i t was assumed that the plant consisted of  PINE STREET T. S. LADYSMITH T. S.  CHEMAINUS T. S.  DUNCAN T. S.  A  A PARKSVILLE  < I.96 + JI.70  "15." 9.90+j 1080  4.69 + ) 5.11  5-30+] 5.79  •536+J-586  —*\*v  0.53+).586  -2.47+) 2.70 I2.39+JI3.50  I.74+JI.90  LAKE COWICHAN T. S.  »-> •*  2.68+J2.93  o  GEORGIA G.S. EACH UNIT 25,600 KVA 3j0T  13-8 KV 85 PF 3600 RPM  CROFTON T. S.  1.41+1 1.54  A  |. j 3 J  132 KVY 63 KVY 7.2 KVA  18,750 KVA ONS 25,000 KVA ONP 3-1/6 5800 KVA 76.2/132 KVY-10^ 60KVY 12 KVA  138 KVY 7^-2^1 +  6.57+J 21.40 'W>  2.02I + J6.58  I.43I + J4.65  5.34 + ) 15.30 HARM AC T. S. 132 K VY  - | _ ^ 30,000KVA0NS g ^ K V Y t 5 2 40,000 KVA ONP 12 KVA ASH RIVER G.S. 28,000 KVA I3.8KV 0.9 PF 514 RPM  " f t "  ^ 6.91+) 22.40  II.42+)37.I5  6  +2i%-7J% LTC ± 10%  13.8 KVA  -AA/V-  ~  25.000 KVA ONS 3 3,333 KVA ONP  FIG. I SINGLE LINE DIAGRAM  OF A PORTION OF THE BCPC SYSTEM.  B.C. ELECTRIC INTERCONNECTION  7  f o u r u n i t s w i t h the input-output The  c h a r a c t e r i s t i c shown i n F i g . 2.  f u r t h e r assumption was made t h a t these u n i t s are a l l  continuously  on the l i n e , which i s q u i t e f r e q u e n t l y the case.  These two assumptions g r e a t l y s i m p l i f y the computer s o l u t i o n of the scheduling theory,  equations.  I t i s to be pointed  out that i n  both of the methods used i n t h i s t h e s i s to schedule a  power system can take these two f a c t o r s i n t o account.  However,  any r e s u l t i n g s o l u t i o n becomes much more complex. The  operating  costs i n v o l v e d are only those c o s t s which  v a r y d i r e c t l y with the plant!s power output.  A l l f i x e d c o s t s such  as d e p r e c i a t i o n or c a p i t a l costs are not i n c l u d e d .  Maintenance  costs are omitted s i n c e they.are u s u a l l y r e l a t i v e l y small and are v e r y hard to d e f i n e . cost of f u e l .  The dominant cost i s t h e r e f o r e the  A d e t a i l e d d i s c u s s i o n of the above p o i n t s can be  found i n r e f e r e n c e 24.' The  method of l e a s t squares was used t o f i t a quadratic  f u n c t i o n to the curve shown i n F i g . 2.  The f i v e p o i n t Gram  approximation was programmed f o r the Alwac I I I - E computer to f i n d the c o e f f i c i e n t s of t h i s quadratic  function.  The c o e f f i -  cients are: 0.017857 4.071428 3 where  F  470.2857 F  l  P  S  2  +  F  2 S P  +  P  3  •  • o  (2-1)  m i l l i o n s of BTU input per hour. E q u a t i o n (2-1) must be m u l t i p l i e d by 0.49 i f i t i s d e s i r e d to f i n d the cost of o p e r a t i o n  i n d o l l a r s per hour.  8  250 200 150 100 50  10 MV OUTPUT Pig. 2  20  15  Input-Output C h a r a c t e r i s t i c of One Generator i n the Georgia Thermal Plant.  Storage  10  20  30  40  50  60  70  STORAGE ( Y d s ^ xx I O ) (Yds 3  g 0  6  • >• AREA, 10 (Yds x 10 ) 2  Pig. 3  V a r i a t i o n of Storage and R e s e r v o i r Area with T o t a l Head f o r the Ash R i v e r R e s e r v o i r .  6  9 •  2.3  The Hydro P l a n t The data f o r the hydro plant was taken from data s u p p l i e d  f o r the Ash R i v e r System. v a r i a b l e area r e s e r v o i r .  T h i s system has a v a r i a b l e head, The curves showing the r e l a t i o n s h i p  between head, area and volume are g i v e n i n P i g . 3.  The curves  showing the r e l a t i o n s h i p between head, discharge and power output are shown i n F i g . 4 . approximated  I t was assumed t h a t the curves could be  by the f o l l o w i n g equation.:  Q = K(ah  2  + bh + c) ( e P  2 h  + gP  h  + i)  ... (2-2)  As b e f o r e , a programme u t i l i z i n g the l e a s t squares method was w r i t t e n f o r the computer to f i n d the c o e f f i c i e n t s of these equations.  The r e s u l t s are as f o l l o w s :  K = 0.019000 a = 0.003600 b = -2.09280 c = 353.4960 e = 0.003710 g = 0.147410 i  = 17.72354  As was mentioned p r e v i o u s l y , the output of t h i s p l a n t has been s c a l e d up by a f a c t o r of f o u r to give the problem basis.  a more reasonable  273.333 ^266.667  I  10  i  I  20  :  .  1  30  1  1  1  40  Turbine Discharge  Fig. 4  1  ..—i  50 (Yds /Sec)  Input-Output C h a r a c t e r i s t i c s of the Ash River Hydro P l a n t .  1—  60  11  CHAPTER I I I SIMPLIFIED SCHEDULING EQUATIONS 3.1  F i x e d Head Case The s i m p l i f i e d scheduling equations were f i r s t developed i n  ' 31 1940 by R i c a r d .  They were improved upon i n 1953  Dandeno, Glimn and Kirchmayer, but with constant head.  who  included transmission losses  Glimn and Kirchmayer  upon these equations i n 1958  by Chandler,  27  further  improved  by s o l v i n g the v a r i a b l e head case. 19  These equations have been d e r i v e d i n s e v e r a l other forms  24 '  by  i  other authors.  I t has been shown that a l l these v a r i o u s forms 24 26 the s c h e d u l i n g equations are e q u i v a l e n t . ' 1  of  The  s c h e d u l i n g equations s t u d i e d i n t h i s t h e s i s are those 26  p u b l i s h e d by Chandler, Dandeno, Glimn and Kirchmayer 27 f i x e d head case and by Glimn and Kirchmayer head case.  f o r the  f o r the v a r i a b l e  I f t r a n s m i s s i o n l i n e l o s s e s are i n c l u d e d i n the  s o l u t i o n of these equations the c o o r d i n a t i o n equations r e s u l t , while i f l o s s e s are n e g l e c t e d the equal incremental cost equations 5  result.  In the f i r s t  case the thermal p l a n t s are scheduled i n  such a manner t h a t the pover i s d e l i v e r e d to the loads on an equal incremental cost b a s i s . . In the second case the thermal p l a n t s themselves are scheduled on an equal incremental cost b a s i s . Both methods of s c h e d u l i n g were s t u d i e d and the r e s u l t s  indicate  that f o r the p a r t i c u l a r problem under study both methods give e s s e n t i a l l y the same r e s u l t s .  ,  These equations are general and are a p p l i c a b l e to an N source system.  They are best s u i t e d to an analog method of  s o l u t i o n , however t h i s method would r e q u i r e a l a r g e amount  12 of equipment not r e a d i l y available.  The solution of these  equations on a d i g i t a l computer i s necessarily limited to the discrete case. These equations are not e a s i l y applied to the case where more than one hydro plant exists on one stream.  However, i f  the time required f o r water to flow between plants i s negligible or i f certain assumptions are made about the p e r i o d i c i t y of the 21 d a i l y load cycles  then these equations can be used to schedule  the plants of a common flow system.  The simplified equations  for the common flow case are derived i n Appendix B, under the assumption that the time delay of flows between hydro plants i s negligible. 3.1.1  Coordination Equations Method The coordination equations to be solved for the optimum  schedule a r e ^ * 2  if  3  df  2 -  +  X  P  L  W~  sn  X  sn  X ' m ^§— Hm + X Hm SP  =  3P a  L  p  = X  n  =  m + 1  m  ' ••••» = 1,  ••• t  n + m  m  3 - 1  )  ... (3-2)  where c^f  c p - = incremental cost of the energy generated i n the S thermal plant i n $ per MW-hr. H  d P  = incremental water rate at the hydro plant i n cubic yds. per sec. per MW.  L  ^p— = incremental transmission loss of the hydro plant H d P  L  ^ p - = incremental transmission loss of the thermal plant  13  A = a Lagrangian m u l t i p l i e r - S/MY h r .  if  ?= a Lagrangian m u l t i p l i e r - ^ / m i l l i o n cubic yds.  The  incremental c o s t o f energy generated  i n the thermal p l a n t  i s given by: = 0.49(2P P +F ) = ^ P g + f  -jp^  1  S  (3-3)  2  2  The t r a n s m i s s i o n l o s s e s are g i v e n by: P  L  = 11 S B  P  2 +  B  22 H P  2  +  2 B  12 S H P  ^  P  Hence 5 P  L  S%  = 22 H  3P dP  ~ = *"11*S 2B-...P + " 2 B12P H „  2 B  P  +  2 B  12 S  ^  P  And u g  C  ...  1 0  (2-2)  Prom  f§3P  = K(ah  w  (3-3)  S u b s t i t u t i o n o f equations (3-2) f  l S P  ...  + bh + c ) ( 2 e P + g) H  2  H  and  (3-6)  t o (3-7)  i n t o equations  (3-7)  (3-1)  gives:  +  f  2  +  11 S P  X ( 2 B  TK(ah + bh 2  +  +'c)(2eP  2B  12 H> = * P  + g) + ( 2 B X  H  2 2  P  H  + 2B Pg) = A 12  Simplifying: 2 ^ ~ ~X ~ f  Pg =  j  "T P  i  _  +  2  2B  P  . . . (3—8)  n  B  _ gKT(ah  2eKT(ali  12 H  2  A.  + bh + c) _  B  p  ^12 S r  + bh + c) _^  O T l  A  2  +  ZJ3  22  Q  x  14  Equations  (3-8) and (3-9) are the f i n a l scheduling  to be solved* while  equations  The choice of A determines the r a t i o of Pg to Pg  the choice of T  at the hydro p l a n t .  determines how much water w i l l be used F o r the two p l a n t case i t i s r e l a t i v e l y -  easy to solve f o r P,p and P^ e x p l i c i t l y .  However, f o r a more  complex system i t becomes somewhat d i f f i c u l t to o b t a i n an e x p l i c i t s o l u t i o n , f o r t h i s reason an i t e r a t i v e procedure was used. Equations  (3-8) and (3-9) can be expressed  i n the f o l l o w i n g  form:  x  2  a"^ ^ 2 " 2 1 l ^  =  c  a  As long as a ^ and  a  x  r  g r e a t e r than a.^  e  of s u c c e s s i v e displacements  (Gauss-Seidel  r a p i d l y t o the c o r r e c t s o l u t i o n . equations a  22  > >  a  a n (  i  a  2i  m e  "t  G 0  procedure) w i l l  ^  converge  In the case of the scheduling  the i t e r a t i o n converges v e r y f a s t since  »  a  12  a n <  ^  21*  The  i n i t i a l approximation made by the computer i s taken  to be:  1 I t i s necessary  to choose A so t h a t the f o l l o w i n g r e s t r i c t i o n  i s met: P  Si  +  P  H i  = Di P  +  P  Li  ( " > 3  10  That i s , the t o t a l g e n e r a t i o n f o r the i t h hour must be equal to the l o a d demand p l u s the t r a n s m i s s i o n l i n e l o s s e s * .  An i t e r a t i v e  * The s u b s c r i p t s i , j and k r e f e r to those used i n F i g . 6 to d e s c r i b e the v a r i o u s loops i n v o l v e d i n the c a l c u l a t i o n s .  15  procedure i s used t o f i n d the c o r r e c t value of A. t f°  power d e l i v e r e d ,  The i n i t i a l  some i n i t i a l assumed value o f A^, i s  r  c a l c u l a t e d from equations (3-8), (3-9) and (3-10), and the d i f ference between the scheduled l o a d , PJJ^, and P j ) i * '  -0  I  S  found.  If  t h i s d i f f e r e n c e i s not l e s s than, or equal t o , some preassigned t o l e r a n c e C, a new v a l u e of A i s found from the f o l l o w i n g equation 5P (SA) D  A 3  = A 3 +  + 1  x  ... (3-11)  i  5P  D  where SP^ = P . - P 3 SA = A. ' - A . l I N  3  3 - 1  S'P - P - P 3-1 ° D ~ Di Di 1  r  The convergence fast.  3  r  to the c o r r e c t value of A was found to be quite  A t o l e r a n c e i n d e l i v e r e d power of 0.001 MW i s reached i n  about f i v e i t e r a t i o n s on the average. It  i s a l s o necessary to choose  c o r r e c t amount of water i s used.  a value of if  such t h a t the  An i t e r a t i v e procedure of  the type d e s c r i b e d above was found to give good r e s u l t s .  The  equation used i s :  /k+i  =  ? k |vli2l  _  +  ( 3  _  1 2 )  where  sv  =v - v  8V  = V  T  The convergence  k T  T  k  - V^-  1  to the c o r r e c t value of X was found to be  16  f a s t i n most cases. a bad estimate of If The  However, the s o l u t i o n may not converge i f i s made.  0  s c h e d u l i n g equations  (3-8) and (3-9) were s o l v e d f o r  the f i x e d head, two p l a n t case, u s i n g the constants g i v e n i n Chapter  II.  The value of head was assumed to be 352.3 y a r d s .  The t a i l w a t e r e l e v a t i o n was assumed to be constant at 88 y a r d s . i  1  The  flow diagram,for  the computer s o l u t i o n o f these  equations  i s shown i n P i g . 5. P i g . 6 shows the v a r i a t i o n of A w i t h power d e l i v e r e d f o r most economic s c h e d u l i n g . PJJ with V  P i g . 7 i n d i c a t e s the v a r i a t i o n of  f o r f i x e d l o a d l e v e l s while P i g . 8 shows the c o r r e s -  ponding v a r i a t i o n s i n Q, the hydro p l a n t discharge w i t h If .  A  l o a d c o n s i s t i n g of seven steps, and the schedule r e s u l t i n g f o r a v a l u e of  equal t o s i x , i s shown i n P i g .  9.  of the t o t a l volume of water used, V^, , with T , p a t t e r n i s shown i n Fig» 10.  The v a r i a t i o n f o r t h i s load  The r e s u l t s of the computer s o l u t i o n  f o r the system l o a d shown i n F i g . 9. are t a b u l a t e d i n Table 1.  r=  6  Period(Hrs). P  g i  P  Hi  2i  h  i  V. I  A. I  0-4  27.5677  34.8956  26.972  264.3  388399  2.6434  5-8  43.2554  41.0288  29.564  264.3  425726  3.0185  9-11  59.3817  47.3565  32.527  264.3  351289  3.4271  12-13  75.9831  53.8964  35.896  264.3  258451  3.8745  13-16  59.3817  47.3565  32.527  264.3  351289  3.4271  17-20  43.2554  41.0288  29.564  264.3  425726  3.0185  21-24  27.5677  34.8956  26.972  264.3  388399  2.6434  TABLE 1.  V  T  S o l u t i o n of the C o o r d i n a t i o n Equations f o r the Load Schedule shown i n P i g . 9 .  17 INPUT »  Constants  INPUT  A. X  Yes  Is Jump Switch 2 Up? No  Yes  Is Jump Switch 1 Up? No Find P 3> P 3  T—*\  i+1  3+1  Si  Yes  H i  INPUT V T INPUT  A t i , PD;  •No  fTs  SZ  At  1  = 24?'  a'+P^-P^.-pf.  Is P  No Find • No  Yes  Is H A t i := 24? Yes  No  F i n d Q±  i+1 Is  = 24?  Z^j  k+1  Yes F i n d V ? .IT Yes  Is Jump Switch 2 Up?  i  OUTPUT i+1  A t  ' Si' Hi>2i>V i P  P  VT^-V?.  <  No F i n d Yk+1  X  OUTPUT  No Is  I l A t i = 24?  OUTPUT Yes  l<e d  Is  7k+i  ,  v  J T  Yes Yi  Jump Switch 3 Up No OUTPUT  Fig. 5  PLJ  Flow Diagram f o r the Computer S o l u t i o n of the C o o r d i n a t i o n Equations f o r the F i x e d Head C a s e .  C?  15  6  A  0 Fig.  6  V a r i a t i o n of Pp with A f o r Constant If . 1  20  40  Fig. 7  60  V a r i a t i o n of the parameters A, Q and P f o r the coordination equation s o l u t i o n . R  20 Fig.  40 8  60  80  Qdds^/sec)  V a r i a t i o n of Q with if f o r Fixed Load L e v e l s .  P (M¥) R  V a r i a t i o n of PJJ w i t h X f o r Fixed Load L e v e l s .  15  0  80  19  150  i  .  i  0  4  8  Pig.  0  9  2 F i g . 10  i  ,  .  .  12  16  20  i  24  T(HRS)  A T y p i c a l Schedule f o r the F i x e d Head Case ( X = 6 ) .  4  6  8  10  V a r i a t i o n of the T o t a l Volume of Water used with T f o r the Load P a t t e r n shown i n P i g . "9»»  Y  20  3olo2  Equal Incremental Cost Method I f the t r a n s m i s s i o n l i n e l o s s e s are n e g l e c t e d then the  f o l l o w i n g scheduling equations r e s u l t s f  ~  2  J  =  S  f  O  >  o  o  ( 3 ~1  3 )  x  A~  ! _ gKT(ah P  H  =  + bh + c)  2  s-^  >  ooo (3-14)  2eK"y(ah^ + bh + c)  A  These equations were a l s o solved f o r the above two p l a n t case, Fig  0  11 shows the v a r i a t i o n of A w i t h d e l i v e r e d power f o r most  economic scheduling* T  Figo 12 i n d i c a t e s the v a r i a t i o n of P^ with  f o r f i x e d l o a d l e v e l s while Figo 13 shows the corresponding  v a r i a t i o n of Q with  •  In order t o compare the two methods of s c h e d u l i n g , i t i s necessary to compare the c o s t s o f the schedules r e s u l t i n g each method f o r a common system l o a d .  