@prefix vivo: . @prefix edm: . @prefix ns0: . @prefix dcterms: . @prefix skos: . vivo:departmentOrSchool "Applied Science, Faculty of"@en, "Electrical and Computer Engineering, Department of"@en ; edm:dataProvider "DSpace"@en ; ns0:degreeCampus "UBCV"@en ; dcterms:creator "Fedoroff, Vitaly L."@en ; dcterms:issued "2011-05-25T22:52:06Z"@en, "1970"@en ; vivo:relatedDegree "Master of Applied Science - MASc"@en ; ns0:degreeGrantor "University of British Columbia"@en ; dcterms:description """A novel, idealized mathematical description is presented of the cooling process of electric power cable system by an integral liquid coolant configuration. A model is first obtained for constant fluid velocities, by the use of Laplace transform and subsequent solution of an ordinary differential equation. A simple integral expression is obtained, which is numerically integrated by the Simpson's rule. The set of curves so obtained, clearly show the delay of the temperature peak vs. the peak of the power demand. A steady state form of control is proposed. Block diagrams of two implemetations are shown. These control systems are designed to maintain the temperature of the effluent constant. An optimization algorithm based on developments given by Sage is presented. This algorithm will optimize the transfer of the cooling system from the initial condition of zero velocity to the steady state operation described above."""@en ; edm:aggregatedCHO "https://circle.library.ubc.ca/rest/handle/2429/34862?expand=metadata"@en ; skos:note "THE OPTIMIZATION OF COOLING OF UNDERGROUND CABLE SYSTEMS by VTTALY L. FEDOROFF Diplome d'Ingenieur E l e c t r i c i e n , U n i v e r s i t e l'Aurore a. Shanghai, 1951. A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE r' REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE i n the Department of E l e c t r i c a l Engineering We accept t h i s t hesis as conforming to the required standard Research Supervisor. Members of the Committee. Acting Head of the Department Members of the Department of E l e c t r i c a l Engineering THE UNIVERSITY OF BRITISH COLUMBIA A p r i l , 1970 In p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t o f t h e r e q u i r e m e n t s f o r an advanced d e g r e e a t t h e U n i v e r s i t y o f B r i t i s h C o l u m b i a , I a g r e e t h a t t h e L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and s t u d y . I f u r t h e r a g r e e t h a p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y p u r p o s e s may be g r a n t e d by t h e Head o f my Depar tment o r by h i s r e p r e s e n t a t i v e s . I t i s u n d e r s t o o d t h a t c o p y i n g o r 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 f i n a n c i a l g a i n s h a l l not be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . Depa r tment The U n i v e r s i t y o f B r i t i s h C o l u m b i a V a n c o u v e r 8, Canada ABSTRACT A novel, i d e a l i z e d mathematical d e s c r i p t i o n i s presented of the cooling process of e l e c t r i c power cable system by an i n t e g r a l l i q u i d c o o l -ant c o n f i g u r a t i o n . A model i s f i r s t obtained f o r constant f l u i d v e l o c i -t i e s , by the use of Laplace transform and subsequent s o l u t i o n of an o r d i -nary d i f f e r e n t i a l equation. A simple i n t e g r a l expression i s obtained, which i s numerically i n -tegrated by the Simpson's r u l e . The set of curves so obtained, c l e a r l y show the delay of the temperature peak vs. the peak of the power demand. A steady state form of c o n t r o l i s proposed. Block diagrams of two implemetations are shown. These c o n t r o l systems are designed to maintain the temperature of the e f f l u e n t constant. An optimization algorithm based on'developments given by Sage i s presented. This algorithm w i l l optimize the t r a n s f e r of the cooling sys-tem from the i n i t i a l condition of zero v e l o c i t y to the steady state oper-a t i o n described above. i i TABLE OF CONTENTS Page Introduction 1 CHAPTER I CONSTANT VELOCITY MODEL 4 . 1.1 General Considerations 4 1.2 The Rectangular Model and I t s J u s t i f i c a t i o n 4 1.3 Heat Losses 5 1.4 Derivation of Coupled Flow and Heat Transfer P a r t i a l D i f f e r e n t i a l Equation . 7 1.5 Computation of C o e f f i c i e n t s . . 13. 1.6 Analogue Model Implementation . 