from  Consider the f o l l o w i n g  case where the combined output o f the two p l a n t s i s s e t a t 40 MW f o r 8 hours, 70 MW f o r 8 hours and 110 MW f o r 8 hours and the p l a n t s are scheduled u s i n g the equal incremental c o s t method. The r e s u l t i n g schedule i s found f o r a value of T equal to f i v e . th The a c t u a l power d e l i v e r e d during the i hour i s g i v e n bys P  Di  =  P  Si  +  P  Hi" L i P  0  0  0  ( 3  1 5 )  where P . = B....P P . P . + B-^P Li 11 S i + 2 B 12 S i Hi 22 H i 2  T  2  10  0  tT  The next step i s to use the c o o r d i n a t i o n equations t o schedule the p l a n t s to meet t h i s d e l i v e r e d l o a d .  F i n a l l y the c o s t of  6  15 •  r 0  20 P i g . 12  40  60  80  P (Mtf) H  V a r i a t i o n of P^ with 7^ f o r F i x e d Load L e v e l s -  V a r i a t i o n of the parameters A ~^ Q, and P^ f o r the equal incremental cost s o l u t i o n . f  f  to  22  o p e r a t i o n of the two schedules must be compared.  I t i s necessary  to use the same t o t a l volume of water i n the c o o r d i n a t i o n equations s o l u t i o n as i n the equal incremental cost s o l u t i o n . Equation (3-12) was used to accomplish t h i s r e s u l t . r e s u l t i n g schedules are t a b u l a t e d i n Table I I .  The  I t i s t o be  noted t h a t i t r e q u i r e d f i v e i t e r a t i o n s to f i n d the proper value of If  f o r the second schedule.  Equal Incremental Cost Method Period hrs.  Si  Si  Hi  Li  3.5338  36.4662  27.6141  2.05884  1.4713  9-16  23.8505  46.1495  31.9441  2.41275  3.3700  17-24  50.9393  59.0607  38.7836  2.88463  7.5214  0-8  T  2832245  C o o r d i n a t i o n Equations Method  r  k+l  V,T  5.00000 5.16537 4.72663 4.75045 4.74853 4.74855 Period h r s .  2741573 26S6724 2840600 2831514 2832251 Li  •"Hi  Si 3.4658  36.5371  27.6436  2.12294  9-16  23.8275  46.1737  31.9558  2.58179  17-24  51.0068  58.9897  38.7427  3.25038  0-8  TABLE I I .  2832251  Comparison of Schedules Obtained by the Equal Incremental Cost Method and the C o o r d i n a t i o n Equations' Method.  23 The  t o t a l c o s t of o p e r a t i o n over the day was found f o r the  two schedules.  In both cases i t was equal to $7496.56.  The  cost of o p e r a t i o n f o r d i f f e r e n t v a l u e s of t o t a l volume of water used was found f o r both methods of s c h e d u l i n g .  The r e s u l t s  are shown i n Table I I I .  Cost $/day Coordination Equations Method  T  Equal Incremental Cost Method  2507401  7496.57  7496.56  2280590  7904.27  7904.28  1898707  8787.46  8484.55  1765489  9189.10  9189.23  1678016  9503.74  9503.90  TABLE I I I .  Comparison of the Cost of Operation of the C o o r d i n a t i o n Method and Equal Incremental Cost Method of Scheduling.  I t i s evident from the r e s u l t s shown i n Table I I I that i t i s immaterial, f o r the p a r t i c u l a r system s t u d i e d , whether the t r a n s m i s s i o n l o s s c o e f f i c i e n t s are i n c l u d e d i n the scheduling equations or n o t .  Other  sources^ have i n d i c a t e d that s u b s t a n t i a l  savings can be r e a l i z e d by t a k i n g " l o s s e s i n t o account.  I t would  appear t h a t t h i s i s not true i n a l l cases and t h a t exceptions to t h i s r u l e do occur. In t h i s p a r t i c u l a r system, since %>22^ l l ' B  ^  e  inclusion  of the l o s s constants f o r c e s the hydro p l a n t to produce more power during low l o a d l e v e l s and l e s s power d u r i n g h i g h l o a d l e v e l s as compared t o the case where the l o s s e s are ignored.  24  The  r e s u l t i n g r e d u c t i o n i n system l o s s e s means t h a t the t o t a l  r e q u i r e d g e n e r a t i o n i s l e s s f o r the second  case.  i s of the same order of magnitude as ^22* of these constants has l i t t l e  For t h i s  ^us ^  e  system  inclusion  e f f e c t on the s o l u t i o n as f a r as  economy of o p e r a t i o n i s concerned. I f l o s s e s are ignored the s o l u t i o n of the s c h e d u l i n g equations can be speeded up by a f a c t o r of a t l e a s t t h r e e . However,if  speed i s not an important f a c t o r , but accuracy i s , i t  i s % e t t e r t o include" the "loss c o e f f i c i e n t s s i n c e the system l o s s e s are then accounted f o r a u t o m a t i c a l l y . 3.2.1  V a r i a b l e Head Case The  equations f o r the v a r i a b l e head case are e s s e n t i a l l y  the same as those used f o r the f i x e d head case and are g i v e n by: 8P  T  | f - ^  Y § £ 3P  =  + A  H  dP  A-  =  A  ... (3-16) A  ... (3-17)  H  P *2  T=T e X o  X  d H  dt ... (3-18) 24 27  These equations were f i r s t d e r i v e d by Glimn and Kirchmayer The one major change i s that If  i s no longer a constant.  ' These  equations are now a p p r e c i a b l y more d i f f i c u l t to solve than those f o r the f i x e d head case.  I t i s necessary t o employ numerical  i n t e g r a t i o n methods when s o l v i n g the v a r i a b l e head scheduling equations. As b e f o r e , the s c h e d u l i n g equations can be w r i t t e n as:  25  *2  P  =  s  "  f  -T  2  +  B  1 2 - 2 —  "  A  2 B  ...(3-19)  11  ! . glCTtah + bh + c) _ 2  P „ = —  =—*  2 B  1^-§-  :  2eK?(air + bh + c)  +  A  p  ...(3-20)  ,  2 B  22  In the f o l l o w i n g i t i s assumed t h a t the r e s e r v o i r area i s e s s e n t i a l l y constant over the time p e r i o d under  consideration.  T h i s assumption i s a good one f o r the p a r t i c u l a r r e s e r v o i r under consideration. Now | § = K(2ah + b ) ( e P  2 H  + gP  H  + i)  -  '  and (eP  2  + gP„ + i ) =  H  H  =-£ K ( a h + bh + c) 2  Also q- Q — j - dt  h ( t ) = ho + /  ... (3-21)  'o therefore h  e  q= T -  .. (3-22)  or Q = - Ah + q hence  r= r  e z o'  where  /^t z=  (2ah b)(.q-Ah) (all + bh + c)A +  ] ^ o  d  t  ...(3-23)  26  B r i e f l y , the s o l u t i o n of these equations was c a r r i e d out as f o l l o w s . and  Knowing h  (3-20) f o r P  g  i t i s p o s s i b l e to solve equations  Q  (3-19)  and P„ . Hence, Q can be found from o o T h i s value of Q i s s u b s t i t u t e d i n t o equation 0  equation (2-2).  Q  (3-22) to o b t a i n h .  (3-21*) cane be%ew!*.itten as f o l l o w s :  o  .  /i+At h  (i+At) =  i  h  +  h  i  d  t  R e f e r r i n g to r e f e r e n c e 38 an approximation to t h i s e x p r e s s i o n i s g i v e n by: = h  + At  ±  (1 + i V + £  v  2  + | V  +  3  ...,)h  ±  ... (3-24) where vh = h. - b .  ( i  _  A t )  .  E q u a t i e n (3—24) cannot be a p p l i e d u n t i l at l e a s t i i n i t i a l o r d i n a t e s , h^,  h^ of the s o l u t i o n have been  o r d i n a t e s h^, hg,  found.  Once  tu have been found the c a l c u l a t i o n  may  be continued by the use of the open type formula given by equation  (3-24).  points.  The d i f f i c u l t y l i e s i n o b t a i n i n g these i s t a r t i n g  Since the r e s e r v o i r head v a r i e s slowly i n most  cases  i t was decided t h a t one s t a r t i n g p o i n t would pe s u f f i c i e n t . E q u a t i o n (3-24) now becomes: h  i+l  =  h  i  +  h  i  A  ••• ( "25)  t  3  T h i s i s the simple E u l e r approximation.  I f At i s not chosen  too l a r g e and i f the head does not v a r y too g r e a t l y over the p e r i o d under c o n s i d e r a t i o n t h i s approximation  should give  a c c u r a c i e s c o n s i s t e n t with the accuracy of the data being Hence, knowing h  Q  and h  i t i s p o s s i b l e to f i n d ( 4.£+y) v-n  Q  0  used.  27  Before the s c h e d u l i n g equations  can be solved f o r the  next p o i n t i n time i t i s necessary to solve equation for z  (3-22)  As before l e t :  (o+At)*  z  i+l = H  +z  i  where (2ah+b)(q -Ah) (ah Knowing  l i  (  o  +  A  t  )  and z  2  + bh + c)  (o+At)  i t i s p o s s i b l e to propagate the  s o l u t i o n of the scheduling equations to a p e r i o d of time At a f t e r the i n i t i a l  instant.  The flow diagram f o r the computer  s o l u t i o n of these equations i s shown i n P i g . 14. Table IV i l l u s t r a t e s the s o l u t i o n of these equations f o r a value of l o a d equal t o 150 MV f o r a 24-hour p e r i o d .  A constant  l o a d was used so t h a t i t would be p o s s i b l e to see how the discharge Q v a r i e d w i t h time.  In the p a r t i c u l a r problem under  c o n s i d e r a t i o n the area of the r e s e r v o i r i s so l a r g e as to make the v a r i a t i o n of head with time very s m a l l . of s o l u t i o n obtained more apparent, was decreased by a f a c t o r of t h r e e .  To make the type  the area of the r e s e r v o i r The v a r i a t i o n s of discharge  and. head with time are p l o t t e d i n P i g . 15 and 16, r e s p e c t i v e l y . As was brought out i n the i n t r o d u c t i o n , the discharge was such t h a t the head i s kept as high as p o s s i b l e f o r as long as p o s s i b l e to conserve  energy.  ^ - V a r i o u s v a l u e s of At where t r i e d i n the s o l u t i o n of the scheduling equations.  F o r the problem under c o n s i d e r a t i o n a  value of At equal to one hour gave r e s u l t s which were accurate to w i t h i n the l i m i t s s e t by the r e s t of the s o l u t i o n . Studies conducted  on t h i s system, with the area modified t o  28  INPUT Constants INPUT, A, T INPUT P e r i o d . Pp  Find p  ic l s  3+1  r  p  i  3+p  i  P ^ Hi H  J_p  _p 0  Yes  S i Hi Di L i <C No + i  r  Find A 3  +  r  1  F i n d Qj i+1  F i n d h i , z±, e i z  Find  hi+i.Yi+i  OUTPUT Time(hrs) ,P ,P ,% ,  I3i  Q  No No  i '  h  l > V  Si  if  I  END of P e r i o d Yes END of 24 hours Yes STOP  Fig. 14  Flow Diagram f o r the Computer S o l u t i o n of the V a r i a b l e Head Problem.  29 P  Hours  Si  P  Hi  %  h  i l +  T  ± + 1  1*9.5305  81.9732  53*9212  26^.2209  3^9876  1.  ^9.5279  81.975 *  53«938^  26l4.ll4.l8  3. ^9627  2.  I4-9.5056  82.OOO5  53.9728  26fc.Q627  3.^925^  3.  1*9.^53  82.0U45  5^.0213  263.9835  3.^756  k.  k9.k079  82.1077  5^.08*i4  263.901*1  3.W13 *  5.  1»-9.3312  82.1920  514.1631  263.82146  3.^7387  6.  U9.2368  82.2955  5^.2565  263.7^50  3.^6516  7.  U9.1233  82.U205  5U.3661  263.6651  3.^5521  8.  U8.9905  82.5669  5^.^923  263.5850  3.^03  9.  I48.8392  82.7328  5^.633^  263.50U7  3.U3162  10.  I48.6699  82.9191  5^.7899  263.1*2^1  3.^1799  11.  U8.I4807  83.127?  5*4.9637  263.3^31  3.to3il4  12.  I48.2729  83.3560  55-153^  263.2618  3.38709  13.  U8.0I468  83.6058  55.3600  263.1802  3.36985  ll*.  U7.7992  83.8782  55.58I4I  263.0981  3.351M  15.  147.5321  8I4.1718  55.8257  263.0157  3.3318I  16.  U7.2U63  8I4..I488O  56.0851  262.9327  3.3HOU  17.  I46.9I4O3  8I4.8262  56.3626  262.8I493  3.28913  18.  I46.6158  85.I85I4  56.6575  262.7653  3-26608  19.  I46.2688  85.5693  56.9729  262.6807  3.2I4192  20.  - I+5.9027  85.9753  57.3070  262.5956  3.21665  21.  I45.51U8  86.U051  57.6612  262.5098  3.19030  22.  I45.IO70  86.8579  58.0353  262A233  3.I6289  23.  I4I4.6778  87.3337  58.14293  262.3362  3.13^2  2I4.  I4I4.2271  87.8361  58.8I467  262.3362  3.13^2  TABLE IV.  1  1  S o l u t i o n of the V a r i a b l e Head C o o r d i n a t i o n Eauations f o r a Constant System Load of 120 MY C T = 3.5)  30  i  0  i  .  .  4 F i g . 15  .  8  ,  1  12  .  .  1  16  .  20  1  •  24 TIME(HRS)  V a r i a t i o n of Discharge with Time f o r the Schedule shown i n Table IV.  266 {  264 h  262 [  12  F i g . 16  16  V a r i a t i o n of Head with Time f o r the Schedule shown i n Table IV.  20  24 TIME(HRS)  .31  give l a r g e head v a r i a t i o n s , i n d i c a t e d that At must be more c a r e f u l l y .  I t appeared that i n the p a r t i c u l a r case  choosing At equal to one The  chosen studied  h a l f an hour would give good r e s u l t s .  stream flows, q^, were assumed to vary l i n e a r l y over  short periods  of time.  The  expression  used to d e s c r i b e  these  was :  *i+i I t was  = *i  i i  +  A t  f e l t t h a t t h i s approximation was  sufficiently  accurate  since i t i s v e r y d i f f i c u l t to p r e d i c t the flows at a l l . The  s o l u t i o n of the v a r i a b l e head case f o r the  shown i n P i g . 17 i s t a b u l a t e d  i n Table V.  f o r the system under study the head may "' 1' I f the head v a r i e s c o n s i d e r a b l y  be  system load  I t i s apparent that considered  constant.  then i t g e n e r a l l y f o l l o w s  the r e s e r v o i r area w i l l also vary.  In t h i s case the above  approach must be a l t e r e d s l i g h t l y .  At each p o i n t i n the  a new  value  of area must be c a l c u l a t e d and  an expression  that  solution similar  to equation (3-25) used to f i n d U s u a l l y , both head and discharge of time.  For the  almost l i n e a r .  functions  case under c o n s i d e r a t i o n the r e l a t i o n s h i p i s  This i s due  head are r e l a t i v e l y s m a l l . discharge  are n o n - l i n e a r  to the f a c t that the v a r i a t i o n s i n I t i s to be noted t h a t the p l o t of  v s . time i s s l i g h t l y concave upward.  140 t  D  120  100  80 H 60 P4  40  20  6  0  10  12  14  16  18  20  22  24 TIME(HRS)  P i g . 17  A T y p i c a l Load Demand and the R e s u l t i n g Hydro P l a n t Schedule f o r a Value of T = 5.0.:  ro  33  1. 2. 3.  P  70.00  28.3677 28,3663  1+5-2053 1+5.2062  31.!+917 31.!+936  261+.3000 261+.2871  10.0000 10.0000  80.00  35.93 *8 35.9312 35.921+2  1+8.635!+ 1+8.6U00 1+8.61+66  33.1677  261+.2871 261+.2732 261+.2593  10.0000 10.0000 10.0000  1+3.6016 1+3.5920  52.121+1 52.1350  3!+. 9627 31+. 9703  261+.2593  10.0000 10.0000  95.00  1+7.1+720 1+7.1+590  53.8921 53.9068  35.9065 35.9166  261+. 21+1+3  26I+.2288  10.0000 10.0000  90.00  1+3.5780 1+3.5611  52.11+88 52.1661+  31+.9797 3!+. 9908  261+.2288 261+.2138  10.0000 10.0000  95.00  1+7. kkl6  1+7.1+209  53.921+7 53-9!+63  35.9282 35-9!+19  261+.2138 261+.1982  10.0000 10.0000  100.00  51.329!+ 51.3063  55.7175 55.71+39  36.9093 36.9261  261+.1982 261+.1821  10.0000 10.0000  120.00  67.21+56 67.2152  62.96U8 62.9967  1+I.IO93  1+1.1315  261+.1821 261+.1631+  10.0000 10.0000  83.6786 83.61+18 83.599!+  70.1+1+92 70.1+879  70.5322  1+5.8509 1+5.8801 1+5.9131  26i+.i63i+ 261+. 11+19 261+.1201+  10.0000 10.0000 10.0000  130.00  75.3016 75.2553  66.7731 66.8211+  1+3.1+792 1+3.5130  26I+.120U 26I+.1003  10.0000 10.0000  135.00  79.3853 79.3338 79.2773  68.69UO 68.71+91 68.8085  1+1+.7133  261+.1003  1+1+.7523  261+.0795  26I+.0586  10.0000 10.0000 10.0000  130.00  75.1J+91+ 75.0878  66.93M 66.9975  !+3.59H 1+3.631+6  26I+.0586 261+.0385  10.0000 10.0000  120.00  66.935!+  I+I.3289  66.8717  63.290I+ 63.357!+  1+1.3721+  26I+.0385 26I+.0197  10.0000 10.0000  58.8527 58.787!+  59.701+8 59.7733  39.1970 39.239!+  261+.0197 261+.0021  10.0000 10.0000  90.00 l+.  ; ' 5  6. 7. 8. 9. 10. 11. 12.  ll+O.OO  13.  ll+.  15.  16.  110.00 17.  Qi  Di  p  Hours  b  S i l  1  33.1717 33.1767  l+l+. 791+2  261+. 21+1+3  95.00  1+6.9960 1+6.9307  5*+. 3920 5I+.I+60I+  36.209U 36.21+87  261+.0021 263.9861+  10.0000 10.0000  100.00  50.8291+ 50.7596 50.681*9  56.21+23 56.3162 56.391+7  37.2295 37.2729 37.3189  263.9861+ 263.9701 263.9537  10.0000 10.0000 10.0000  90.00  1+2.9218 1+2.81+73 U2.7692  52.8372 52.9153 52.997*  35.3796 35.^233 35.^690  263.9537 263.9385 263.9232  10.0000 10.0000 10.0000  80.00  35.1269 35.01+86  1+9.1+829 1+9.561+9  33.6397 33.6833  263.9232 263.9091  10.0000 10.0000  70.00  27.5193 27.UM9  1+6.091+6 1+6.1756  31.9636 32.001+2  263.909I 263.8959  10.0000 10.0000  18. 19. 20. 21. 22. 232k.  53ABLE V:  1  Solution o f the Coordination Equations, Corresponding to the Load Demand shown i n F i g . 17, f o r the Variable Head Case.  35  CHAPTER IV SOLUTION BY INCREMENTAL DYNAMIC PROGRAMMING 4.1  Discussion of Dynamic Programming Techniques • • v  In t h i s chapter the two-plant scheduling problem i s solved using dynamic programming techniques. The v a r i a b l e head, v a r i a b l e t a i l w a t e r case i s considered with a l l r e s t r i c t i o n s applicable to the problem being r i g o r o u s l y accounted f o r . The use of dynamic programming techniques i n f i n d i n g the most economic 20 mode of operation of a hydro-thermal system i s not new.  Little  solved the long-range scheduling problem using t h i s type of 30 procedure.  A paper to be published by Bernholtz and Graham  describes the use of dynamic programming procedures i n the s o l u t i o n of the short-range problem f o r the f i x e d head case. The procedure followed i n t h i s chapter to solve the v a r i a b l e head case i s s i m i l a r to that followed by Bernholtz and Graham.  In theory, the method as described i n t h i s chapter  could be applied to the long-range problem.  However,- the  untenable assumption must be made that the stream flows are known functions of time. Furthermore, other problems such as s p i l l a g e occur i n the long-range problem.  Over short periods of time l i t t l e can be  done to avoid s p i l l a g e i f an excess of water occurs.  The only  a v a i l a b l e a l t e r n a t i v e to s p i l l i n g i s to operate the hydro plants at maximum discharge. In the long-range problem i t may be p o s s i b l e to avoid excess spillage^ by s t a r t i n g to draw down the r e s e r v o i r s at an e a r l i e r date and by operating the plants at higher discharges throughout the year.  A computer s o l u t i o n can  36  be f o r c e d to f o l l o w these a l t e r n a t i v e s a u t o m a t i c a l l y by a p p l y i n g 19 high p e n a l t y cost f a c t o r s  to any s p x l l a g e which may  occur or  by d e s i g n i n g the programme so that the r e s e r v o i r heads are not allowed to exceed c e r t a i n l e v e l s throughout the y e a r . I t i s very d i f f i c u l t to prove t h a t a true extremal i s obtained when u s i n g dynamic programming procedures.  No  attempt  42 i s made to do so i n t h i s t h e s i s .  Bellman  proves t h a t a t r u e  extremal i s reached f o r s e v e r a l simple examples.  A brief  d e s c r i p t i o n of dynamic programming procedures i s g i v e n i n Appendix C. 4.2 Method of S o l u t i o n The problem  i s to minimize the cost of thermal power over  some predetermined p e r i o d , say one day. to  That i s , i t i s necessary  minimize t H i=l  f.(P )At  •  q  1  ...  (4-1)  S  s u b j e c t to the r e s t r i c t i o n s 0<Po • <Po, <P S min oi S max Q  0< Q. . l mm v  l  +  v  2  * *  +  P„. + P .= Hi Si c  P . Hi max TJ  where  £ Q. < Q, l l ,  v  i  max a  v  * ** 24  +  v  =  V  T  **•  (~^ 4  2  P ^ . + P, . Di Li  ^ P-p,. + P . Di Li  v. = Q.At  m  T  P  q  o  . mm  = volume of water discharged d u r i n g the i t h p e r i o d .  = the t o t a l volume of water scheduled f o r use.  37  f u r t h e r l i m i t s , which correspond t o two extremes of o p e r a t i o n , may be p l a c e d on the o p e r a t i o n of the hydro p l a n t .  These  limits  apply to V\ , the volume of water l e f t f o r use a t the beginning of  the i t h hour.  i s r e s t r i c t e d by the f o l l o w i n g  limits  V. . < V. < V. x mxn x x max  ... (4-3)  I f the hydro p l a n t i s run at f u l l d i s c h a r g e u n t i l the scheduled amount of water i s used then shut down, V. . results. ' x. mxn is left  i n o p e r a t i v e u n t i l there i s j u s t enough time l e f t to  discharge the scheduled amount of water, V. to  If i t  results.  s h i p p i n g or other r e s t r i c t i o n s the range of  Due  may be f u r t h e r  l i m i t e d by the f a c t t h a t a zero discharge c o n d i t i o n i s not p e r m i s s i b l e and t h a t some water must be c o n t i n u o u s l y d i s c h a r g e d . This i s a convenient method of i n c l u d i n g these r e s t r i c t i o n s . The method of s o l u t i o n i s based on the i d e a of approximation i n p o l i c y space r a t h e r than i n f u n c t i o n space.  This i s a l o g i c a l  approach since the main problem i s t o determine the s t r u c t u r e of the  optimum p o l i c y , V ^ ( t ) , and not the r e t u r n f ( V \ ( t ) ) .  type of procedure has two advantages: "' (a)  i t always leads to monotonic  This  43 approximations,  and,  (b) i t i s u s u a l l y one p a r t of the problem about which a c e r t a i n amount i s known from experience.  .  The c a l c u l a t i o n s are based on the f o l l o w i n g assumptions: (1) One hydro p l a n t and one thermal p l a n t are t o be considered. (2) The system l o a d i s assumed t o be a s e r i e s of step f u n c t i o n s of  time.  (3) The hydro p l a n t has a v a r i a b l e head r e s e r v o i r and a variable tailwater  elevation.  38  (4)  The cost curve for the thermal plant i s assumed to be s t r i c t l y convex and one which can be approximated by a quadratic function.  (5) The period, At, i s assumed to be one hour and the calculations are based on average hourly discharges and elevations. This l a t t e r assumption should give a solution accurate to within the l i m i t s set by the input data.  I f the head varies  very rapidly, i t may be necessary to choose At to be smaller than one hour i n order to have a reasonably accurate solution. The f i r s t step of the solution i s to assume an i n i t i a l set of average hourly discharges.  These must s a t i s f y a l l the  r e s t r i c t i o n s stated i n 4-2 and 4-3. Having specified these discharges i t i s possible to f i n d the average value of head for each hour from the following expressions: H. + H. i+1 1 2 1  H. x ave  AH. x 2  = H. x  (4-4)  where At  AH. = x  A quadratic function was used to describe the variation of t a i l water elevation with turbine discharge. H = Ti  TQ  2 ±  + sQ + t A  TJhat i s 000  (4-5)  The average net head i s given by h. = H. -  AH.  x  - H,  4-6  39  Having found the net head and knowing the average discharge, the average hydro generation can be found from the f o l l o w i n g equation: P  =  H i  K U l ^  + bh. + c)(eQ.  2  + gQ  2  + i)  i  ...  The hourly thermal generation must now be found. be accomplished P  Si  also  P  Hence  Pg.  B  +  L i  P  i n the f o l l o w i n g manner.  Hi = Di P  = B P  S i  + P  = P  n  ll Si P  2  H i  +  +  — 1 2  S i  + B^g.  D i  This can  I t i s known that  L i  + 2B P P .  2  ( 2 B  P  12 Hi " P  H  2 2  + 2B  2  P  B P .  +  Si  +  ( B  (4-7)  <-> 4  8  2  H  1 2  P .P g  22 Hi P  2  +  H i  +  P  B^.  2  D i " Hi> = P  0  Solving f o r Pg^ gives • 12 Hi 2 B  P  Si  =  P  2B 11  +  1  ~  1  ± 2  2 B  12 Hi B 11 P  •  )2  -^ 22 Hi '11 B  P  2 +  P  ...  -Di ~  x  H -i>' P  (4-9)  The r e s t r i c t i o n s on the system force the two roots of t h i s equation to be p o s i t i v e and r e a l . Only one of the two roots obtained from equation (4-9) w i l l be p h y s i c a l l y r e a l i z a b l e , hence there i s no d i f f i c u l t y  i n choosing the c o r r e c t root of t h i s  equation. 4.3  D e r i v a t i o n of C r i t e r i o n Function From equation (4-8) i t f o l l o w s that  40  dP . H  +  dPg.  = dP. Li 3P  apTT Si  Li  ^ S i  1 or  d P  Si  aP 9P77 Hi  L i  = -  1 -  +  L  d P  l  3P. Hi 3P  9P  Hi  d P  Li  ... (4-10)  Hi  Si  I f the o r i g i n a l hydro generation i s changed by some small amount dPg^ then the corresponding g e n e r a t i o n i s given by equation  change i n the thermal  (4-21).  C o n s i d e r i n g the d i s c r e t e  .30 i s made"  case, the f o l l o w i n g approximation 3P . Lx S>PHi T  1 AP . S  = 1 -  (P„.' - PHx°) Hx ' x  Li  3 P  * Si P  p  Hi  o  o  p  '"Si  (  4 _ 1 1  )  where PJJ^° denotes the o r i g i n a l value and PJJ^ ' the new value of hydro g e n e r a t i o n .  This change i n thermal  generation i s worth  approximately  AF  AP  - -  AP a  r  AF A P  si  1 -  Si  3 P. Li Hx  3P  ( P Hi  Lx  Si p r  -  o o Hi ' S i p  r  P.  Hi (4-12)  Define  G(P .° -pS i, ) = Q  H  f  1 -  AF  H  AP  Si  1 -  3P. Li 9P. 9P 9P  Hi  Li Si 3  o o Hi ' S i p  r  ... (4-13)  41  = desired c r i t e r i o n function where  ^C^si^ The  will  i  =  ncremen  "*' l a  cost  o f thermal power.  p r o b l e m i s t o choose a n a l t e r n a t i v e  hydro schedule which  minimize  *T  f(P  S i  °)AP  °  s i  . . .  ( 4 - 1 4 )  i=l or  e q u i v a l e n t l y t o maximize 24 G(P  H i  °  P  (  s i  °)(P  H i  '  "  V  i=l or  since Pg^'  i s independent  f  G  < Hi°' P  o f JPJJ£°> t o maximize  Si  P  0  )  P  Hi  —  ( 4  1 5  >  i=l  The  fact  that maximizing  from e q u a i i o n hydro  f  < ^Hi°'  > Pg..• , e q u a t i o n  valid  e  ( 4 - 1 2 )  <  l  minimizes  i t c a n be s e e n t h a t  g e n e r a t i o n means a d e c r e a s e  versa. . If,PJJ^ PJJ'  where  ( 4 - 1 1 )  ( 4 - 1 5 )  u  a  +  'i  o  n  ( 4 - 1 4 )  follows  an i n c r e a s e i n  i n t h e r m a l g e n e r a t i o n and v i c e ( 4 - 1 2 )  i s negative.  i s positive,  while i f  I n t h e above  i t i s equally  and more l o g i c a l t o d e a l w i t h t h e d i s c h a r g e s .  i n s t e a d o f d e a l i n g w i t h P J J ^ ° and P ^ ' ,  ° and  Thus,  ' w i l l be  c o n s i d e r e d i n t h e f o l l o w i n g work. 4.4  C h o i c e o f a New P o l i c y ' f  The  original policy  P^i  9  $2  9  i s given by  • • • • Qj[ oo.o Q24  The  p r o c e d u r e a d o p t e d above  f o r the s o l u t i o n of t h e problem l i m i t s  the  s e a r c h f o r a new s e t o f d i s c h a r g e s  to within a r e l a t i v e l y  42  narrow range about the o l d d i s c h a r g e s . .... 0-24° i w i l l be Y.  0  If a  f o l l o w e d , then the corresponding  s  o  schedule  ••• «  values of v\  where V ° l+l V V  0  24  V ° 0  = V.° - v.° l l - v ~ 24  0  v  = V  ...  T  (4-16)  The hydro p l a n t can be i n any of the f o l l o w i n g three s t a t e s at the' beginning  of the i t h i n t e r v a l :  V - ' V' V v  ••• -  +v  (4  17)  Here v represents a small change i n the q u a n t i t y of water scheduled- f o r use during the i t h hour.  For the f i r s t  the only p e r m i s s i b l e s t a t e i s  For the other periods  = V^,.  any of the above three s t a t e s or any two i b l e . Table VI  of them may  i 1' +  4.5  V  i 1 +  +  v  be  permiss-  i n d i c a t e s t h a t there are at most nine ways i n which  i t i s p e r m i s s i b l e to go from the s t a t e s V^-v, V  period  V^, V^+v  to Y^ ^ +  - v,  '  S o l u t i o n by Incremental  Dynamic Programming i  The problem i s to be An i n i t i a l  solved by u s i n g an i t e r a t i v e  trial  s e t of discharges  i n the s t a t e s Y/  at the beginning  3  from t h i s i n i t i a l  approximation  i s chosen which r e s u l t of each hour.  ±  T  = Max  The  return  i s given by the f u n c t i o n a l  equation : f °(Y )  procedure.  +  f ° ±mml  (V-v)  43  Prom State  To State i  v  Y V  i  ..  .  V  "  +  Qi  v  - v  i + 1  i  V  i  +  Resulting Discharge  Q - AQ i  1  2i  - v  i + 1  Q. -  2AQ  i  V  i +  v  V V  fi.  i + 1  l 1  +  - AQ %  T  +  TABLE VI  P o s s i b l e States of Hydro Plant and R e s u l t i n g Discharges.  where f ^ V  = Max ( G ( v ) ) a  1  The next step i s to choose some transformation which maximizes the r e t u r n .  The new r e t u r n can be denoted by  f - j ^ ) where 1  f  i  1 (  V  ~ i° Vf  (  In general, t h i s process i s continued u n t i l the s o l u t i o n converges.  A d e t a i l e d d i s c u s s i o n of t h i s method and i t s . 42  convergence i s given by Bellman.  I t can be seen that the  44  process c o n s i s t s ,of a sequence of maximizations, each i n one variable. The word "incremental" i s used as d e f i n e d by B e r n h o l t z 30 and Graham.  That i s , i t r e f e r s to the f a c t that f o r each  i t e r a t i o n the changes i n the value of d i s c h a r g e are l i m i t e d to a narrow range about the o r i g i n a l d i s c h a r g e . l i m i t the process i n t h i s way  I t i s necessary t  since the v a l u e s of G^ are only  a p p l i c a b l e to incremental changes i n output. For each hour of the day i t i s necessary to f i n d the p e r m i s s i b l e s t a t e s i n which the hydro p l a n t can e x i s t .  Having  found these, i t i s then necessary to f i n d the optimum mode of o p e r a t i o n f o r each of these s t a t e s .  As i s common w i t h many  dynamic programming problems the s o l u t i o n proceeds from the end to the b e g i n n i n g of the p e r i o d under c o n s i d e r a t i o n .  That i s ,  the maximization s t a r t s a t the 24th hour and proceeds t o the 1st- hour. For the 24th hour the p e r m i s s i b l e s t a t e s are "V^ ~ s> v  ^24*  V24 + v and the r e s p e c t i v e optimum and only allowable d i s c h a r g e are  - AQ,  Q^ 2  f  R  Q4 2  AG  +  24 24" ( V  E  24< 24>  R  24 ?24  -*  v )  ie  m  a  x  i  m  G  P  weighted outputs ares  m  (  A  )  2  G  + V )  u  = 24 H 2 4 - 2 = 24 H^24  Y  (  '^  P  )  = 24 H^24 G  P  For the i t h hour, 2 ^ i ^ 2 3 ,  + A  2  )  and the s t a t e "V^-v, the  f o l l o w i n g d i s c h a r g e s are p o s s i b l e , Q^,  Q^-AQ, Q^-2AQ.  corresponding weighted outputs over the hour ares  The  45  G.P (Q.) H  R  +  i + 1  (Y  G.P (Q.-AQ)•+ B H  i + 1  I + 1  -v)  (V  )  I + 1  ... (4-19)  The optimum discharge i s the one producing the maximum output and i s denoted by R^("V\-V). the s t a t e s V. and V.  State  V.-v  Allowable Discharges  Weighted Outputs  Qi  ViM)  Q.-AQ  G P (Q.-AQ) + R  Q.-2AQ  G.P (Q.-2AQ)  i  G  Qi-AQ  V.+v  i  + A  2  +  TABLE "VII  1  P  ( V  I  (V  i l  ( V  +  +  B  H  i  V G  - >  1  +  +  ^(V.-.v)  )  1  i + 1  +v)  i r  7 )  +  +  R  R  H  ( ^ f l ) + B  i H^i) P  +  W  B  +  i  +  1  V  i  + 1  (V  ( T  +  (V  i + 1  G P (Q+2AQ) + R .  i+  Qi  +  +  i HfQi>, ' i l^i+l> i  Q AQ  Maximum Weighted Output  i  v  I  ) + E  G P (Q.-AQ)  -  i (  R  H  %(2i  Qi  Q.+2AQ  +  H  +  i  Proceeding i n t h i s way f o r  , Table Vllmay be drawn up.  Q. AQ V  weighted  l  i  +  +  V  i + 1 +  i + 1  1  )  i  (  Y  i  )  v)  -v) R^+v)  >  Allowable States and Corresponding Weighted Outputs.  46  4.6  Computer S o l u t i o n The  is  flow diagram f o r the computer s o l u t i o n of t h i s problem  shown i n F i g . 1 8 .  The  same constants were used i n t h i s  s o l u t i o n as were used i n the previous the hydro p l a n t input-output had  s o l u t i o n , except f o r  constants.  to be found f o r equation (4-17) •  the u n i t s of the thermal p l a n t are  A new  I t was  at equal incremental  assumed t h a t a l l  continuously  so t h a t maximum e f f i c i e n c y i s obtained operating  set of constants  on the  line  when the u n i t s  are  costs. 3  The The  volume of water to be used was  l i m i t s on the thermal p l a n t and 12 < Q  hydro p l a n t  .  are:  < 60 y d / s e c 3  ±  10 - P  set at 3270298 yd  S I  ^  80  MW  PTT. as shown i n Table VII Hi max V. . , V. as shown i n Table V I I . a mm' I max The the  system l o a d i s shown i n F i g . 20.  