15 1.7 D i g i t a l Implementation and Temperature Curves 16 CHAPTER II VARIABLE VELOCITY OPERATION 21 2 .1 General Considerations 21 2.2 Scale Changes 21 2 .3 Change of Variable 22 2.4 Variable V e l o c i t y . . 23 2 .5 The .Degenerate Performance Functionals 24 2.6 Proposed Cooling Controls 26 CHAPTER I I I OPTIMAL TRANSFER TO EOJJILIBRUM OPERATION 30 3.1 General Considerations . 30 3.2 I n i t i a l Conditions 31 3.3 The Perturbed P a r t i a l D i f f e r e n t i a l Equation . . . . . . . . 33 3.4 Performance Functionals • 33 3.5 The. Hamiltonian 35 3.6 The F i r s t V a r i a t i o n and the Co-state 36 i i i <. V 3.7 The Gradient Condition & The Steepest Descent of the Hamiltonian • 38 3.8 T r a n s v e r s a l i t y Conditions 38 3.9 The Computing Algorithm 40 CONCLUSION 42 APPENDIXES A. Water Conductivity 43 B. Representative Load Curve . 44 General References . . . 46 References 46 i v LIST O P P R I N C I P A L S Y M B O L S -Notes: l ) Bars over a symbol denotes the same v a r i a b l e i n d i f f e r e n t u n i t s . Q, Q. I t i s dropped i n subsequent development. 2) When the ommision of arguments does not introduce any ambiguities, these w i l l be dropped to s i m p l i f y the notation. 3) T i l d e i s used to d i f f e r e n t i a t e between d i f f e r e n t v a r i a b l e s using the same l e t t e r symbol, which i s u n i v e r s a l l y accepted. Ex., ^ = density; ^ = r e s i s t i v i t y ; c = constant; c = s p e c i f i c heat capacity. 4 ) Same l e t t e r symbols are used f o r dependent variables having d i f -f e r e n t arguments. Ex., u ( x , t ) , u(y), u ( t ) , or simply i n d i c a t e d by\"u(---). ' c,c,,C c . h . H . H . k . _k ... J , J A L . P • -q • -Q\" t d • u Constants S p e c i f i c heat capacity Height of the rectangular cooling channel The Hamiltonian Heaviside f u n c t i o n of i t s arguments .. Running index - — S p e c i f i c heat conductivity Performance i n d i c e s * Length of the cooling run Costate v a r i a b l e - Heat f l u x Laplace transform of the heat f l u x Time delay f o r constant v e l o c i t y = L/U 0 Coolant v e l o c i t y Steady state coolant v e l o c i t y v Perturbed coolant v e l o c i t y w Width of the rectangular cooling channel lOOw/m Transferred to the rectangular duct model the heat f l u x on one wall i s : 50 x 100 — 2 x 10' watts/cm^ *;//////// P I P E A N D GROUND X2 i COOLING CHANNEL 3 cm x2, CABLE SHEATH X2iJ r 0 8 ~~ 50 cm COOLING CROSS-SECTION Fig. 1-1 7 The above numerical data w i l l be used i n the t h e s i s . Further information on constants used can be found i n Appendix A. 1.4 Derivation of coupled flow and heat t r a n s f e r p a r t i a l d i f f e r e n t i a l equation The purpose of t h i s s e c t i o n i s to derive a simple model f o r the cable and the c o o l i n g duct system. I t i s possible to derive more r e a l i s -t i c models using a d i s c r e t e approximation to the p a r t i a l d i f f e r e n t i a l equa-t i o n f o r the heat flow i n three dimensions. However, such an approach would g r e a t l y complicate the problem of dynamic optimization. In p a r t i c u l a r , the boundary conditions are such, that the numerical computations would re-quire extensive use of i t e r a t i o n s . Furthermore phy s i c a l i n s i g h t i n t o pos-s i b l e p r a c t i c a l suboptimal controls would be l o s t . The model chosen i s i l l u s t r a t e d i n F i g . 1-2. The coolant flows i n a rectangular channel such, that the height i s l e s s than the width, which i n turn i s much l e s s than the length. h <. w 4- L ' This model i s u s e f u l i n that i t approximates a possible physi-c a l s i t u a t i o n where the transmission cable i s c e n t r a l l y placed i n the c o o l -8 q(t)'=fR LOSS COOLANT VELOCITY AND HEAT FLOW F i g . 1-2 y i n g pipe. In t h i s s i t u a t i o n the heat f l u x has r a d i a l symmetry, so that a u n i t width need be considered i n the model (see F i g . 