s o l u t i o n the  constants and  computer expects the  At the  s t a r t of  input of a l l p e r t i n e n t  then awaits the  input of Q., P ., V. . , V.. * i ' D i ' I mm' I max A subroutine was w r i t t e n which w i l l take i n V , 0 , 0 - , P-n• 1 *max' * * i JJi and c a l c u l a t e V. . , V. . I t then p r i n t s out a tape l mm' I max n  x  m  7  r  containing e  r  the q u a n t i t i e s Q. , P-p.. , V. . , V. ^ * i ' D i ' i mm* l  max  i n the  format  r e q u i r e d by the main programme.  I t i s p o s s i b l e to have the  computer place V. . * l mm  d i r e c t l y i n t o main memory i f  r  and V. i max  J  J  this i s desired. The  i n i t i a l p a r t of the  diagram as boxes 4-5"J» and  tabulates  The  s o l u t i o n i s shown i n the computer f i r s t  the r e s u l t s i n main memory.  flow  solves f o r PTT. P™.  i s found  47 INPUT Hydro Constants, L i n e Loss Constants, Cost C o e f f i c i e n t s ,  START  INPUT  2i»  V - VT&QV Ah T  Calculate  H , i  P . U  i  +  1  , H  T  , h.  :  I  :  -  Calculates  8 OUTPUTS  y~t ^At, incrje&e.iftal  •Calculates V. x Outputs  At  c o s t s and weights G.  i=l  x  10  11  Calculate f o r 24th hour V-v, V, V+v Q-AQ,  Check a l l r - e s t r i c t i o n s and make r e q u i r e d changes. F i n d maximum weighted output f o r each permissible state.  Q , Q+AQ  h-Ah, h, h+Ah P -AP , H P A P H  1  H  H +  12 Calculate f o r i t h hour h.-Ah x  Q-J+AQ, Q +2AQ  Qi»  i  Hi' Hit H' P . 2AP . P  P  H  +  Calculates  and PTT. xii Prom t r i a l discharges  Hx max  t  H  Di'  x mxn V, x max  t  Calculates  P  A P  H  Check' a l l restrictions.  13 F i n d maximum  Turn Flag 1 on, to 13.  weighted output and t a b u l a t e i n main"memory along with corresponding Quantities.  14 Flag 1 on-to 15. Flag 2 on-to 16. Flags o f f to 17.  48 15 Calculate f o r i t h hour V\ , h^, Q.-AQ, Q., Q.+AQ P -AP , P , P A P . Check a l l restrictions. H  R  H+  16 Calculate f o r i t h hour Vi+v» h^+Ah, Q.-2AQ, Q AQ, Q P -2AP , P -AP Check a l l restrictions.  Turn Flag 2 on to 13.  H  i+  Hi  18 Test i ' IP >1 to 10 =1 continue  1 7  Calculate and tabulate Table V I I I .  H  HP  v  Qj+AQ, Q  1?  h -Ah, h ]L  1 ?  1+v  Qj_+AQ hj^+Ah  H i - H ^ Hi> A P  P  P  Hi  + A P  21 Jump Switch 1 up Outputs Table V I I I . 23 Jump Switch 1 up Outputs "Hi' V.1 , R. x for optimum schedule.  22 Finds and t ab-ula_t.es Optimum Schedule.  9  25 Input new value AQ', Outputs and stores AQ , v', A h . 1  Hi  19 Calculate f o r 1 s t hour  P  20 Check a l l restrictions. Find weighted output f o r 24th hour and add to appropriate ones f o r 23rd hour.  Flags off to 13.  ±  1  24 Jump Switch 3 up to 5. Jump Switch 3 down. Proceed.  To 5  F i g . 18 Computer Plow Diagram f o r Dynamic Programming S o l u t i o n of Short-Range Scheduling Problem.  3.0  VOID iH  x max 2.0 i i V. • x min  O l>  1.0  final 5-  0  10 P i g . 19 6  original  12  14  16  18  A p l o t of the l i m i t s V.  20  . and V. corresponding mm • I max to a t o t a l discharge of 3270298 yds-* a t a maximum r a t e .  V  r  I  of 60 yds /sec.  The i n i t i a l t r i a l p o l i c y V. f f i n a l optimum p o l i c y are also shown. 1  e  and the  22  24 TIME(HRS)  50  from the f o l l o w i n g equation + (2B Po • 12 b mm  B JP . 22 H i max 0  The  2  n  1 0  ' H i max  + (B  Pc. 11 oi  • +P ,.-Pc. 2  1 1  T  ram  Di  • )  £>i mm'  computer i s programmed so t h a t i t s e l e c t s the p h y s i c a l l y  correct root.  Next the q u a n t i t i e s H., H. H. „ , h. . and ^ l ' i+l l ave l net 7  P^i  are found from equations ( 4 - 4 ) -  ( 4 - 7 ) and t a b u l a t e d i n  main memory. The h o u r l y thermal generations are found from equation (4-9).  The weights G^ and the h o u r l y cost of o p e r a t i o n are  found from equations ( 4 - 1 3 )  and ( 2 - 1 ) .  E q u a t i o n ( 4 - 1 3 ) may be  r e w r i t t e n as G(P  H i  ,P  S i  )  =  (2*^4*,,)  1 -  2(B  1 -  2(B  2 2  1 1  P  H i  P,  S i  +B  1 2  P .)  +B  1 2  P .)  s  ...  (4-20)  H  F i n a l l y , .the s t a t e of the hydro p l a n t , V ^ , a t t h e beginning of each hour i s found and t a b u l a t e d i n main memory. At  24_  t h i s point ) i=i  p r i n t e d out.  The f i r s t  f . (Pc;.)At_ and V™ are c a l c u l a t e d and 1  S l  as a check on the r a t e of convergence  and the second as a check t h a t no e r r o r s have occurred d u r i n g the  calculations.  hence, not  V^ i s found from the h o u r l y d i s c h a r g e s and  i f any e r r o r s have o c c u r r e d , the value of Vrr, found w i l l  agree w i t h the o r i g i n a l v a l u e .  I f Jump Switch 2 i s a t jump,  a l l the q u a n t i t i e s d e s c r i b e d above are p r i n t e d out as they are calculated. The maximization process i s c a r r i e d out as shown i n steps 10-22 of the flow diagram.  I n e f f e c t the q u a n t i t i e s shown i n  Table VI are c a l c u l a t e d f o r each hour and t a b u l a t e d i n main memory of the computer. for  the f i r s t  A s p e c i a l subroutine had t o be w r i t t e n  and l a s t hours as the s o l u t i o n f o r these two  51  hours d i f f e r s from t h a t f o r the r e s t of the p e r i o d . In  order to f i n d the optimum hydro schedule the computer  f i n d s the maximum weighted  (Bo (V  output  A  0 A  ))  f o r the e n t i r e  day and the corresponding optimum value of d i s c h a r g e ,  ^OA^OA^  along with the corresponding v a l u e s of hydro p l a n t d i s c h a r g e , head and megawatt output are t a b u l a t e d i n main memory.  The  corresponding optimum d i s c h a r g e s f o r the other hours are then found.  The  c r i t e r i o n used i s as f o l l o w s , g i v e n V^ the computer  solves f o r  from the equation ' V. _ = V.-Q.At i+l 1* i  Having  found the c o r r e c t value of  i t i s easy to p i c k out  the corresponding v a l u e s of head, discharge and The  l o a d demand s t u d i e d along with Q., *i to  t a b u l a t e d i n Table V I I I . The is  initial  output.  V. . , V. l mm' I  and f i n a l hydro  schedule  shown i n F i g . 20 along with the t o t a l l o a d demand.  v a l u e s of discharge are shown i n Table I X corresponding f i n a l v a l u e s of H.,  is max  The  final  along with the  V. and R.(V.).  This f i n a l  i i i i i optimum schedule was obtained from Table X which was t a b u l a t e d i n the computer f o r each hour of the day as p r e v i o u s l y mentioned. 3 3 The value of v used was 7200 yd. , hence AQ = 2 yds /sec. 43  The The  c o s t of o p e r a t i o n f o r the i n i t i a l  s o l u t i o n converged  i n 20 i t e r a t i o n s .  schedule was As can be seen  Table X there was  l i t t l e saving  five iterations.  The f i n a l cost of o p e r a t i o n was  g i v i n g a net saving  i n c o s t s gained i n the  i n costs of $96.23.*  $8938.40. from last  $8842.17,  Table XI shows the  * T h i s savings i n c o s t s can be reduced i f a b e t t e r approximation to the i n i t i a l hydro p l a n t schedule i s made. As i t i s , t h i s savings i n o p e r a t i n g c o s t s i s small and as i s pointed,out i n the c o n c l u s i o n , the use of a d i g i t a l computer to schedule t h i s system i s not r e a l l y f e a s i b l e .  140  "D  120  100  _Ho_  80 o  „ ,o  60  0  40  AQ = 2 AQ = ,3  20  10  0  14  16  18  20  22  24 TIME(HRS)  Pig.  20  A T y p i c a l L o a d Demand a n d t h e R e s u l t i n g H y d r o Schedule Obtained by I n c r e m e n t a l Dynamic Programming.  ro  P  Di  V  v  imin  imax  Himax  p  70.0000  3270298  3270298  6I+.869  21.5520  80.0000  3051+298  3270298  76.676  21.5520  80.0000  2838298  3270298  76.676  21.5520  90.0000  2622298  3270298  88.838  1+0.0000  95.0000  21+06298  3272098  95.062  Uo.oooo  90.0000  2190298  3270298  88.838  1*0.0000  95.0000  1971+298  3270298  95.062  1»0.0000  loo.oboo  1758298  3270298  IOI.388  52.0000  120.0000  151+2298  3270298  127.821+  52.0000  ll+O.OOOO  1326298  32U0000  156.1+02  52.0000  11+0.0000  1110298  3021+000  156.1+02  52.0000  130.0000  891+298  2808000  11+1.812  52.0000  135.0000  678298  2592000  11+9.027  52.0000  135.0000  1+62298  2376OOO  lf+9-027  52.0000  130.0000  21+6298  2160000  ii+i.812  52.0000  120.0000  30298  I9I+I+OOO  127.821+  1+0.0000  110.0000  1728000  lil+,368  1+0.0000  95.0000  1512000  95.062  1+0.0000  100.0000  1296000  101.388  1+0.0000  100.0QOO  1080000  101.388  21.5520  90.0000  861+000  88.838  21.5520  90.0000  61+8000  88.838  21.5520  80.0000  1+32000  76.676  21.5520  70.0000  216000  61+.869  21.5520 <-, |  •  ..  ,. i . , , . . .  1  TABLE VI11  The Hourly System Load J S a ^ i e f t l ^ a the I n i t i a l Discharges. The Hourly Values imin> imax ^ Himax are also shown.  o  f  v  v  p  P  Hi  v  i (BTU x 1 0 °  1+1.3108  29.5520  352.3000  3270298  7927.657  1+5.23^1  31.5520  352.2883  3163910  7712.70*4  Ul.3078  29.5520  352.2753  3050323  7*+7*+.706  1+9.0663  33.5529  352.2636  291+3936  7257.363  53.6323  36.OOOO  352.21+95  28231*49  6987.085  1+9.9076  3I+.OOOO  352.2339  26935^9  6682.278  53.627**  36.OOOO  352.2195  25711^9  6I+O9.27I  53.6250  36.0000  352.2039  21+1+151+9  6113.590  67.6033  ¥+.0000  352.1883  23119*+9  5815.32l4  7l4.0i+6l  1+8.0000  352.1679  21535*+9  5*+02.2l42  71+. 0 1 + 1 0  1+8.0000  352.11+51  19807*49  1+917.127  67.5900  1+1+.0000  352.1223  18079*49  I4I432.026  70.8552  1+6.0000  352.1019  I6I495U9  *»009.763  71+.0266  1+8.0000  352.0803  11+839*49  3561.685  70.81+58  1+6.0000  352.0575  131UI49  3082.379  6U.215I4  1+2. OOOQ  352.0359  ll*+55*49  2639.729  6O.76I+3  1+0.0000  352.0167  99*+3*>-9  2257.257  53.5920  36.0000  351.9987  85031+9  1902,865  53.5893  36.0000  351.9831  72071+9  1598.19*4  57.2153  38.OOOQ  351.9675  5911^9  I3OO.033  1+9.0203  33.5520  351.9507  1+5*43*49  972.232  1+9.0180  33.5520  351.9366  333562  702.IOI4  I+I.2639  29.5520  351.9921+  21277*4  i+31.985  1+1.2625  29.5520  351.9107  IO6387  2114.783  TABLE  IX  Optimum Hydro Schedule for the System Load Shown i n Fig. 2 0 .  Allowable States  P  Hi  (BTU x 10°)  3270298 3270298 3270298  1+1.3108 37.2950 33.1870  29.5520 27.5520 25.5520  352.3000 352.3000 352.3000  7927.657 7927.MI 7926.679  3X63910 3171110 3178310  1+5.231+1 1+5.231+1 1+5.231+1+  31.5520 31.5520 31.5520  352.2883 352.2895 352.2907  7712.701+ 7733.351* 7753.997  3050323 3057523 3062723  1+5.2321+ 1+5.2326  1+1.3078  29.5520 31.5520 31.5520  352.2753 352.2765 352.2777  7l+7l+. 706 7U95.355 7515.997  2936736 29U3936 295>H36  1+5.2305 1+9.0663 I+9.O663  31.5520 33-5520 33.5520  352.2621+ 352.2636 352.261+8  7236.23!+ 7257.363 7278.001+  28l'59+9 282311+9 283O3U9  1+9.9098 53.6323 57.261+1  3I+.0000 36.0OOO 38.OOOO  352.21+83 352.21+95 352.2507  6965.929 6987.O85 7OO7.726  26935^9  27007I+9 27079U9  1+9.9076 1+9.9079 53.6300  31+. oooo 31+.0000 36.0000  352.2339 352.2351 352.2363  6682.278 6702.719 6723.O8O  25711^9 257831+9 25855^9  53.6273 53.6276 53.6279  36.0000 36.0000 36.0000  352.2195 352.2207 352.2219  61+09.271 61+29.711 6I+I+9.909  2W15U9 21*1+871*9 21*5591+9  53.621+9 53.6252 57.2563  36.0000 36.0000 38.0000  352.2039 352.2051 352.2063  6113.590 6131+.028 615!+. 221+  2301+71+9 23H9I+9 2319l!+9  6l+. 2l+l+7 67.6033 67.6036  1+2.0000 1+1+.0000 1+1+.0000  352.1871 352.1883 352.1895  5791+.802 5815.321+ 5835.760  215351+9 216071+9 216791+9  7l+.0l+6i 7I+.0I+65  71+. 01+66  1+8.0000 1+8.0000 1+8.0000  352.1679 352.1691 352.1703  51+02.21+2 5I+22.676 51+1+2.896  I9807I+9 198791,9 199511+9  71+.01+10 7^.01+11+ 77.1275  1+8.0000 1+8.0000 50.0000  352.11+51 352.11+63 352.11+75  1+917.127 1+937.558 i+957.778  180791+9 l8f5il+9 l8223!+9  67.5900 70.8600 70.8601  1+1+.0000 1+6.0000 1+6.0000  352.1223 352.1235 352.121+7  1+1+32.026  161+231+9 161+951+9 165671+9  70.8552 71+.0317  70.8551  1+6.0000 1+6.0000 1+8.0000  352.1007 352.1019 352.1031  3989.335 I+OO9.763  1  1+1+52.1+55  1+1+72.5I+3  1+029.851  56  1+6.0000 1+8.0000  352.0779 352.0791 352.0803  3520.692 35M.257 3561.685  352.0563 352.0575 352.0587  3061.952 3082.379 3102.377  352.031+7 352.0359 352.0371  2619.195 2639.729. 2659.726  38.GOOO 38.0000 1+0.0000  352.011+3 352.0155 352.0167  2215.911 2236.607 2257.257  1+9.8722 53-5917 53.5920  31+.0000 36.0000 36.0000  351.9963 351.9975 351.9987  1861.02U 1882.169 1902.865  71351+9 72071+9 72791+9  1+9.8700 53.5893 57.2182  31+.0000 36.0000 38.0000  351.9819 351.9831 351.981+3  1577.500 1598.191+ 1618.382  57671+9 58391+9 59111+9  53.5866 53.5866 57.2153  36.0000 36.0000 38.0000  351.9651 351.9663 351.9675  1258.361 1279.21+3 1300.033  1+1+711+9 l+5»+3l+9 U&l 51+9  1+9.0200 1+9.0203 1+9.0203  33.5520 33.5520 33.5520  351.9*+95 351.9507 351.9519  972.232  326361 333562 31+0762  1+9.0180 1+9.0180 1+9.0183  33.5520 33.5520 33.5520  351.9351* 351.9366 351.9378  681.221+ 702.101+ 722.71+2  205571+ 212771+ 219971+  1+1.2639 1+1.2639 1+5.I8U5  29.5520 29.5520 31.5520  351.9212 351.9221+ 351.9236  1+11.105 1+31.985 1+52.622  91987 99187  33.11*80 37.2513 I+I.2625  25.5520 27.5520 29.5520  351.9083 351.9095 351.9107  172.5^5 193.901+ 21I+.783  H+695^9 IU767I+9 l»*839*9  70.8502 7^.0265 7U.0266  130391.9 13183^9  m  67.5765 70.81+58 70.81+62  11383^9 111+55^9 11527^9  60.7677 6U.215U 67.5727  1+1+.0000 1+6.0000 1+6.0000 1+0.0000 1+2.0000 1+1+.0000  9799^9 98711+9 99^3^9  57.2235 57.2236 60.761+3  83591+9 81+311+9 85031+9  1  1311  IO6387  TABLE X  1+8.0000  951.350 992.870  Tabulation of Sable VI1 for the Final Iteration.  57  c o s t of o p e r a t i o n at the end of each i t e r a t i o n and the r e s u l t i n g net s a v i n g s :  I t e r a t i o n Number 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20  TABLE XI  Cost of Operation  Net Savings  8938.40 8925.06 8916.50 8907.93 8900.52 8893.95 8887.40 8880.36 8874.56 8869.26 8864.80 8860.18, 8856.63 8853.24 8850.70 8848.09 8846.10 8844.72 8843.80 8842.57 8842.17  '  13.34 8.56' 8.57 7.41 6.57 6.55 7.04 5.79 5.31 4.46 4.62 3.55 3.39 2.54 2.61 1.99 1.38 0.92 1.23 .40  Cost of Operation and Net Savings at the End of each I t e r a t i o n .  One of the major problems i n u s i n g t h i s method i s to choose the proper value of AQ to use.  There seems to be no s e t method  of determining the optimum value of AQ, i t s choice depends on the system being studied.' Table XII shows the number of i t e r a t i o n s r e q u i r e d f o r the s o l u t i o n to converge f o r d i f f e r e n t v a l u e s of AQ and the r e s u l t i n g c o s t s o f o p e r a t i o n .  58  AQ  Number of I t e r a t i o n s  Cost of  Operation  ($)  2  20  8842.17  3  11  8846.60  4  10  8845.01  5  10  8853.71  6  No  TABLE XII  convergence  -  Dependence of the Speed of Convergence and of the Cost of Operation on AQ.  Although higher values of AQ give good convergence i t was found t h a t the r e s u l t i n g hydro schedule Instead of the hydro p l a n t , o u t p u t  was  very  irregular.  i n c r e a s i n g smoothly f o r the  f i r s t twelve hours, i t i n c r e a s e d i n an e r r a t i c manner. leads to poor o p e r a t i o n .  The  schedule  This  r e s u l t i n g f o r a value  of  3 AQ = 3 yd /sec was  found to give the best  results.  Only three p e r m i s s i b l e s t a t e s of the hydro p l a n t were allowed chapter.  at each p o i n t of time i n the s o l u t i o n g i v e n i n t h i s As can be  seen from Table XII, i f the p e r m i s s i b l e domai  t h a t i s , the allowable range of AQ  i s too l a r g e the accuracy  of the s o l u t i o n f a l l s o f f v e r y r a p i d l y .  ,  I f i t i s d e s i r e d to  consider a l a r g e r p e r m i s s i b l e domain i t becomes necessary have more p e r m i s s i b l e s t a t e s at each p o i n t of time.  It is  apparent t h a t f o u r or more p e r m i s s i b l e s t a t e s could be This would r e s u l t i n f a s t e r convergence but the  to  considere  computational  d i f f i c u l t i e s become extreme on the Alwac I I I - E computer. Another p o i n t which i s very important  when s o l v i n g these  59  equations  i s the degree of accuracy  sustained i n the s o l u t i o n .  I t was found that the convergence of the s o l u t i o n t o the most economic schedule was  2M>4^Be'e-eJ3fea^  was taken i n s c a l i n g the computer programme.  This  presented  some d i f f i c u l t i e s since i t was necessary to have a programme capable of handling a wide v a r i e t y of input data with a minimum r i s k of s c a l i n g alarms occurring. As i t i s , the s o l u t i o n obtained f o r the system load shown i n P i g . 20 does have some i r r e g u l a r i t i e s .  One example of  i r r e g u l a r operation can be seen during the t h i r d hour when the hydro plant output drops instead of remaining  constant.  I t was found that the behavior of the s o l u t i o n could be improved upon i f the permissible domain was made p r o g r e s s i v e l y smaller as the s o l u t i o n converged.  That i s , the incremental  discharge,  AQ, i s made smaller as the s o l u t i o n proceeds*. The. a p p l i c a t i o n of t h i s type of procedure to the common flow problem would lead to few extra d i f f i c u l t i e s unless the time lags of r i v e r flow between plants were appreciable. I t i s p o s s i b l e t h i s d i f f i c u l t y could be overcome i f i t i s assumed i  that the system load i s c y c l i c , that i s , today's power demand i s of the same pattern as yesterday's.  This type of approxima21  t i o n has been made i n the long—range o p t i m i z a t i o n problem. Since the v a r i a t i o n i n head was so s m a l l , the area of the r e s e r v o i r was assumed to be constant.  I f t h i s i s not the case  i t i s necessary to write an expression f o r the area i n terms of the t o t a l volume of storage i n the r e s e r v o i r . For any p a r t i c u l a r case a second order approximation  of the form  60  A. = mV.  2  1  should be s u f f i c i e n t .  1  + nV. + 0 x  ... (4-21)  Using t h i s equation i t i s p o s s i b l e to  solve f o r the value of A^ to use i n equation (4-4) f o r each p e r i o d of time.  61 CHAPTER V  •^•5.1  D i s c u s s i o n of the Previous The  Solutions  optimum mode of o p e r a t i o n f o r the two p l a n t system  has been found u s i n g two d i f f e r e n t approaches. approach was considered f i r s t . are used to d e r i v e equations  The c l a s s i c a l  Here, v a r i a t i o n a l  techniques  which when solved f o r a given  set of c o n d i t i o n s gave an optimum mode of o p e r a t i o n .  An  analytic  exceedingly  s o l u t i o n of these  equations  i s , i n general,  d i f f i c u l t to o b t a i n even f o r the simpler cases. methods were used to f i n d a s o l u t i o n to these  Thus,  numerical  equations.  The use of c a l c u l u s to solve t h i s problem leads to other d i f f i c u l t i e s also.  A l l the system c o n s t r a i n t s and boundary  c o n d i t i o n s must be s a t i s f i e d .  These r e s t r i c t i o n s can be s a t i s f i e d 24  by the use of both Lagrangian  m u l t i p l i e r s and i n e q u a l i t i e s .  However, the s o l u t i o n to t h i s type of problem i s o f t e n a boundary p o i n t of the r e g i o n of v a r i a t i o n and thus i t i s necessary to t e s t a l l p o s s i b l e combinations of boundary values and i n t e r n a l extremals. Furthermore, i t i s necessary  to run through a s e t of c o n d i -  t i o n s s u f f i c i e n t t o i n s u r e t h a t a t r u e extremal and n o t a l o c a l 24 one has been obtained. even f o r a simple  These problems can become q u i t e complex  case.  Many of the d i f f i c u l t i e s mentioned above can be avoided i f dynamic programming methods are used to f i n d a s o l u t i o n to the scheduling problem.  The r e s t r i c t i o n s on the system  o p e r a t i o n can be e a s i l y s a t i s f i e d ; the only d i f f i c u l t y i s to prove t h a t the r e s u l t s obtained represent a true extremal mode  62  of o p e r a t i o n .  T h i s method compares system performance f o r  d i f f e r e n t modes of o p e r a t i o n of the hydro,plants.  In order  to  accomplish t h i s a s c a l a r f u n c t i o n of the s t a t e v e c t o r i s introduced.  I t i s c a l l e d a c r i t e r i o n or r e t u r n f u n c t i o n and  process i s c a l l e d optimal As was  pointed  i f i t optimizes  this c r i t e r i o n function.  out e a r l i e r , depending upon the choice  the c r i t e r i o n f u n c t i o n , the o p t i m i z a t i o n may a minimization  desirable situation  take the form of  of c o n s t r a i n t s i s a  since they l i m i t the range of v a r i a t i o n s  over which the process i s to be The v a r i a t i o n a l  optimized.  procedure as used here may  be regarded  a dynamic programming process of a continuous and  the  of  or maximization with r e s p e c t to the c o n t r o l v e c t o r .  I t i s to be noted that the existence  type.  the  To b r i n g out the  s i m i l a r i t i e s of the two  as  deterministic  methods  consider  following:  From equations 3-1 df dP  and  3-2:  +  X  c  Tip"H +  x  9P  S  3lT = * H  Solving for X gives:  A  df  = dP  c  1 -  or  7  df  1 1  !5 SPr  "  9P  H  i  -  BP 8P  T  H  !5 apH dP  c  (5-1)  63  Comparing equation ( 4 - l ) to equation (4-13), i t f o l l o w s that ^  v. G  i( Hi' Si>  T i  P  r  SL  =—IQ, S P  8  =±  P  i ^7 .^9 Q  1  (  (.49)  ...(5-2)  Hi  The two methods use e s s e n t i a l l y the same equations to f i n d an optimum mode of o p e r a t i o n .  This must be so since both methods  schedule the hydro p l a n t on the b a s i s t h a t the incremental worths of water f o r each hour are equal except f o r the hours where the hydro p l a n t (or thermal p l a n t ) i s operating at an induced end p o i n t , t h a t i s , at a p o i n t where e i t h e r hydro or thermal r e s t r i c t i o n s are  broken.  In the c l a s s i c a l approach,  a value of If  is arbitrarily  chosen and the optimum schedule f o r t h i s value of ~jf from the scheduling equations.  An i t e r a t i v e procedure must then  be found which w i l l f i n d the c o r r e c t value of T correct t o t a l discharge.  to give the  In the dynamic programming  a f e a s i b l e s o l u t i o n i s a r b i t r a r i l y chosen and an procedure  i s found  approach,  iterative  set up which w i l l tend to e q u a l i z e the h o u r l y i n c r e -  mental worths of water.  The h o u r l y incremental worths of water  corresponding to the problem solved are shown f o r the  first,  t w e l f t h and t w e n t i e t h i t e r a t i o n s i n Table X I I . The c o o r d i n a t i o n equations were used to f i n d the optimum schedule f o r the system l o a d shown i n F i g . 20.  Since the v a r i a t i o n  i n head i s so s m a l l , the f i x e d head equations were used to schedule the two p i A n t s .  The head was  assumed to have an  average  64  v a l u e of 352o096Q5 y a r d s .  The optimum schedule obtained i s  shown i n Table X I I I . The  c o s t of o p e r a t i o n was $8830<»29 by the c o o r d i n a t i o n  equations method as compared to $8842.17 f o r the dynamic programming method.  As i s to be expected,  schedules are obtained.  nearly identical  The small d i f f e r e n c e i n c o s t s i s due  to the d i s c r e t e nature of the second s o l u t i o n p l u s t h e f a c t that d i f f e r e n t equations must be used to d e s c r i b e the hydro p l a n t input-output c h a r a c t e r i s t i c s f o r the two cases. I t can be seen from Table X I I t h a t the d e v i a t i o n from the average value o f X ^ grows l e s s as the s o l u t i o n converges, and the average value of if ^ approaches t h a t obtained i n the coord i n a t i o n equations method.  Hourly Worth of Water Hour  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 Average .....  Iteration 1  $/ Y d s  I t e r a t i o n 12  6.15016 6.62038 6.62027 7.10744 4.45048 4.27532 4.45025 4.62813 3.90324 4o52901 4.52887 4.20996 4.36768 4.36754 4.20958 3.90251 4.99192 4.44857 4.62635 4.62623 7.10137 7.10125 6.61432 6.14440  5.48438 5.59729 5.59718 5.70906 4.96773 • 4.77832 4.70218 4.88722 4.58481 4.77273 4.52887 4.68035 4.85202 4.85187 4.67994 4.58580 4.73170 4.96549 4.88539 4.88525 5.70533 5.70517 5.59299 5.80301  5.16564  5.06384  3  x 10  I t e r a t i o n 20  4.88747 5.00144 5.29125 5.11331 4.96766 5.05015 4.96739 5.15984 4.83471: 5*02772 5.02756 5.19708 5.11155 41 85190 4,93209 5.09581 4.99207 4.96-556 5.15789 4.88531 5.11046 5,11031 5.28756 4.88382 5  •  TABLE XIII Hourly Worths of Water.  6  5.03791  65  3271230. 3271897. 3270268. 3270321. 3270317.  5.00000 4.99858 5.00199 5.00193 5.00196 At  P  Si  P  A. I  Hi  1.00  28.4117  45.1587  31.4931  2.68330  2.00  35.9823  48.5857  33.1662  2.86495  1.00  43.6622  52.0606  34.9503  3.05450  1.00  47.5441  53.8164  35.8854  3.15241  1.00  43^6627  52.0608  34.9505  3.05451  1.00  47.5442  53.8164  35.8854  3.15241  1.00  51.4557  55.5851  36.8502  3.25254  1.00  67.4085  62.7939  41.0191  3.67683  2.00  83.8860  70.2315  45.7187  4.14396  1.00  75.5784  66.4827  43.2993  3.90460  2.00  79.7138  68.3491  44.4910  4.02275  1.00  75.5791  66.4830  43.2995  3.90462  1.00  67.4085  62.7939  41.0191  3.67683  1.00  59.3698  59.1623  38.8714  3.45975  1.00  47.5447  53.8166  35.8856  3.15243  2.00  51.4558  55.5851  36.8503  3.25254  2.00  43.6622  52.0606  34.9503  3.05450  1.00  35.9823  48.5857  33.1662  2.86495  .1*00  .28.4118  45.1587  31.4931  2.68330  TABLE XIV  Optimum Schedule f o r the System Load shown i n -Fig. 20 as Obtained from the C o o r d i n a t i o n Equations.  66 \ \  It  i s d i f f i c u l t to compare the speed of s o l u t i o n of the  two methods.  For the c o o r d i n a t i o n s equation, the speed of  s o l u t i o n depends upon the i n i t i a l the  dynamic  approximation of T •  For  programming method, the speed of s o l u t i o n depends  upon the i n i t i a l  t r i a l p o l i c y chosen.  The programmes w r i t t e n f o r the f i x e d head s o l u t i o n and f o r the  dynamic programming s o l u t i o n were both optimized.  For  the  system l o a d shown i n F i g . 20, i t r e q u i r e d 2 minutes to  complete an i t e r a t i o n f o r a p a r t i c u l a r value of 1f~ f o r the f i x e d head s o l u t i o n . to  On the average i t r e q u i r e s s i x i t e r a t i o n s  o b t a i n a c o r r e c t value of  initial  estimate of T  f o r t h i s method, even i f the  i s f a r out.  This means t h a t on the  average i t r e q u i r e s twelve minutes to o b t a i n a s o l u t i o n f o r the f i x e d head case.  Here, input of d a t a and output of r e s u l t s  r e q u i r e s about 3 minutes.  The time taken f o r one  complete  i t e r a t i o n f o r the dynamic programming method i s 2 minutes "and 40 seconds.  Since i t r e q u i r e s about eleven i t e r a t i o n s to o b t a i n  a s o l u t i o n , the time taken i s about 30 minutes.  Input and  output of data takes 5 minutes. No attempt was made to optimize the programming of the c o o r d i n a t i o n equations f o r the v a r i a b l e head case. programme takes about  s i x minutes per iteration.-to f i n d the  schedule f o r the load l e v e l shown i n F i g . 20. ikL^^'i^<m^:i35-vi^^v^s  The r e s u l t i n g  Hence i t would  to reach a complete s o l u t i o n .  could no doubt be much improved upon.  This time  re  67  5.2  General S o l u t i o n of the Scheduling Equations  5.2.1  Introduction No attempt was made i n t h i s t h e s i s to solve the case  where more than two p l a n t s e x i s t e d i n a system.  As was mentioned  p r e v i o u s l y , i t would r e q u i r e a v e r s a t i l e computer to solve the s c h e d u l i n g equations economically f o r the general case. the f o l l o w i n g paragraphs  an attempt i s made to i n d i c a t e the  e x t r a steps i n v o l v e d in- s c h e d u l i n g a more complex 5.2.2  In  system.  The C o o r d i n a t i o n s Equations Method The general scheduling equations are g i v e n by equations  3.1  and 3.2.  These can be expressed as f o l l o w s :  1 Sn  P  2n  f!  =  l  A  1  Hm  YZ  - 2  B„;P,  ^  n  + 2Bnn  " gm m^ K  —  a h 2 + b h + c  )m  "  2  ( 5  ~  3 )  H mk k k^m B  P  P  2e OMali +bh+c) - 2B m m°m m mm  These equations are i d e n t i c a l t o those f o r the two-plant  case  except t h a t an e x t r a term e x i s t s i n the numerator f o r each e x t r a power p l a n t i n the system.  Thus, i n a system c o n t a i n i n g  ra hydro and n thermal p l a n t s there w i l l be n+m equations to solve.  The value of A used i n these equations must be chosen  to s a t i s f y the f o l l o w i n g system c o n s t r a i n t :  68  n+ra  m  m+_n  JZ p.. + H p - p - Ht k  j=m+l  J  k=l  m+n  D  k=l  K  p  Ad j p  0  j=l .... (5-4)  The c o r r e c t value of A to be used can be found from equation  (3-11).  I t i s f e l t t h a t the number of i t e r a t i o n s r e q u i r e d to f i n d the c o r r e c t value of A should be r e l a t i v e l y independent of the number of p l a n t s present i n the system. In order to s a t i s f y the hydro; p l a n t d i s c h a r g e c o n s t r a i n t s , the c o r r e c t value of  must be found.  For the system studied  i t was found t h a t an i t e r a t i v e ' procedure based on equation (3-12) could be used w i t h good r e s u l t s to f i n d the c o r r e c t value of "f to use.  I f more than one hydro p l a n t e x i s t s i n the system the  s i t u a t i o n becomes d e c i d e l y more complex.  In t h i s case the  water used at the i t h hydro p l a n t depends not only on T .  but  a l s o i s some complex f u n c t i o n of the/Q^'s of the other hydro p l a n t s as w e l l . I t i s p o s s i b l e t h a t an i t e r a t i v e procedure which uses equation (3—12)to f i n d the c o r r e c t value of T p l a n t might s t i l l  give good r e s u l t s .  f o r each hydro  However, there i s no  guarantee that the above i t e r a t i v e procedure w i l l converge i n the. general case and i f i t does the convergence may be slow. Another p o i n t t o consider here i s t h a t the hydro p l a n t i n p u t output curves used were smooth a n a l y t i c f u n c t i o n s . always the case, q u i t e o f t e n they are d i s c o n t i n u o u s .  This i s not This w i l l  not cause many e x t r a problems i n programming but there i s the p o s s i b i l i t y that the convergence of the system to the c o r r e c t value of S w i l l be e f f e c t e d .  Such d i f f i c u l t i e s may  occur i f  69  the  p l a n t s are o p e r a t i n g i n the r e g i o n of these d i s c o n t i n u i t i e s .  5.2.3  The Incremental Dynamic Programming  Approach  This approach can be e a s i l y a p p l i e d to a more complex system, equations s i m i l a r to those used i n t h i s t h e s i s b e i n g applicable.  C o n s i d e r i n g a power system c o n s i s t i n g of two  hydro  p l a n t s and n thermal p l a n t s the f o l l o w i n g procedure could be followed. (1) S t a r t out with a f e a s i b l e s o l u t i o n f o r each of the two hydro p l a n t s , that i s , one which s a t i s f i e s a l l of the system  constraints.  (2) Schedule the thermal p l a n t s on an equal incremental cost basis.  