1-2). The heat 2 f l u x r e s u l t s from the i R l o s s in.the transmission cable. Because the heat los s through the pipe i s neglected as well as the thermal r e s i s t i v i t y and the heat capacity of the'cable, the f o l l o w i n g boundary conditions w i l l p r e v a i l . At the outer cooling pipe surface, where y=1i, the temperature gradient i s zero: f o r y= h *Q(x,h,t) _ Q < x: ,0,L > (1.1) *y - w 2 ) 3y f o r a uniform cable c r o s s - s e c t i o n the heat generated w i l l be independent of the s p a t i a l coordinates so that q ( x , t ) = q ( t ) . As such, these boundary values define, what i s known i n mathe-matical physics, the problem of Neumann. I t admits a m u l t i p l i c i t y of so-l u t i o n s , and i t i s the intake temperature of the co o l i n g f l u i d which defines the s p e c i f i c one. The i n i t i a l temperature of the cooling f l u i d . i s taken as a r e f e r -ence. For convenience, t h i s i s taken to be zero: ©(0,y,t) = 0 f o r \" (1.4) _o,L 1U The formation of the p a r t i a l d i f f e r e n t i a l equation can follow the de-velopment shown i n Kay [9] or i n Knudsen & Katz [io], f o r a more modern approach see Rojdestvensky & a l [ l l ] . . The Laplacian i s equated to the t o t a l time d i f f e r e n t i a l , with s u i t a b l e c o e f f i c i e n t s on both sides of the equa l i t y : t ( & + & ) = ? f 2 8 ( 1 . 5 ) dx 3y Dt Expanding the t o t a l time d i f f e r e n t i a l , and noting that the com-ponents of the f l u i d v e l o c i t y i n the Oy and Oz d i r e c t i o n are zero: Eq ( l . 3 ) reduces to: * ( ^ f + * - f ) = cf ( u « + S S ) ( 1 . 6 ) dy Qx dt To further s i m p l i f y the model the fol l o w i n g assumptions w i l l be made: a) the v e l o c i t y u(.) i s independent of time., t h i s i s the constant velo-c i t y case, and due to the i n c o m p r e s s i b i l i t y of the f l u i d i t w i l l be independent of the x coordinate, so that: u(.) = u(y) (1.7) b) a temperature average over the y coordinate can be taken. This i s i n t u i t i v e l y j u s t i f i e d because h i s so much less i n value than the length of the co o l i n g duct^L. consequently 9(x,t) = - I 9(x,y,t) dy (1.8) h j S u b s t i t u t i n g Eq. (l-8) i n t o Eq. (l-6) one obtains the following: 2 h \" ' * ) - = & ( 1 . 9 ) dx dt ,d e , 1 de cy(—~2 + - * dx h dy o Noting the boundary value given i n Eq. ( l - l ) , the Eq. (l-9) reduces to: a2e be _ djz _ o-^ _ _ _ g _ * 2 - Udx • ^ - - , Z - ^ (1.10) ,dx d X d t hk hep Take the Laplace transform of Eq. ( l - l O ) , a2® a © c r — - u dx dx 5 © = -Q_ hep (1-11) Equation ( l - l l ) i s an ordinary d i f f e r e n t i a l equation (O.D.E.) of second order. The c h a r a c t e r i s t i c equation i s given by: a - oL - — - 0 I u \\2 u s _ (a - — ) - — 9 - - - o 2 1.0) and that f o r a f i x e d range i n s, one w i l l make the f o l l o w i n g approximation 4 r s « l . u then: 1 / u + u / AO'S *>2 cr o ? V . A [ u ± u ( 1 + 2^5 ) ] <*\" 2.; 2 2 u and: a, = - —- a0 ~ — 1 U £ KJ the value of ( l . s ) can now be expressed i n the form - i i r u L • (9(L,s) -2-- + C.e u \" + C ?e « -hep For i n t e r e s t l e t us take a dimensionality check: • 2 m - l Dim|> ] = L f Dim[u] = L T - 1 Dim[s] =• T\" 1 Dimensionality of 6L_ u Dim [ — ] — ——^ — — 1 (dimensionless) Dimensionality of uL a-Dim [^p ] — ^ = 1 (dimensionless) L T 1.5 Computation of C o e f f i c i e n t s C]_ and C?. Rewriting Eqs (l-13) & (l-14) i n the form: J L . IL. hep s C i a i + C2*2 •c. - f c, I • ' h£p s y i e l d s 14 C (a„ - a.) = — • ^ (a n - u) and C 2 C = r n - - -i hep s q 2 - a 1 J 1 S *l ~ u a 2 - a. [ 1 + a l h 5 j > , (1-16) or C ^ _ J a . . a2 0 - u hc$>s a 2 - a (l-17) with: a n = — / OC, L -, \\ . a r L 1 e l - 1 + ue 1 OL a2 = - ( e a 2 L - 1) + u e a 2 L a 2 s u b s t i t u t i n g the previously -found approximations f o r and y i e l d s = - - ; a 2 = - • s L U _«U • a = - ^ - ( e -1J + u e — u u ^ H i -a 2 = ^ (e * - 1) + ue « u • i t follows that C 2= 0 and 0^ = The temperature f u n c t i o n i s : @ ( L s) = - -S- • e ^ 1 heps heps 1 . £ ' (1-18) hep S L u (l-19) ® ( L , s ) = - ± - Q ( i - ± e U S ) hep def i n i n g : u t i s the time f o r a hypothetical f l u i d element to transverse the length d of the cooling duct. The inverse of the ® ( L,s) can be given as a convolution i n t e g r a l t 6 (itr^j rj ( r ).£ (t - * ) d? (1-21) or where t e(L,t) = J ^ ( t _ r ) | ( T ) d r ( 1_ 2 2 ) n ( t ) = oC\"1 [ J Q(s) ] ^ q ( t ) ' (1-23) cp cp-?(t) \"= X\" 1 [ t ( J \" ^ \" ' S ) ] . (1-24) Equation (l-24) y i e l d s ... £ ( t ) = | [ U ( t ) - U ( t - £ ) ] where U.(t) .is the u n i t step fun c t i o n . S u b s t i t u t i n g Eq.