That i s  P  Sn =  f.  -4 A  s  .  ^  •••(5-4)  +2B nn  (3) Leaving the o p e r a t i o n of no. 2 hydro p l a n t constant, determine the most economical p a t t e r n of o p e r a t i o n f o r no. 1 hydro p l a n t and the combined thermal system.  The  procedure used would be e x a c t l y the same as t h a t f o l l o w e d i n t h i s t h e s i s except t h a t i n t h i s case, the thermal p l a n t s must be scheduled by u s i n g equation 5.4.  The  value of F ( P . ) used i n equation (4.13) would be t h a t ol 0  found i n step 2 above.  >  (4) Step 3 i s repeated f o r hydro p l a n t no. 2 with the o p e r a t i o n of  the f i r s t hydro p l a n t l e f t  at that found i n step 3.  (5) T h i s process i s repeated u n t i l the s o l u t i o n converges.  70  By f o l l o w i n g the above technique the s o l u t i o n of the m u l t i - d i m e n s i o n a l problem has been reduced to the s o l u t i o n of a s e r i e s of s i n g l e dimensional problems hydro p l a n t s ) .  (with r e f e r e n c e to the  In e f f e c t t h i s s o l u t i o n i s a r e l a x a t i o n method  which g i v e s monotonic  convergence towards an optimum mode of  operation. I t i s estimated t h a t the time per i t e r a t i o n f o r each hydro p l a n t would be about 4 minutes, making the time f o r one i t e r a t i o n equal to 8 minutes.  This time i s based on the  t i o n t h a t there are only two thermal p l a n t s p r e s e n t .  complet suppos  It is  obvious that the time taken f o r one i t e r a t i o n depends upon the number of thermal p l a n t s p r e s e n t .  71  CONCLUSIONS Two  d i f f e r e n t methods, the c o o r d i n a t i o n equations method and  the incremental dynamic programming method, have been used to optimize the o p e r a t i o n of a two-plant hydro-thermal was  system.  It  found t h a t both methods gave n e a r l y i d e n t i c a l schedules as  could be expected.  Furthermore, both methods r e q u i r e d a p p r o x i -  mately the same l e n g t h of time to give the f i n a l  schedules.  I t i s to be noted that the c o o r d i n a t i o n equations are more likely  to give a t r u e minimum than the incremental dynamic  programming method.  However, i f the s o l u t i o n i s such t h a t the  hydro or thermal p l a n t i s o p e r a t i n g a t an induced boundary point for  p a r t of the day, then i t i s d i f f i c u l t to prove that the former  method does give a t r u e minimum. For it  system,  i s f e l t that the incremental dynamic programming approach i s  the b e s t . it  the case where more than two p l a n t s e x i s t i n a  There are s e v e r a l reasons f o r t h i s c o n c l u s i o n .  i s f e l t t h a t the problem  use w i l l be v e r y d i f f i c u l t  Firstly,  of f i n d i n g the c o r r e c t value of IT to i n the general case.  Secondly, i f the  thermal or hydro p l a n t s have d i s c o n t i n u o u s input-out c h a r a c t e r i s t i c s , or i f i t i s necessary to shut down u n i t s i n the thermal s t a t i o n f o r p a r t of the day, then the s o l u t i o n of the c o o r d i n a t i o n equations becomes very d i f f i c u l t .  These problems are not n e a r l y  so s e r i o u s i f dynamic programming procedures are used to solve the problem. I t appears that the use of a general purpose schedule a hydro thermal system i s not f e a s i b l e . s o l u t i o n of the s c h e d u l i n g problem  computer to The  would r e q u i r e the  economic u s e of  72  s o p h i s t i c a t e d analog or analog d i g i t a l  computers.  The  coordina-  t i o n equations are i d e a l f o r analog methods of s o l u t i o n s i n c e i t i s a l s o p o s s i b l e to i n c o r p o r a t e u n i t s i n t o the computer to c o n t r o l system  frequency.  The  savings i n costs r e s u l t i n g from o p t i m i z i n g the o p e r a t i o n  of the system s t u d i e d i n t h i s t h e s i s are s m a l l .  I t would appear  that the use of computers f o r scheduling purposes i s not j u s t i f i a b l e u n l e s s the t o t a l system l o a d i s i n the order of 5000 MW or h i g h e r , and t h a t the thermal g e n e r a t i o n i s at l e a s t 50% of the total  generation.  73  REFERENCES A. O p t i m i z a t i o n of Thermal Systems 1. S t e i n b e r g , M.J. and Smith, T.H., Economic Loading of Steam Power P l a n t s and E l e c t r i c Systems. McGraw-Hill Book Company, Inc., New York, 1953. 2. S t e i n b e r g , M.J., "How to C a l c u l a t e Incremental Production Power. Y o l . 90, J u l y 1946, pp. 76-78, p. 146.  Costs",  3. T r a v e r s , R.H., Harker, D.C., Long, R.V. and Harder, E.L., "Loss E v a l u a t i o n - P a r t I I I . Economic D i s p a t c h Studiee of Steam E l e c t r i c Generating Systems", AIEE T r a n s a c t i o n s , V o l . 71, p a r t I I I , February 1958, pp. 1545-1552. 4. 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B r u n d e l l , R.N. and G i l b r e a t h , J.H., "Economic Complimentary Operation of Hydro Storage and Steam Power i n the I n t e g r a t e d TVA System", AIEE T r a n s a c t i o n s . V o l . 78, p a r t I I I , June 1959, pp. 136-156.  75  E. Methods of Short-Range O p t i m i z a t i o n of Hydro-Thermal Systems 24. Arismunandar, R.A., "General Equations f o r Short-Range O p t i m i z a t i o n of a Combined Hydro-Thermal E l e c t r i c System", M.A.Sc. T h e s i s . The U n i v e r s i t y of B r i t i s h Columbia, May 1960. 25. Kirchmayer, L.K., Economic C o n t r o l of Inter-Connected Systems. John Wiley and Sons, Inc., New York, 1959. 26. Chandler, W.G., Dandeno, P.L., Glimn, A.P. and Kirchmayer, L.K., "Short-Range Economic Operation of a Combined Thermal and H y d r o - E l e c t r i c Power System", AIEE T r a n s a c t i o n s . V o l . 77, p a r t I I I , October 1953, pp. 1057-1065. 27. Glimn, A.F. and Kirchmayer, L.K., "Economic Operation of V a r i a b l e Head H y d r o - E l e c t r i c P l a n t s " , AIEE T r a n s a c t i o n s , V o l . 77, p a r t I I I , December 1958, pp. 1070-1078. 28. Carey, J . J . , "Load A l l o c a t i o n on a Hydro-Thermal E l e c t r i c System", S.M. T h e s i s . Massachussetts I n s t i t u t e of Technology, Cambridge, Mass., June 1953. 29. Watchorn, C.W., "Coordination of Hydro and Steam Generation", AIEE T r a n s a c t i o n s . V o l . 74, p a r t I I I , 1955, pp. 142-150. 30. B e r n h o l t z , B. and Graham, L . J . , "Hydro-Thermal Economic Scheduling", AIEE T r a n s a c t i o n s . No. 51, December 1960, pp. 921-932. 31. R i c a r d , J . , "The Determination of Optimum Operating Schedule f o r Inter-cbnnected Hydro and Thermal S t a t i o n s " , Revue Generale de 1 ' E l e c t r i c i t e , P a r i s , Prance, September 1940. P. P r o b a b i l i t y Theory 32. Hold, A., S t a t i s t i c a l Theory with E n g i n e e r i n g A p p l i c a t i o n s . John Wiley a n d ^ o n s , Inc., London, 1952. 33. Du B o i s , P.H., M u l t i - V a r i a t e C o r r e l a t i o n a l A n a l y s i s . Harper and B r o t h e r s , P u b l i s h e r s , New York, 1957. 34. Kenney, J.P. and Keeping, E.S., Mathematics of S t a t i s t i c s , p a r t I and p a r t , I I , Second E d i t i o n , D. van Nostrand Company, Inc., P r i n c e t o n , New J e r s e y , 1951.  76  G. Transmission Loss Equations 35. George, E.E., "A New Method f o r Making Transmission Loss Formulae", AIEE T r a n s a c t i o n s August 1959, pp. 583-588. P  36. Glimn, B.H. and Kirchmayer, L.K., "Loss Formulas Made Easy", AIEE T r a n s a c t i o n s . V o l . 72, p a r t I I I , 1953, pp. 730-737. 37. Reference 5, pp. 48-159. H. Power Flows 38. Ward, J.B. and Hale, H.W., " D i g i t a l Computer S o l u t i o n of Power Flow Problems", AIEE T r a n s a c t i o n s . V o l . 75, p a r t I I I , June 1956, pp. 398-404. 39. Van Ness, " I t e r a t i o n Methods f o r D i g i t a l Load Flow S t u d i e s " , AIEE T r a n s a c t i o n s . August 1959, pp. 583-588. 40. Glimn, A.F. and Stagg, G.W., "Automatic C a l c u l a t i o n of Load Flows", AIEE T r a n s a c t i o n s . V o l . 76, p a r t I I I , June 1957, pp. 817-825. 41. J u l i e n , K.S. and Smith, B.R , "Power Flow Study", Department of E l e c t r i c a l E n g i n e e r i n g , Report, The U n i v e r s i t y of B r i t i s h Columbia, Vancouver, B.C. 0  I. General References 42. B o l z a , Oskar, L e c t u r e s on the C a l c u l u s of V a r i a t i o n s . U n i v e r s i t y of Chicago P r e s s , Chicago, 1904. 43. Bellman, R., Dynamic Programming. P r i n c e t o n U n i v e r s i t y Press, P r i n c e t o n , New J e r s e y , 1957. 44. H i l d e b r a n d , F.B., I n t r o d u c t i o n to Numerical A n a l y s i s . McGrawH i l l Book Co., Inc., 1956. 45. Noakes, F., and Arismunandar, R.A., " B i b l i o g r a p h y of Optimum Operation of Power Systems, 1919-1959". Mimeographed Report, Department of E l e c t r i c a l E n g i n e e r i n g , The U n i v e r s i t y of B r i t i s h Columbia, Vancouver, B.C.  77  NOMENCLATURE A  = surface area at the hydro p l a n t (yd )  K,a,b,c, = constants of the input-output f u n c t i o n of the hydro p l a n t f3jk  = d e r i v e d l o s s formula c o e f f i c i e n t (MY)  f  = f u e l c o s t a t the thermal p l a n t  ($/hr.)  f^,f »f-j = constants of the c o s t f u n c t i o n a t the thermal p l a n t 2  G(Pjj,Pg) = c r i t e r i o n f u n c t i o n used i n the incremental dynamic programming s o l u t i o n H  = t o t a l head a t the hydro p l a n t (yds) = t a i l w a t e r e l e v a t i o n a t the p l a n t (yds)  h  = net head a t the hydro p l a n t (yds)  m,n,o  — constants of the f u n c t i o n used to d e s c r i b e the v a r i a t i o n , of area with head  Pj^  = system l o s s e s (MY)  Pg  = thermal g e n e r a t i o n (MY)  PT|  = hydro generation  P  = l o a d demand (MV)  B  (MW)  Q  —- d i s c h a r g e a t the hydro p l a n t (yds / s e c )  AQ  = incremental change i n discharge (yds /sec)  q  = r i v e r i n f l o w i n t o the r e s e r v o i r  R^(V\)  = maximum weighted  V,j,  = volume of water scheduled f o r use (yds )  v  = incremental change i n volume of water scheduled f o r use d u r i n g the i t h hour ( y d s )  (yds /sec)  output a t the i t h hour  3  V\  = volume of water remaining a t the beginning of i t h hour  7  = Lagrangian m u l t i p l i e r used i n t h i s t h e s i s as incremental c o s t a t the hydro p l a n t ( $ / m i l l i o n y d s ) 3  X  = Lagrangian m u l t i p l i e r used i n t h i s t h e s i s as incremental cost of d e l i v e r e d power ($/M¥-hr.)  !T  = time d e l a y of flow between hydro p l a n t (hrs.)  78  APPENDIX A»  LOSS FORMULA  S e v e r a l methods of f i n d i n g the B constants have been 35 36 37 published."'-''  The most s t r a i g h t f o r w a r d method and the 35  one used i n t h i s t h e s i s i s t h a t d e s c r i b e d by E.E. George This procedure  i s n o t a p r a c t i c a l one f o r l a r g e power  systems,  a s i x - s o u r c e system r e q u i r i n g 21-1oad flow s t u d i e s and the i n v e r s i o n of a 21-order matrix.  However, f o r smaller systems  i t g i v e s accurate r e s u l t s q u i t e q u i c k l y and uses only standard computer programmes. Steps i n C a l c u l a t i n g the B Constants For a complete 35.  d e s c r i p t i o n of t h i s method see r e f e r e n c e  The f o l l o w i n g i s a b r i e f summary of the steps i n v o l v e d i n  f i n d i n g these constants;  »-  N(N+1) l ) Perform — ^ *- power flow s t u d i e s , where N i s the number 2 of sources and a c t i v e i n t e r c o n n e c t i o n s . The accuracy of these power flows should be h i g h , having an accuracy of at l e a s t 0.1^.  I n performing these flow s t u d i e s the  f o l l o w i n g p o i n t s are important? a) keep magnitude of source voli'&ges constant, b) keep system l o a d constant, c) keep generator outputs w i t h i n  - 2Qfo  of the base case.  2) C a l c u l a t e the l o s s e s from each study by the I R method. Set up a system of  ^If"^- simultaneous 4  the B constants as unknowns.  equations, u s i n g  Place the t o t a l  calculated  l o s s e s on the r i g h t hand side of the corresponding equations.  Set the square of t h e s e l f - g e n e r a t e d powers  as c o e f f i c i e n t s of t h e s e l f B constants and two times  79  -the product of the corresponding generations as the coefficients  of the mutual B constants-  3) Solve the s e t of equations obtained by a good matrix i n v e r s i o n method. 38 The Ward and Hale method used.  of performing power flows was  The b a s i c program u t i l i z i n g t h i s method, as w r i t t e n f o r  the Alwac I I I E , i s d e s c r i b e d i n r e f e r e n c e 41.  The equations  used are. b r i e f l y * d e s c r i b e d here: Let  1.. = G. . + IB.. = the s e l f admittance 11 11 ix J  Y.. l  a t node i  = G.. + riB.. = the mutual admittance ^ V i and 3 .  k  between nodes  I. x  = a. + i b . = the net c u r r e n t at node i . x x  U\  = P^ + jQ^ = the l o a d a t node i .  0  Now I. = Y . . E . + x xx x  y 4jr  Y . . E .  2.3 3  r  U.x =x E .x* I. = the complex power* 1 Solving f o r E ^  *  j-  +  E.* E.  =  1  Y. .E.  4—r. i j 3  ^  ii  ... ( A - l )  Instead of s o l v i n g equation ( A - l ) d i r e c t l y to f i n d the node v o l t a g e s i t i s approximated  by l i n e a r equations.  These l i n e a r 44  equations are s o l v e d by the method of s u c c e s s i v e displacements. As the s o l u t i o n proceeds  the o r i g i n a l t r i a l node v o l t a g e s are  r e p l a c e d by s u c c e s s i v e l y b e t t e r approximations to38 the c o r r e c t * The n o t a t i o n used here i s used by Ward and Hale  38 The e q u a t i o n s u s e d i n t h e i t e r a t i o n a r e :  node v o l t a g e . For  a l o a d node p. = a.(e.G..+f.B..+a.) + 3.(-e.B..+f.G..+b.) \ x 1 11 1 11 1 l 1 11 1 11 1 q. ='a.(-e.B..+f.G..-b.) + 3.(-e.G..-f.Bii+a.) i l l i i l i i l * i l i i l l r  ...(A-2)  r  u  For  a generator p.  node  = a (e G. +f B i  E. « - E , l l 2  i  2  i  i  +a )  i i  i  + B.t-e^.+f.G.^+b.)  ..o(A-3)  = 2e,a, + 2 f , 3 . i i l' l  whe r e a. l  = c o r r e c t i o n t o be added t o e. l  8. = c o r r e c t i o n t o be added t o f . i l r  p^ = d i f f e r e n c e b e t w e e n s c h e d u l e d and a c t u a l r e a l  power  q^ •= d i f f e r e n c e b e t w e e n s c h e d u l e d and a c t u a l r e a l  power  E^' = a c t u a l v a l u e  of the generator  E^  = scheduled value  UL  = t h e c o m p l e x power  node  of the generator  voltage  node  voltage  The s o l u t i o n i s b a s e d on t h e f a c t t h a t a t e a c h l o a d node t h e p.  scheduled r e a l node t h e r e a l  and r e a c t i v e powers a r e known, w h i l e power and t h e r e q u i r e d v o l t a g e  at a  generator  m a g n i t u d e a r e known.  Thus a t a l o a d node t h e p r o b l e m i s t o f i n d t h e c o m p l e x v o l t a g e , while and  a t a generator  node t h e p r o b l e m i s t o f i n d t h e r e a c t i v e power  t h e r a t i o between t h e r e a l  One g e n e r a t o r  i s designated  and r e a c t i v e p a r t o f t h e v o l t a g e s .  as a s w i n g n o d e , t h a t i s , t h i s g e n -  e r a t o r must p r o d u c e t h e r e q u i r e d d e l i v e r e d by the other  e x t r a power t o meet t h e l o a d n o t  generators.  The s y s t e m s t u d i e d h e r e i s shown i n f i g . 2 1 . are  g i v e n , on a b a s e o f 138 KV and 500 MVA.  o f l o a d p o w e r s and g e n e r a t o r  voltage  The i m p e d a n c e s  The s c h e d u l e d  values  m a g n i t u d e s a r e shown i n  a  10  Oct  A  A °<  A  Oc  0a  Ob  09  08  0.0732 +30.0746  1.5745+J1.7155M  Of-<-  -o-  -Q-  0.1257 +J0.1372  1.3383 • +j 1.4588  0.5960 +30.6493  0.2211 +30.2407  0.6735 +j0.7358  0.0681 +30.0745  j'3.0316 f  02  O  •  0.2901 +j2.6751 0.0337+J0.0372  o-  0.1792 +J0.1957  jl>9787 0.1814+J0.5881  01  0.0531+3*0.1728  0.376 +J0.1221 1:1.05 05  06  jl.558  Off Nominal Turns Ratio  07 00  M  F i g . 21. The Modified Version of the BCPC System used i n the Power Plow A n a l y s i s .  82  Table  XV.  The system s t u d i e d represents a p o r t i o n of the  B.C.P.C. system.  The network has been somewhat a l t e r e d .  To cut  down the number of nodes the impedances of a l l branches l e a d i n g from the main d i s t r i b u t i o n system t o the a c t u a l l o a d centers have been n e g l e c t e d .  F u r t h e r changes were r e q u i r e d since the  number of branches per node i s l i m i t e d to f i v e due to storage l i m i t a t i o n s of the computer program.  The system was assumed to  have two p l a n t s , one thermal (Georgia P l a n t ) and one hydro (Ash  River P l a n t ) .  GENERATOR NODE  1  1.0400  2  1.1819  Node No.  Scheduled Power  Scheduled Voltage Magnitude  Node No.  Scheduled  Power  0.4  Node No.  Scheduled Power  3  0.16+J0.04  10  O+3 0.  4  0+jO  11  O.O6+3O.OI  5  0.Il+;j0.03  12  0.06+J0.02  6  0+j 0  13  0+3 0  7  0+3 0.  14  O.O6+3O.OI  8  O.l+jO.03  15  O+3 0  9  O.O6+30.02  16  TABLE  XV.  0.2+J0.06  Scheduled Voltages-and Powers.  The scheduled powers aire* shown i n Table s o l u t i o n i s shown i n Table XVI.  XV; the f i n a l  The o v e r a l l accuracy i s 0.066$  ,1  >de  E  f  e  0  a  b  P  Q  01  1.040000  .000000  1.039993  .000  .436681  -.146899  .454148  .152775  02  1.181608  -.026255  1.181885  1.266  .330973  -.339731  .400000  .392739  03  .980643  -.183620  .997681  10.608  -.150258  .068953  -.160010  -.040028  04  .981866  -.172812  .996948  9.982  -.000010  .000043  -.000017  -.000041  05  .972531  -.183020  .989594  10.665  -.103643  .050419  -.110024  -.030065  06  .967940  -.188078  .986038  11.000  .000004  .000023  -.000001  -.000023  07  .956577  -.288619  .999161 " 16.797  -.000124  .000326- -.000213  -.000276  08  .950792  -.290008  .994034  16.965  -.087465  .058371  -.100089  -.030133  09  .950653  -.290029  .993896  16.965  -.051871  .036949  -.060028  -.020081  Oa  .954275  -.289235  .997131  16.869  .000026  .000116  -.000009  -.000118  Ob  .919529  -.298823  .966858  18.013  -.055826  .029037- -.060010  -.010019  Oc  .899242  -.303464  .949066  18.657  -.053163  .040195  -.060004  -.020012  Od  .879634  -.309147  .932373  19.366  -.000067  .000230  -.000130  -.000182  Oe  .878294  -.309595  .931259  19.419  -.057177  .031596  -.060000  -.010048  Of  .880104  -.308912  .932739  19.345  .000008  .000037  -.000005  -.000035  10  .764031  -.324670  .830139  23.023  -.193462  .160745  -.200000  -.060003  P  G  = 0.854148  P-n = 0.810540 P  L  = 0.043608 00  TABLE XVI.  F i n a l Results of the F i r s t Power Flow A n a l y s i s .  84  for  the r e a l power and  0.48%  f o r the r e a c t i v e power.  This i s  w i t h i n the d e s i r e d accuracy; however, i t i s to be noted t h a t the accuracy of the r e s u l t s at some of the nodes are not t h i s good.  Several d i f f i c u l t i e s  s o l u t i o n of t h i s problem. (l) Scaling:  occurred  Some of these  This system contains  i n the  nearly  computer  are:  some very  short  transmission  l i n e s ; thus, the admittances vary g r e a t l y i n magnitude. P a r t of the program was  r e w r i t t e n so as to minimize  problem as much as p o s s i b l e . was  Furthermore, a  subroutine  i n c l u d e d which i n e f f e c t permits r e s c a l i n g of  program at w i l l . the MVA  I t i s based on the i d e a t h a t by  base of the  system, i t i s p o s s i b l e to vary  magnitude of the c u r r e n t s , powers and pondingly.  admittances  To i l l u s t r a t e t h i s problem more f u l l y  the c o r r e c t i o n to the v o l t a g e  this  the changing the corresconsider  at node 07 f o r a t y p i c a l  iteration. -0.000142 = <x(0.964259 x 79.6036 + 0.249933 x 90.9303 - .000192) + 8(0.964259 x 90.9303 - 0.249933 x 79.6036 - .000173) -0.000142 = <x(99.48454) - 8(67.78462) -0.000215 = ct(67.78500) - 8(99.48492) S o l v i n g gives a = 0.000000031 6 = 0.000002140 The  b a s i c s c a l i n g of the programme i s 8:24  t h a t the accuracy Obviously  which means  s u s t a i n a b l e by the computer i s 0.000000059.  the above c o r r e c t i o n s do not mean too much even though  a large error s t i l l  e x i s t s i n the power.  The  computer s o l u t i o n  cannot give much b e t t e r than f o u r - f i g u r e accuracy p a r t i c u l a r problem.  for this  85  (2) The convergence of the s o l u t i o n was v e r y slow. o r i g i n a l computer  solution u t i l i z e d  The  no a c c e l e r a t i o n  techniques and i t appeared t h a t a s o l u t i o n would not be obtained i n a reasonable l e n g t h of time. a c c e l e r a t i o n techniques were examined. used was  Different The one  finally  chosen f o r two reasons  a) i t was simple t o i n c o r p o r a t e w i t h the e x i s t i n g  program,  b) i t appeared to have g i v e n good r e s u l t s elsewhere. The a c c e l e r a t i o n formula used E  1  = E 1  1  + K(E  i - 1  was  - E )  ...  1  (A-4)  This method, when used on the system d e s c r i b e d i n r e f e r e n c e 38, reduced the number of i t e r a t i o n s r e q u i r e d by a f a c t o r of three. "the  The improvement w i l l depend upon the value of K used,  optimum range f o r K being between 1.2  l a r g e r than 1.8 g i v e s no convergence. a c c e l e r a t i o n should be a p p l i e d u n t i l have been  and 1.7.  Any value  I t was found that no at l e a s t 3 i t e r a t i o n s  completed.  Even though the above techniques were used on the problem under c o n s i d e r a t i o n , 380 i t e r a t i o n s were r e q u i r e d to o b t a i n an acceptable degree of accuracy*.  ( F i g . 21 shows the convergence  of the power at node 07 to the c o r r e c t v a l u e .  The value of K  used was v a r i e d throughout the s o l u t i o n as shown).  The  f o l l o w i n g reasons c o n t r i b u t e to t h i s slow convergence: (a)  The r a t i o of the l a r g e s t to the s m a l l e s t impedance connected to any node should not be too l a r g e .  Nodes  The o r i g i n a l i d e a was to use e x i s t i n g programmes t o do the power f l o w s . Probably, i f a l l the d i f f i c u l t i e s could have been f o r e s e e n completely new programmes would have been w r i t t e n , thus r e s u l t i n g i n a s u b s t a n t i a l savings i n time.  P,Q  -0.005 I 0  i 50  . 100  . 150  200  250  . 300  350  NO. OF ITERATIONS F i g . 22  Convergence of P and Q to the C o r r e c t Value of Node 07.  87  06, 07 and 13 a l l break t h i s r u l e and i n every case thes three nodes g r e a t l y delayed the convergence of the solution.  Furthermore,  f a i l e d to converge  the v o l t a g e s a t these three node  p r o p e r l y to any s t a b l e v a l u e .  One  c r i t e r i o n which can be a p p l i e d here i s t h a t the s e l f admittances  should be s u b s t a n t i a l l y l a r g e r than the  mutual admittances.  I t i s f e l t t h a t any v e r y short  s e c t i o n s of t r a n s m i s s i o n l i n e  should be omitted i n the  solution. (b) The problem of s c a l i n g becomes important as the s o l u t i o n s t a r t s to converge.  As p r e v i o u s l y i n d i c a t e d v o l t a g e  c o r r e c t i o n s can be very small and f o r t h i s reason the s o l u t i o n may f a i l  to converge p r o p e r l y .  Many problems remain to be answered on the s u b j e c t of power flow s t u d i e s .  D i f f e r e n t a c c e l e r a t i o n f of mulae-have. beleni'i  v  39 suggested it  is felt  as an a i d to o b t a i n i n g b e t t e r convergence,  however  t h a t a c c e l e r a t i o n techniques w i l l not h e l p i f the  system under study i s i l l - c o n d i t i o n e d .  One p o s s i b i l i t y i s the  d i r e c t s o l u t i o n of equation ( A - l ) f o r the bus v o l t a g e s . could be accomplished  by a f a i r l y simple i t e r a t i v e  This  scheme and  should not r e q u i r e too much time on a f a s t e r computer. The f i n a l r e s u l t s of the three s t u d i e s made are shown i n Table XVI. The three equations to be solved f o r the. B constants are 0.181659B « + 0.160000B  22  + 0.206250B  n  = 0.043608  0.178868B  + 0.249996B  22  + 0.127976B  n  = 0.047302  0.158574B ' + 0.358767B  22  + 0.070087B  11  = 0.0 53090  12  f 12  12  S o l v i n g for the B constants g i v e s  88  B 22 = 0.103105 B 11 = 0.068817 B » 1 2  = 0.071111 = 2B  12  It, i s to be noted that the actual power delivered and not the scheduled power was used i n calculating the B constants. These B constants f i t the data f o r the given load l e v e l s exactly.  No attempt was made to see how closely they would  check with other load l e v e l s .  I t was found f o r the system  described i n reference 38 that the B constants as found above gave the losses to three place accuracy f o r a constant load l e v e l and d i f f e r e n t d i v i s i o n s of the load between the two generators.  However, i f the load level was varied substantially  from the base case the accuracy of the losses as given by the B constants dropped o f f considerably.  The B constants were  found to be quite sensitive to variations i n the voltage levels at the generators. I t i s assumed i n the derivation of these constants that the t o t a l system load can be replaced i n effect by a single equivalent load at some star point and that the r e l a t i v e position of t h i s f i c t i t i o u s load i s constant.  This assumption i s ob-  v i o u s l y not correct, however, i t i s a necessary one.  I t appears  that the best method available to describe system losses i s the B constant method.  89  APPENDIX B.  COMMON FLOY PLANTS.  *1  p  is.  r  O P  ?  +p +v> Hl H2 S + r  s  A common flow system such as that shown above difficulties  i n scheduling.  of water a v a i l a b l e f o r use on q  2  k  u+/  ^he  o n  discharge  = time f o r water to flow from p l a n t 1 to 2.  The main reason  presents  i s t h a t the amount  at hydro p l a n t No.  2 depends not  from hydro p l a n t No.  1.  The  only  problem  i s s i m p l i f i e d to a great degree i f both hydro p l a n t s have c o n s i d e r a b l e storage  c a p a b i l i t i e s and CT* i s s m a l l .  In t h i s case i t i s p o s s i b l e to assume t h a t the time delay is negligible  and t h a t the discharge  immediately a v a i l a b l e to the  from the f i r s t p l a n t i s  second p l a n t .  As the stream  are assumed to be known f u n c t i o n s of time f o r the problem i t i s now The  equations  will  The  reason  scheduling  f o r t h i s i s t h a t the equations  then be d i r e c t l y comparable to the equations thesis.  equations.  f o l l o w along the l i n e s of  that used i n reference 26 to o b t a i n the o r i g i n a l equations.  short-range  p o s s i b l e to solve f o r the scheduling  d e r i v a t i o n of these  flows  obtained  considered  in this  will  90  The problem i s to minimize t f d t = minimum  ...  )  (B-l)  s u b j e c t to the r e s t r i c t i o n s ... (B-2) t  rt Q dt = V 2  +j  T 2  (Q - )dt = V  or  2  Q l  Q dt 1  ... (B-3)  T 2  These r e s t r i c t i o n s are s a t i s f i e d when f d t +T where If. = a Lagrangian -t  Q d t +T  /  1  26 I (Q -Q )dt  2  x  2  1  = 0  multiplier.  1  S f d t + ^(T1-T2)Q1^ or  S(T -T )Q ^  gfdt +  1  2  1  + S?2Q dt  = 0  2  + S/ Q dt = 0 2  2  Finally  (B-4)  H2  Also  P S r  + +  p  HI  p H2  + +  _p L  r  hence  _o D~  „ p  u  ap - ^p- S rP  aP  T  SP + SP S  Solving f o r ( l (1 -  3P  H1  T  HI 3P  T  'HI  + SP SP  H  H2  d J  s  3P  L  T  - ^3 P - S PH 2 " 9 P "£E>H 1 = 0 U i  0  H 2  T  X  H1  gives  H1  SP  L  1  9P = - (1 - ^ )  T  S  £P  . <9P - (1 - ^ p ^ - ) S P T  G  H2  H  2  ... (B-5)  91  Prom eq. (B-4)  3P^*  S P  H1  " 3P^  =  P  S " ^ 2 S2 P9PTT^. ^ ^ H2  L  through by 1 - ^ 5 — and s u b s t i t u t i n g Hl 3  Multiplying  S  P  (B-5) i n t o the  d r  result  gives  3P ^1-^2)  HI  3P  (1  or  ^  5P~  3P  Ql~*P  9 Q  " C5PT  9f ^  T  3P  HI  9P  T  SP  ) S P  S  S  P  H2  H 2  U i  H 2  C  A  ^  ^^2 + <fo 2 3 P - SP, H2  0  L (1 - gp^) -  1  "  +  T  P  (1 -  9  3P  H  3P  1  S  S  +  T  2 9P.H2  3 B HI  H2  SP  t  3 P  S B  0  H2  But the c o e f f i c i e n t o f each v a r i a t i o n must be equal to zero. Hence 9P  9Qi  <*i-?2> 5 ^  "^  ( 1  90^ ^1~~^2^ 9P HI  c5P  T  "  ( 1  9P ~ 3p_ ) 112  " a4  :>>  T  %  }  =  ° -  ( B  "  6 )  9P 9Q 4 9 ( l - =VD ) ~ = 0 H1 H2  L  L  -  5 P  2  9 P  (B-7)  From eq. (B-6) and ( B - 7 )  3Qn  3f  8P  (  S  ^l-'^2  )  Sp  HI  c9P  (1 - g j ^ )  L  (1 9Q7) 2 SP.  H2  (1 =  A.  -  3P  )  HI  112  92 Therefore  fcjr fr- +  A  our  s  =  3P-  SP^-  3 P 9P  \/ ^ 2 +  A  ... (B-8)  A  g  T  L  TP^ =  ••• (B-10)  X  Equations (B-8) - (B-10) are the r e q u i r e d scheduling equations. ^  determines the volume of water to be used at hydro p l a n t No. 1  while  determines the amount of water to be used a t hydro p l a n t  No. 2. I f the second hydro p l a n t i s a r u n - o f - t h e - r i v e r type, then 0^=0.  the above a n a l y s i s holds true only f o r the case where  I t i s not l i k e l y that an a n a l y t i c s o l u t i o n such as that d e s c r i b e d i n t h i s section-can be obtained f o r the case where  0.  In  t h i s case d i f f e r e n t procedures must be used. The dynamic  programming technique d e s c r i b e d i n t h i s  thesis  21 or a procedure d e s c r i b e d by Peck range problem may be a p p l i c a b l e . at l e a s t two approximations.  f o r the s o l u t i o n of the l o n g I t i s s t i l l necessary to make  The f i r s t  i s that the d a i l y load  c y c l e i s p e r i o d i c and the second i s t h a t the r e s e r v o i r l e v e l i s at approximately the same e l e v a t i o n at the beginning of each day. These two assumptions may be j u s t i f i a b l e i n some cases. The s o l u t i o n of the common flow problem i s v e r y complex it  i s obvious that a great deal of work remains to be done on  t h i s problem.  I t appears t h a t the use of dynamic  programming  and Monte C a r l o methods should give good r e s u l t s here.  and  93  APPENDIX C.  STREAM FLOWS STUDIES IN THE BCPC SYSTEM .  I t has been s t a t e d i n t h i s t h e s i s  (Chapter l ) t h a t the  c o r r e l a t i o n between stream flows i n the B.C. Power Vancouver I s l a n d System  i s extremely low.  i s p r a c t i c a l l y no c o r r e l a t i o n at a l l .  Commission  In some cases there  T h i s chapter wishes to  prove t h i s statement and shows a p o r t i o n of the work done~'"aTontg this  line. One of the most d i f f i c u l t problems i n the long-range schedul-  ing,  of a hydro thermal system i s f o r e c a s t i n g the stream flows.  I t i s impossible to t e l l  exactly' how  over some f u t u r e p e r i o d of time.  much water w i l l be  available  The best s o l u t i o n to the problem  appears to be to use p r o b a b i l i t y methods to give estimates of the stream f l o w s .  