(l-23) and (l-24) i n t o (l-2l) y i e l d s 6(L,t) = — f q(t) dT (1-25) hep ^ d 1.6 Analogue Impementation The equation (l-25) can be instrumented i n several ways, those shown below require the use of a delay l i n e . These have been instrumented previously, see f o r instance the works of Sarkar [l2] or Schone [l3]« For r e a l time implementation use can be made of tape :y disk or drum recorders. I f the analogue s i g n a l i s recorded on tape i n the form of pulses a v a r i a -ble v e l o c i t y \"could even be instrumented. Figures 1-4 and 1-5 show these delay l i n e s i n block form. 16 F i g . 1-5 uses two i n t e g r a t o r s the e f f l u e n t temperature i s the d i f f e r e n c e of the outputs of the two i n t e g r a t o r s . The delay l i n e i n t h i s case stores the h i s t o r y of the cable losses f o r the time period t ^ . A f t e r a f a u l t condition, i f the data i s destroyed, then only approximate values can be i n s e r t e d into the delay l i n e . This w i l l require a reduc-t i o n of the maximum allowable current i n the cable f o r a time. Figure 1-6 shows a sin g l e i ntegrator, the delay l i n e i s i n i t s output. The i n i t i a l conditions i n the delay l i n e would store the tem-perature d i s t r i b u t i o n i n the cooling duct. I t could be r e i n s e r t e d i n t o the delay l i n e by thermocouples placed i n the cooling duct, providing a better i f somewhat more expensive arrangement. The computation of Eq. (l-25) by d i g i t a l methods using a t y p i c a l load curve i s given i n the next paragraph. The ease of computation and the s i m p l i c i t y of the algorithm i n d i c a t e s a possible a p p l i c a t i o n of a D i g i t a l D i f f e r e n t i a l Analyser with i t s inherently better p r e c i s i o n vs. the analogue computer. 1.7 D i g i t a l implementation and temperature curves. Given a load curve (see appendix B) points are taken from i t at i n t e r v a l s h. Using S t i r l i n g ' s i n t e r p o l a t i o n formula and taking x ( i ) L x \\~ x ( i + l ) h = x(i+-l) - x ( i ) u = [x - x ( i ) ] /h y i e l d s .. S ( x ) = y ( i ) + f [ y ( i + l ) - y ( i - l ) ] + § [ y ( i + l ) - 2 . y ( i ) + y ( i - l ) ] These values are computed by a subroutine. tCt) + e a,t) HEAT INPUT DELAY LINE MODEL Pig. 1-4 DELAY LINE TIME = td TTT TEMPERATHRE DISTRIBUTION DELAY LINE MODEL Fig. 1-5 i y The computation of the i n t e g r a l V,c f ; t + t , / 1 d S ( t ) d t t. 1i s made by the trapezoidal r u l e , using the. same time i n t e r v a l as f o r the load curve: Q(Ut)= — • - f q(t) + 2q(t +At)t...-h..+2q[t + (n-l)At] 4 q ( t + nAt) J hc.j) 2 1 ' : This graph i s traced using a master program developed i n the E l e c t r i c a l Engineering Department by Dr. H.R. Chinn and i s shown i n Pig. 1-6. The time i n t e r v a l -At i s defined by At ar -•• d n Figure (l-6) can be u s e f u l i n evaluating various cooling schemes and t h e i r cost. I t i l l u s t r a t e s the delay i n achieving the maximum temper-ature f o r various coolant v e l o c i t i e s . Approximate optimization methods could be based on the curves by assigning a cost to the amount of the coo l -ant used and a cost to excessive temperature. The approximation would be i n the use of the curves to predict a sub optimum piecewise constant coo l -ant v e l o c i t y . The next chapter w i l l present an exact treatment of o p t i -mization f o r the case of a time-varying v e l o c i t y . - 1 0 1 2 3 4 5 6 7 8 9 10111213141516171819202122232425262728293031323334353S373839404142434445454743495051 TIME OF DRY Fig 1-6 DIURNAL TEMPERATURE CURVES CHAPTER II VARIABLE VELOCITY OPERATION 2.1 General Considerations. The main l i m i t a t i o n of the process model described i n Chapter I, where the v e l o c i t y i s maintained constant, i s i n the f l u i d tempera-ture v a r i a t i o n . To t h i s temperature v a r i a t i o n should be added the tem-perature drop i n the i n s u l a t i o n . I t i s much l a r g e r than the v a r i a t i o n encountered when the cable i s placed underground. A method f o r reducing t h i s temperature spre-ad i s obtained by using a go-return pipe. I t has been analysed i n reference [3]. Another method would be to keep the temperature d i s t r i b u t i o n along the cable run constant. This mode of operation i s proposed and analysed i n t h i s chap-t e r . I t i s of course a f i r s t approximation, which