The i d e a l way  to f o r e c a s t stream flows would  be t o base these p r e d i c t i o n s , i n p a r t at l e a s t , on long-range weather f o r e c a s t s .  However, weather f o r e c a s t s are n o t a b l y  i n a c c u r a t e and as y e t cannot be r e l i e d upon to any l a r g e extent. Even i f i t were p o s s i b l e to p r e d i c t how much p r e c i p i t a t i o n could be expected w i t h i n the next few weeks there would s t i l l the d i f f i c u l t problem of determining how river  remain  t h i s would a f f e c t the  flows. Another approach to t h i s problem i s to assume t h a t there i s  a r e l a t i o n between the monthly (weekly) average flows of a p a r t i -  20 21 cular river.  P h y s i c a l l y , t h i s assumption can be j u s t i f i e d .  Ground water and water from m e l t i n g snow and from r a i n f a l l the r i v e r i n a manner which i s a d e f i n i t e c h a r a c t e r i s t i c of the t e r r a i n of the water shed.  The problem i s t o f i n d the  c o r r e l a t i o n between these f a c t o r s and the stream f l o w s .  ' feed  94 Two methods, e i t h e r r e g r e s s i o n applied The  to t h i s problem.  or c o r r e l a t i o n , can be  Both methods were used i n t h i s study.  stream flows were p r e d i c t e d  w i t h the use of the r e g r e s s i o n  c o e f f i c i e n t s while the c o r r e l a t i o n c o e f f i c i e n t was used t o determine the s i g n i f i c a n c e indicate  of the r e g r e s s i o n  the degree of c o r r e l a t i o n present.  difference  c o e f f i c i e n t s and to There i s a b a s i c  between these two c o e f f i c i e n t s which should be  remembered. Regression c o e f f i c i e n t s are appropriate i f one v a r i a b l e , y, may be designated as being dependent on the other v a r i a b l e , x. Values of y may be p a r t l y c o n t r o l l e d subsequent to x. two  or caused by x, or y may be  T may a c t u a l l y be considered as the r e s u l t of  elements, one of which i s f u n c t i o n a l  i n nature.  I f the l a t t e r d i d not e x i s t the p o i n t s would l i e on  some p a r t i c u l a r l o c u s .  The e f f e c t of the random p a r t  s c a t t e r these p o i n t s about t h i s l o c u s . it  and the other i s random  i s to  Under the above c o n d i t i o n s  i s a p p r o p r i a t e to attempt to estimate th,e value of the dependent  v a r i a b l e y from a knowledge of the independent v a r i a b l e x. the  On  other hand, c o r r e l a t i o n i s a measure of the two-way average  of r e l a t i o n s h i p between two q u a n t i t i e s . flows, as i n many p h y s i c a l  situations,  In the case of stream e i t h e r method i s a p p l i c a -  ble. Little  significance  can be p l a c e d upon the r e g r e s s i o n or  c o r r e l a t i o n c o e f f i c i e n t s i f th<e d i s t r i b u t i o n of stream flows i s 34 not normal.  I t was necessary to transform the flows xn order  to o b t a i n a normally d i s t r i b u t e d flow d i s t r i b u t i o n .  A trans-  formation of the type u =  fU ) ±  ...  (C-l)  95  was used. Several d i f f e r e n t  transformations were t r i e d and the r e s u l t s  i n d i c a t e that the flows had a l o g normal d i s t r i b u t i o n .  The t r a n s -  formation used was u  i  = l o g (x.^ + a)  where a i s an a r b i t r a r y p o s i t i v e  ... (C-2) or negative  constant.  The data used i n t h i s study was obtained from the BCPC. The average monthly flows i n t o Comox Lake were s t u d i e d f o r a p e r i o d of 37 y e a r s .  The histogram  of the untransformed  i s g i v e n i n F i g . 23, and of the transformed example.  June flows  flows i n F i g . 24, f o r  I t i s t o be noted t h a t the d i s t r i b u t i o n i n the second case  i s d e c i d e d l y more normal. The The  equation of the r e g r e s s i o n l i n e i s given by y = a + bx.  coefficients  a and b can be found by the method of l e a s t  squares,  The q u a n t i t y b i s the slope of the r e g r e s s i o n l i n e and i s u s u a l l y c a l l e d the r e g r e s s i o n c o e f f i c i e n t .  I t can be shown t h a t  Z"x E vA - N—Z *x y— ^ a = — (Ix) - NZ(x) :  ==  (Lx) ..2  The c o r r e l a t i o n r  where  =  ... (C—3)  J  2  N'E(x)  2  2  c o e f f i c i e n t i s g i v e n by S  x  v  Sx Sy  Sxy = the covariance of the values x^, y^.  ' _ £(x-x)(v-y) S x  N • = the standard d e v i a t i o n of x 1  2  96  sy= The most convenient N  r =  (iZCvi-y) ) 2  formula f o r r i s ^ £*y  - jIxEy  (Nrx -(Xx) ) 2  4  2  (NXy -(Iy) ) 2  2  _  r  2  ( c  _  4 )  2  Tests have been devised which give an i n d i c a t i o n of the s i g n i f i c a n c e of the r e g r e s s i o n and c o r r e l a t i o n  coefficients. 34  Student's  t t e s t i s used f o r t h i s purpose.  I t can be shown  that  * - (w)  fj(^  -  rt-iof'f^)*  ... (c-5)  il-r / where 0 represents the t r u e p o p u l a t i o n value of the r e g r e s s i o n coefficient.  In order to see i f an observed value of b d i f f e r s  s i g n i f i c a n t l y from zero, set p = 0 and  calculate  I f t h i s i s l a r g e r than a set value of t f o r any  assigned  s i g n i f i c a n c e l e v e l and N-2, as found from standard t a b l e s , then the g i v e n value of b i s s i g n i f i c a n t .  A l t e r n a t e l y (C-5)  can be  solvedl to f i n d confidence l i m i t s f o r p .  e =MI=f¥  -  ± b  <0  -  The value of b i s not s i g n i f i c a n t i f t h i s i n t e r v a l i n c l u d e s zero.  The  above equations are based on s e v e r a l assumptions as  d i s c u s s e d i n r e f e r e n c e 34.  The main requirement  i s t h a t the  d i s t r i b u t i o n must be normal. Confidence  l i m i t s f o r the true r e g r e s s i o n l i n e and f o r an  estimated value of y f o r a g i v e n value of x may  be found.  These  7)  97 are given by Sey (l + N" \  a + bx ± t a  1 1  + ( ^ ) ) NSx^ / 2  •••  2  ( C  "  8 )  where Sey = v a r i a n c e of y values around the r e g r e s s i o n l i n e . 2  N S x  t  2  . £( . x  .  = value of t corresponding to some given l e v e l a.  significance  The r e g r e s s i o n and c o r r e l a t i o n c o e f f i c i e n t s f o r the r i v e r flows a t Comox Lake are shown i n Table XV.  MONTHS  b  r  a  Jan-Feb  0.3992  1.7611  0.4433  Feb-Mar  0.2406  2.2303  0.3826  Mar-Apr  0.3505  2.0171  0.3621  Apr-May  0.2545  2.4001  0.2980  May-June  0.8194  0.5142  0.6478  June-July  0.8196  0.2754  0.7696  July-Aug  0.7175  0.5249  0.6747  Aug-Sept  0.7553  0.6090  0.5508  Sept-Oct  0.3145  2.1040  0.2616  Oct-Nov  0.2534  2.3459  0.3149  Nov-Dec  0.1387  2.7189  0.1716  Dec-Jan  0.1349  2.5776  0.0970  TABLE XVII.  Regression and C o r r e l a t i o n C o e f f i c i e n t s f o r the River Flows a t Comox Lake.  The r e g r e s s i o n l i n e f o r the June-July flows i s p l o t t e d i n Fig.  25 .  The 95f° and 80% confidence l i m i t s as found  from  98  12 1Q  200  300  400  F i g . 23  500  600  700  800  900  1000  1100  D i s t r i b u t i o n of the R i v e r Flows i n t o Comox Lake. C l a s s Length = 200  12 10  S  8  W  6  2.35  2.45 2.55 2.65 2.75 2.85 2.95 3.05 3.15 3.25 3.35  F i g . 24  D i s t r i b u t i o n of the R i v e r Flows when Normalized on Log B a s i s . C l a s s Length = 0.1  River Flow (July)  3.2  «  2.6  1  2.7  *  2.8  1  •  2.9  3.0  n  3.1  1  3.2  i 3.3  i 3.4  JUNE FLOWS (Log o f ) F i g . 25.  The Regression L i n e f o r the June-July R i v e r Flows and the Corresponding 95 and 80$ Confidence. M m i t s . -0  -a  100  equation (C-8) are a l s o shown*  I t i s to be noted t h a t these  confidence l i m i t s have a h y p e r b o l i c shape. S o l v i n g f o r t from equation (C-6) f o r the June-July flows gives t = (.7696) /  3  5 0  )» =  7.116  \l-(.7696r / The value of t f o r N-2  = 37 and f o r a s i g n i f i c a n c e .level of  95%  34 i s about 2.040.  I t can be seen that the value of t as found  from equation (C-6) i s s u b s t a n t i a l l y l a r g e r than t h i s v a l u e .  It  i s thus safe to assume t h a t the value of b found i s s i g n i f i c a n t . As can be seen from Table XV, i n d i c a t e t h a t there i s l i t t l e  the r e s u l t s of t h i s  study  or no c o r r e l a t i o n present between  the monthly flows of the r i v e r s t u d i e d f o r the m a j o r i t y of the year. to  In t h i s case i t appears  t h a t i t would be very  difficult  t r y to f o r e c a s t these r i v e r flows by u s i n g r e g r e s s i o n technique I f t h i s i s the case, the s o l u t i o n of the long-range  schedulin  problem becomes extremely d i f f i c u l t . I t i s c e r t a i n t h a t the 20 21 method d e s c r i b e d by L i t t l e and enlarged upon by Peck would not work.  101  APPENDIX D.  BRIEF OUTLINE OF DYNAMIC PROGRAMMING PROCEDURES.  i The f o l l o w i n g i s only a b r i e f o u t l i n e of the more  important  0  p o i n t s i n v o l v e d i n dynamic programming methods. c r i p t i o n of dynamic programming procedures r e f e r e n c e 42. method.  A full  can be found i n  Three terms occur r e g u l a r l y i n d i s c u s s i o n s of t h i s  These terms and t h e i r d e f i n i t i o n s f o l l o w :  Policy:  des-  ;  Any r u l e f o r making d e c i s i o n s which y i e l d s an allowable sequence of d e c i s i o n s .  Optimal P o l i c y :  A p o l i c y which maximizes (minimizes) a p r e -  assigned f u n c t i o n of the f i n a l C r i t e r i o n Function:  state v a r i a b l e s .  A f u n c t i o n of the f i n a l  state variables  which i s used as a - c r i t e r i o n of the system o p e r a t i o n . Dynamic programming deals with the mathematical  theory of  i  m u l t i - s t a g e d e c i s i o n processes. approach,  In c o n t r a s t to the c l a s s i c a l  ( v a r i a t i o n a l c a l c u l u s ) , which takes an N stage process  of M d e c i s i o n s per stage and deals with a MN dimensional stage process, t h i s procedure number of dimensions, stage.  single  reduces the problem to the proper  namely the dimension  of the d e c i s i o n a t each  In p l a c e of determining the optimal p o l i c y , or sequence  of d e c i s i o n s , from some f i x e d s t a t e of the system, we wish to determine system.  the optimal d e c i s i o n to be made a t any s t a t e of the An important aspect of t h i s procedure  h i s t o r y of the system i s of no importance  i s t h a t the past  i n determining f u t u r e  actions. Consider a simple m u l t i - s t a g e a l l o c a t i o n process. there e x i s t s a q u a n t i t y V which i s d i v i d e d i n t o two p a r t s of v and V-v.  Assume  non-negative  Assume t h a t a r e t u r n of g(v) i s obtained  from v and h(V-v) from V-v.  To maximize the t o t a l r e t u r n , the  102  problem  i s t o maximize R(V,v)  Consider v  is  = g ( v ) + h(V-v)  a twa^stage  0  <, a ^  0  ^  process,.  f o r obtaining g(v) ,  t o b(V-v)  q u a n t i t y , av + b(V-v),  a ( v ) + b(V-v) = v  R (V,v, 2  f o rN  As a p r i c e  (D-1)  where  1  +  1  r e c e i v e d a t t h e second  or  . . .  b < 1  the remaining  Setting  function  t o a ( v ) a n d (V-v) i s r e d u c e d  i s reduced  ¥ith  the analytic  V ] L  ( "" )» v  v  1  1  stage.  the process  hO^-v^  the return g ( v ) + 1  The p r o b l e m  now i s t o m a x i m i z e  = g ( v ) + h(V-v) + g (  )  i s repeated.  V  l  ) +  hO^-v^  stages  B (V,v v ...v ) N  f  1  N  = g(v)  + h(V-v) +  ... g ( v . _ ) N  1  +  h  (  v N  _  1  ~  v N  _  1  )  . . . (D-2) where  the quantities v  available  f o rfuture a l l o c a t i o n a r e :  = av + b(V-v)  x  0 =v< V (D-3)  V  RJJ m u s t  solution  is  the value  and  N,  b ( Y  over  N-2 " N-2 v  and l o c a l  }  t h e N-dimensional  Recognizing  (calculus, etc)  a l l that  +  K  (D-3).  solution  Since  = ^ -2  be m a x i m i z e d  relations  of  K_1  the d i f f i c u l t i e s  and t h e r e s u l t a n t  maxima, t a k e  i s r e q u i r e d a t any p a r t i c u l a r  of v i n terms  o f V, t h e t o t a l  t h e number o f s t a g e s  left,  one-dimensional  solved,  keeping  choice  as a goal  a t each  i t should  stage.  of an  problems  a different  A a  region given  stage  analytic  of  view  by the  uniqueness  of the problem. of the process  resources  available,  b e p o s s i b l e t o make  1  The p r o b l e m  will  be  the preservation of one-dimensionality.  103  Define: f (V)  the maximum r e t u r n from an N-stage process, with an i n i t i a l q u a n t i t y Y.  N  starting  Max R ( V , v , v . « . o V ^ j ^ ) N  1  N = 2,3....  f (v)  Max  x  It  ... (D-4)  g(v) + h(V-v)  i s necessary to f i n d an equation  f o r l^Y)  F i r s t d e f i n e the P r i n c i p l e of O p t i m a l i t y .  i n terms of f ^ ( V ) .  43  "An optimal p o l i c y has the property that whatever the i n i t i a l s t a t e and i n i t i a l d e c i s i o n are, the remaining d e c i s i o n s must c o n s t i t u t e an optimal p o l i c y with respect to the s t a t e r e s u l t i n g from the f i r s t d e c i s i o n " . Considering  a two-stage process,  r e t u r n from the f i r s t  the t o t a l r e t u r n i s the  stage plus that from the second stage, a t  which time there i s an amount a(v) + b(V-v) l e f t to a l l o c a t e . This remaining amount must now be used i n an optimum manner to maximize the t o t a l r e t u r n .  I f v^ i s chosen o p t i m a l l y , the r e t u r n  from the second stage w i l l be f ( a ( v ) + b(V-v))  (D-5)  x  Thus the t o t a l r e t u r n i s R ( V , v , v ) = g(v) + h(V-v) + f ( a ( v ) + b(V-v)) . 2  1  1  Since v i s to be chosen to y i e l d a maximum the f o l l o w i n g recurrence  r e l a t i o n may be d e r i v e d :  0< v or i n general  0 ^v^ V for It  N >2 with f ( Y ) d e f i n e d as above. 1  i s thus p o s s i b l e to solve f o r the sequence of values  .iD-6)  104  v , v ^ , . . „ . V j  ^ which give the optimum r e t u r n as w e l l as f i n d i n g  T  f (T). t  Consider the simple problem of determining the maximum of the f u n c t i o n F(v ,v 1  N = Z_g (v )  v )  2  N  i  ...  i  (D-8)  over the r e g i o n (a) i v  + v  (b) v.> Each f u n c t i o n g.(v)  2  + v  3 * * * 24 +  v  =  V  0 i s assumed to be continuous  f o r v . ^ 0.  Since  the maximum P(v) depends only on V and N, d e f i n e the sequence of functions f ( V ) = Max N  (g (v) + ^ ( V - v ) ) N  0>v^V  ...  (D-9)  N = 2,3 =  g]_(v)  In the preceding i t has been assumed t h a t the processes were time independent.  That i s , the t o t a l r e t u r n depends only on N,  the number of stages, and V, the i n i t i a l the time the process i s s t a r t e d .  q u a n t i t y , and not upon  I f t h i s i s not the case, the  procedure  i s q u i t e s i m i l a r to t h a t above, and i s d e s c r i b e d i n  reference  43.  There i s always the problem of p r o v i n g , f o r any given process, whether an optimum mode of o p e r a t i o n i s obtained or not. number of cases i t i s easy to see t h a t an optimal p o l i c y  In a exists.  Vhen t h e r e are only a f i n i t e number of allowable choices a t each stage, or where these choices are l i m i t e d to some f i n i t e plane  or  r e g i o n , i t can be shown t h a t an optimal s o l u t i o n e x i s t s f o r some simple  cases.  

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