neglect the tempera-ture drop i n the i n s u l a t i o n . This strategy which we w i l l explain l a t e r , i n essence, provides a memory device i n the cooling duct i t s e l f . I d e a l l y , and t h i s word has to be used advisedly, the measurement of the temperature at one l o c a t i o n , would provide the temperature d i s t r i b u t i o n throughout the whole run. The second advantage, no l e s s important, i s the constancy of the temperature at various locat i o n s which reduces the mechanical fatigue, due continuous expansion and contraction of the sheath and i n s u l a t i o n . The deleterious e f f e c t s of t h i s l a s t phenomenon have been discussed i n various papers of r e f . [4]. The l o g i c a l basis of such an operation i s presented here. 2.2 Scale Changes. The equation (l.io) i s somewhat more d e s c r i p t i v e of the process 22 than need be. I t can be reduced to a simpler form by the f o l l o w i n g change of v a r i a b l e s (or s c a l i n g ) . x = x sfcr u = u icj q = c y q thus (l-3) can be written: 3 § - } 0 q C P h(B x f Mfcx) dt hcf and taking v from under the d i f f e r e n t i a l sign and s i m p l i f y i n g : 7^ \" u ( t ) , \" 17 - \" ^ (2-1) h As a f u r t h e r n o t a t i o n a l s i m p l i f i c a t i o n we w i l l drop the bars over the v a r i a b l e s and c o e f f i c i e n t s . This parabolic p a r t i a l d i f f e r e n t i a l equation (P.D.E.) w i l l be the mainstay of t h i s t h e s i s . I t has two v a r i a b l e parameters which are under co n t r o l of two d i f f e r e n t , s h a l l we say, e n t i t i e s or adversaries. The f i r s t one u(.t) i s under c o n t r o l of the u t i l i t y engineer, the second, q(t) i s under c o n t r o l of the cutomer. 2.^ Change of v a r i a b l e . The parameter u(t) i n the equations (l-25) does not enter i n the integrand but i s one of the l i m i t s of the i n t e g r a l . However, with the a i d of a change of v a r i a b l e , the following ex-pression i s obtained. From the modified Eq. (l-25) Q(L,t)=-3- \\% sfe&H d(*(r)] ( 2 1 4 ) h % )o u[x (T ) ] ( 2 - U ) given the previous dis c u s s i o n on games, the following strategy i s a v a i l -able to the u t i l i t y engineer: a i k u l = H = c t e Taking M out from under the i n t e g r a l .L Q(L,t) = - i L ( dx = -f =CL (2-16) hep J hep or f o r a v a r i a b l e upper l i m i t : Q ( x t ) = i ( X dx ^ ^ = CX (2-17) hc^ J hep o computing the value of C from eq(7) i n terms of 0 ( l , t ) (2-18) c = M £ e ( u , t ) U and p l a c i n g i t i n (2-17) u [ x ( t ) ] = I/C * q [ x ( t ) ] Should we wish to keep 0(L,t) constant then C i s also a constant and we have obtained a p a r t i c u l a r strategy. 2.4 Variable v e l o c i t y . The r e s u l t s which have been obtained i n the previous paragraph are rather h e u r i s t i c i n t h e i r approach. But these methods do give a pos-s i b l e s o l u t i o n of the equation (2-1) of the forms 0=Cx. Thus r e w r i t i n g Eq(2-l) with u(t) on the l e f t hand side y i e l d s and i n s e r t i n g i n t o i t the s o l u t i o n 0 =Cx one obtains - U = tc (2-20) From Eq.(2-20) the c o e f f i c i e n t C i s : 24 (2-21) (2-22) hu The temperature along the channel: hu The maximum temperature occurs at the end of the channel 0 = max hu recomputing C i n terms of the maximum temperature gives 0 £ - max L Thus the temperature 0 can be given i n terms of the 0 - - - - max Q „. . max , • 8 0 0 = • * L 2.5 The Degenerate Performance Functional. Considering that the highest f l u i d temperatures occur at the end of the cooling duct and that f o r an i d e a l incompressible f l u i d the v e l o c i t y u(t) i s the same throughout the whole length of the duct, the performance f u n c t i o n a l can be given by an expression using a sing l e i n -t e g r a l : ^ f 2 2 * J±- / [A u ( t ) + i>(H(9(L,t )-a o ) ] dt (2-24) (2-23) *1 . where 0 q i s a s p e c i f i e d maximum temperature at the hot end of the cable. The second term i s i n the nature of a Heaviside function and i s defined by the equations: I(e(L,t)-9 ) = 0 f o r e(L,t) L 0 .o o H(0(L, t)-© o) = 0 ( L , t ) - 0 Q f o r 0 ( L , t ) > QQ Because both terms i n Eq (2-24) are p o s i t i v e and u(t) does not vanish f o r non t r i v i a l values of q(t) one would expect that the second term w i l l also be greater than zero. The performance f u n c t i o n a l can also be thought to be expressed i n d o l l a r equivalent terms. The f i r s t term expresses the cost of water used i n cooling the cable. The second •term represents the d o l l a r - e q u i v a l e n t l o s s of investment due to the de-creased l i f e of the i n s u l a t i o n due to excessive temperatures. One should include, f o r a complete d o l l a r balance statement, a term i n q(t) This term a r i s e s due to the f a c t that the heat generated i n the conductor i s r e l a t e d to the power sold to the consumers. Consequently, t h i s enters i n t o the cost f u n c t i o n a l with a negative sign. Adding t h i s t h i r d term and grouping y i e l d s such as C • q(t) (2-25) f o r m = 2 and a s u i t a b l e choice of u ( t ) , the integrand of the f i r s t i n t e -g r a l may be rendered zero: a - u ( t ) 2 - c - q ( t ) 2 - 0 thus (2-27) I t i s seen from (2-22) and (2-27) that (2-28) Consequently, Eq(2-28), f o r given_a,c,h and Q , defines the length (L) of the cooling duct. 26 The above approach r e s u l t s i n a very simple s o l u t i o n because of the s p e c i a l form of the performance f u n c t i o n a l . A d i f f e r e n t form can be obtained by physi c a l considerations which r e s u l t s i n a performance f u n c t i o n a l more c l o s e l y r e l a t e d to actu a l costs. Simple solutions are then no longer p o s s i b l e . The next chapter discusses a general method •for obtaining s o l u t i o n s . 2.6 Proposed Cooling Controls. The Equation (2-18) suggests an arrangement f o r an open loop ( type c o n t r o l . This c o n t r o l , even then, need be open with respect to the process only. The re g u l a t i o n of the. coolant v e l o c i t y to a value u(t) requires feedback i n the valve s e t t i n g vs measured v e l o c i t y . An a d d i t i o n a l requirement i s introduced-by the rate of change of u(t) which has to be l i m i t e d to avoid a hydraulic ram e f f e c t . O r i g i n a l work was done by Solodovnikov the main r e s u l t s of h i s fin d i n g s are given i n r e f [ l 4 ] . P ig. 2-1 shows the block diagram of such an arrangement. A more so p h i s t i c a t e d c o n t r o l updates the value of Act i n the ex-pression f o r the c o e f f i c i e n t , (see F i g . 2-2). k 4 L ... 0 M e M - Aoc) u(t) = k Q q(t) The adaptive c o n t r o l l e r senses the p o s i t i v e d e viation A@ from the allowed maximum temperature 9 . I f AG |Aa], where lAal i s a spe-c i f i e d p o s i t i o n tolerance, then the c o n t r o l l e r output i s 9^ - 'Aa. This r d s u l t s i n an increased k and hence an increased value of water flow. . o When A6=- . JAa| that i s the water temperature i s below the acceptable LJ COOLING SECTION e(0,t)-0 t(t) r < r y > U(t) ea,t) 100 k= he q(t) OPEN TYPE CONTROL Fig. 2-1 28 1 COOLING SECTION i ( t ) G(0,t)=0 \\100 ( X U(t) 1 100h ADAPTIVE CONTROLLER -Hg) a max OPTIMALIZING CONTROL Fig. 2-2 . . . 29 maximum value, the adaptive c o n t r o l l e r ; slowly updates the value of the c o e f f i c i e n t : k by reducing i t s value. 30 CHAPTER III OPTIMAL TRANSFER TO EQUILIBRUM OPERATION . 3.1 General Considerations. The purpose of t h i s s e c t i o n i s to demonstrate the a p p l i c a b i l i t y of the modern c o n t r o l theory to the problem as i t had been developed ear-l i e r . The conventional approach, such as i s given i n Sage (vide i n f r a ) i s somewhat more general than i s required and i s based on the premises that the-control i s a function of time and p o s i t i o n . In the cable cooling pro-blem, however, the c o n t r o l v a r i a b l e i s the cooling f l u i d v e l o c i t y . Be-cause of the i n c o m p r e s s i b i l i t y hypothesis i t i s a function of time only. Suboptimal c o n t r o l , where a constant v e l o c i t y i s used i s also of p r a c t i c a l i n t e r e s t . Consequently, the general expressions must.be modified. I t i s the purpose of t h i s chapter to give these modifications. As has been shown i n the previous chapter, the a n a l y t i c a l ex-pression of the system model contains a term i n u(t) i n the denominator of the integrand. This precludes the d i r e c t extension of the previously obtained r e s u l t s to the zero v e l o c i t y case. Furthermore, we no longer can assume that the t o t a l i t y of the heat generated i n the cable w i l l be c a r r i e d away by the c o o l i n g f l u i d . The f l u i d , b arring convection e f f e c t s , can be considered at r e s t , i n t e r p o s i n g another medium between the conductor and the s o i l . Considerable work has been done i n t h i s d i r e c t i o n , and can be found i n references [ l ] , [ 2 ] . The time constants encountered f o r ground buried cables are expressible i n weeks. Thus any abnormal operation would s t i l l tend to.maintain a slope on the temperature d i s t r i b u t i o n s , with higher temperatures toward the f a r end. While a'temperature slope i s a more favourable condition, the assumption w i l l be 'made that the i n i t i a l temperature d i s t r i b u t i o n 9 Q ( X ) i s a constant. This constant should take the worst case design value, or 31 i t should be determined on s i t e . Pertinent references to t h i s chapter are Sage [ l 5 ] and an ear-l i e r work given i n r e f . [ 8 ] . For a general exposition of perturbation methods see Bellman [ l 6 ] . 3.2 I n i t i a l Conditions. The assumed i n i t i a l and boundary values of the temperature fu n c t i o n 0(x,t) and of the temperature perturbation ^ ( x , t ) are given i n F i g . 3-1-At time t=0 the perturbation i s given i n the. form: f(x,0)=c - Cx (3-1) The temperature and the perturbation at the boundary are taken to be zero: 6 ( o , t ) = vj>(0,t) = 0 (3-2) I t i s also consistent with the model used to take 1 the value of the temperature perturbation gradient at the entrance of the cooling duct to equal zero: ' 3f(o,t) =. 0 or- f ( 0 s t ) ^ 0 (3-3) F i g . 3-1 3.3 The perturbed p a r t i a l d i f f e r e n t i a l equation. Repeating the P.D.E. obtained i n the Chapter II b2Q \"bQ \"bQ _ 1 .x — 2 \" - u — - — _ - - . q (3-4) ox dx 2>t h The st a t i o n a r y s o l t u i o n of the equation i s ©(x)-Cx. Assume that the actual temperature and v e l o c i t y i s expressed i n terms the st a t i o n a r y temperature and normal v e l o c i t y plus perturba-tory terms: © = © + vf = cx + v f (3-5) U =. U + V = £ _ (3-6) he where © and u are the stationary values. S u b s t i t u t i n g (3-5) and (3-6) i n t o (3-4) and substracting the o r i g i n a l (3-4) y i e l d s 3 2(e + *f ) . r \\ (e^f) a(© + f ) 1 /, _\\ 5 x dx h and — P - (u + v) — - V — - — - O (3-8) 3 X ^x i t noting that 5© _ %(Cx) _ c dx d x i t i s seen that Equation (3-5) can be written i n the following form: * t 5 x 2 ' (u4-v) — - >/c (3-9) x 3«4- Performance Functionals. The performance f u n c t i o n a l s encountered i n optimization problems f o r one independent.variable can be given i n various forms, forms which have been named a f t e r Lagrange, Mayer or Bolza. The v a r i e t y of represen-ta t i o n s f o r many independent v a r i a b l e s should be f a r greater, as even a cursory examination of Sage [l5]> Denn [7] or Degtiarev & a l . [17] w i l l show. Considerable l a t i t u d e can thus be given to the formulation of the performance f u n c t i o n a l . The e s s e n t i a l requirement would be the ease of treatment, or a greater p h y s i c a l correspondence to the problem at hando Given two dependent v a r i a b l e s ^(x,t) and v(x,t) the most natural performance f u n c t i o n a l i s obtained by assigning a cost to t h e i r d e v i a t i o n from zero over the time period and length: • Jl = I j6 J L [ W u ( t ) ^Ul t) •+ ¥ 1 2 ( t ) v 2 ( X l t ) ] dxdt (3-10) 0 0 However because the o r i g i n a l temperature d i s t r i b u t i o n i s a p o s i -t i v e non-decreasing function we could consider the deviation at the end only. The coolant v e l o c i t y depends only on time, so does i t s perturbation v(.) and Eq.O-lC) can be given as: J 2 = \\ [W n(t)v|> 2 ( L , t ) t » i 2 ( t ) v 2 ( t ) ] dt . (3-11) o But, because the integrand of the performance f u n c t i o n a l can be added only i f i t i s under the double i n t e g r a l we rewrite t f L J = 2 { j ,[W 1(t)^ 2(L,t) + W 1 2(t) v 2 ( t ) ] dxdt (3-12) 0 0 Of i n t e r e s t i s the p o s i t i v e deviation of the temperature and s p e c i f i c a l l y ( L , t ) . For the subsequent development we w i l l use the form of the performance f u n c t i o n a l given by Eq.(3—12). The augmented performance f u n c t i o n a l w i l l include the constrain-3b ing P.D.E. by using a Lagrange m u l t i p l i e r : J -a 0 '0 ^ - v c - ^ l f dxdt (3-13) 3«5 The Hamiltonian. Introducing the Hamiltonian, which i s obtained by grouping to-gether a l l the terms except the time d i f f e r e n t i a l : -(H - p ) dxdt (3-14) The Hamiltonian i s thus defined by A 1 ,„2 . „ 2 H - 2 ( ¥ l l f + + ¥ 1 2 V > + P ^ \" < f i + V).>, \" V ^ ^ For f u r t h e r study i t i s usual to form the f i r s t v a r i a t i o n of the augmented performance f u n c t i o n a l . The use of the Hamiltonian introduces considerable n o t a t i o n a l s i m p l i f i c a t i o n s : 0 0 ' J x I xx The increments of the p a r t i a l s of the state v a r i a b l e are defined i n the following manner: ^ « f ) and i * ( J f ) (3 •17) Integrating by parts the t h i r d term of Eq.(3-13) y i e l d s ^ ( t y) ax . f }H 0 f x lo A s i m i l a r operation to the fourth term y i e l d s ^ (' — ) dx (3-18) T )x difx rL .2 T ~ P (S f) • — • dx = — dx J r \\ ^H ^ x x ^ ' J ' ( t y XX A p p l y i n g the i n t e g r a t i o n by p a r t s t o the i n t e g r a l e x p r e s s i o n on t h e R.H.S. o f (3-16) y i e l d s l SH 'f. xx XX + A f u r t h e r i n t e g r a t i o n by p a r t s y i e l d s J - p d t = - pSu? } t 1 o S f - ^ d t St (3-21) 3.6 The F i r s t V a r i a t i o n and the C o s t a t e . An e x p r e s s i o n f o r t h e f i r s t v a r i a t i o n c a n now be formed by i n -s e r t i n g Eq/. (3-17 t o j . 20) i n t o e q u a t i o n ( 3 - 1 6 ) . J A -SH o o H \\ 6>p_ X 0\\f 2) t Ixx 2 1 2 v £v d x d t 7- 1 xx bx 0 f: XX .dt The f i r s t v a r i a t i o n can be p a r t i t i o n e d i n t o f o u r p a r t s : (3-22) \"J = J a a. 11 12 a 2 1 l22 Where t h e f i r s t and t h e second p a r t s a r e M.1 x12 rdE fir [— _ A ( J L ) + ^ ( ^ _ ) + i L ^ f d x d t 0 0 f 2 \\ - bf ,xx d x d t (3-23) (3-24) (3-25) o o 0 I The t h i r d and the f o u r t h p a r t s a r e : l21 J = a 2 2 ' f ] d x (3-26) 3 f r r O 2H 3 xx *xx [ d t (3-27) o E q u a t i o n s (3-24) and (3-25) y i e l d t h e c o s t a t e e q u a t i o n and the equa-t i o n o f the g r a d i e n t o f the H a m i l t o n i a n w i t h r e s p e c t t o the c o n t r o l v. E q u a t i o n s (3-26) and (3-27) y i e l d the t r a n s v e r s a l i t y c o n d i t i o n s . G i v e n t h e e q u a t i o n (3-15) f o r t h e H a m i l t o n i a n , (H) the p a r t i a l o f H w i t h r e s p e c t t o y » ^ > ^ > a n < 1 y i e l d s — = - p ( u + V _ = W i 2 V _ p f x (3-28) (3-29) (3-30) (3-31) S u b s t i t u t i n g Eqj. (3-28) t o ( 3 - 3 l ) i n t o Eq£. (3-24) t o (3-27) one o b t a i n s , t h e ' n e c e s s a r y e x p l i c i t f o r m s . -: The c o s t a t e : • . . W l l f (u + V ) —^ H £ 4- ^ - - o a x ' S t o r i n n o r m a l f o r m t ay s t — o ~ U + v ) — ^ ox W n f (3-32) 38 3.7 Gradient Condition and the Steepest Descent of the Hamiltonian. The minimization of the performance f u n c t i o n a l i s accomplished by the steepest descent of the Hamiltonian, obtained by an i t e r a t i v e process incrementing the co n t r o l by a f a c t o r proportional to the Av: ' v ( i + l ) = v ( i ) - p L AV (3-33) Where p\\ i s a c o e f f i c i e n t s u i t a b l y determined i n a subsidiary i t e r a -t i v e loop. The increment A i s computed in'the following manner. I f the c o n t r o l v i s both time and space dependent v(x,t) as i n Sage [ 1 5 ] , then: A v ( x,t) = y i | | (3-34) I f the c o n t r o l i s space dependent oniy a mod i f i c a t i o n to the above expression i s required, and Av can take the. f o l l o w i n g form: A v ( t ) - | J dxdt (3-35) When the control depends on a parameter a then the increment of t h i s parameter i s computed by the f o l l o w i n g expression: A a = f, t f (L — dxdt (3-36) For example a suboptimal co n t r o l of the form: v= v M e ~ a t ' (3-37) could be used and Ecj_.(3-36) can be used to determine the optimum value f o r a . 3.8 T r a n s v e r s a l i t y Conditions. Given the structure of the i n i t i a l conditons at the o r i g i n at time t = t shown i n F i g . 3-1 and the succeeding graphic development f o r t> t c o i t i s natural to take the following value f o r the temperature v a r i a -t i o n : ' 5^(0, t) = 0 f or t ,> t (3-38) s i m i l a r l y the slope of the v a r i a t i o n : ( t y (0,t)) = f y x ( 0 , t ) = 0 (3-39) The i n i t i a l v a r i a t i o n , f o l l owing Sage, w i l l also be set to zero: