!! #!!! # % &!! ( )!

Size: px

Start display at page:

Download "!! #!!! # % &!! ( )!"

Lesley Hart
5 years ago
Views:

Roscoe, Andrew (2004) Demand response and embedded storage to facilitate diverse and renewable power generation portfolios in the UK. Masters thesis, UNSPECIFIED.

1 Roscoe, Andrew (2004) Demand response and embedded storage to facilitate diverse and renewable power generation portfolios in the UK. Masters thesis, UNSPECIFIED., This version is available at Strathprints is designed to allow users to access the research output of the University of Strathclyde. Unless otherwise explicitly stated on the manuscript, Copyright and Moral Rights for the papers on this site are retained by the individual authors and/or other copyright owners. Please check the manuscript for details of any other licences that may have been applied. You may not engage in further distribution of the material for any profitmaking activities or any commercial gain. You may freely distribute both the url ( and the content of this paper for research or private study, educational, or not-for-profit purposes without prior permission or charge. Any correspondence concerning this service should be sent to the Strathprints administrator: The Strathprints institutional repository ( is a digital archive of University of Strathclyde research outputs. It has been developed to disseminate open access research outputs, expose data about those outputs, and enable the management and persistent access to Strathclyde's intellectual output.

2 # # %& ( )

3 + # ), # ) ) % + ) + & +., +

4 % /0%. 1) 2 23 % % )5 / % % ) ) & % ( % ) 7 % % ) # ) % % ) )5% 8 )

5 & & & & ( ) + ) &, + ) ),. ) / % &0 2. ) # ) ( # / + 6 ) ( )1 2 / : 13 ; ( ) + /& ) 1 2 /) / /( / & /( / ) 34 4 /5 / / 6 4 /5 / ( 7 /8 2 % & < =< > 2 2. ) 2 ) =+ > < ) 6 / 4 ( : :2 ) 6 ; / 4 ( (5 ( & 9 (5 ( ) ## (8 ( / %54 (8 ( ( +& 2? 6 7 ) ( + % +8 ( 1 :;6##<;= 10 ( 5 ; 1) ( 8 7 1( (. % 18 2 ) :2 ( ) : : ( &0 5( 22 % :4 2 Α :4 2 / : ( && 5. ( &) 8) 22 1 ; 2 ( ; ( &/ 8+ ( &( 8+ ( & ( &1 % 81 ( & & 8. + ) &>.&

6 ; + / )?/ Β ( &&) 1 &&/ 1 & 7 % &&1 5 &&5 5 & Α &&5 5 ) ; &&. :2 1 # 22 : 1 #1Χ 22 : 1 # 22 : 1 # 2 :6 1 # 2 :4 1 # 2 :: 1 # 2 :; 1 # 22 : 1 # 22 5 / &)) :2 02 : 3 2 : 3 26 : D 24 :6 2: :4 2 5 ( 4 &/0 5 +? &/+ 8 &()

7 7 D 2 # /%7 : D # : D? 972Ε6Φ24 D D 8% 24 D 6 ( 6 972: D : D : ( 6 972; D ; 97 Γ 2 D 97 Γ D 2 97 Γ 2 D 22 Γ D 2 Γ D 2 Γ ) D 2 97 D 2 D # 4 : 7Ε6Φ 6 D 2 ( : D? D 0 ): 6 D =Γ 2 Γ 2>66 D 6 9 ) Γ = >64 D 4 ( ) Γ = >64 D : # ) Γ = > 6: D ; # ) Γ = > 6; D + 4 D 2 #( 31 9 ) Γ = >42 D 22 # 31 ) Γ = >42 D 2 # 31 ) Γ = > 4 D D D 26 D 46 D D 2: ( ) : D 2; ( ) :2 D 2 : D +& :: D 2 +& ) :; D +& ) 2 :; D + ) ;: D D6 = > D6 2 D6 2 ) D D =257)> 6 D6 : 21Χ =657)2657) > 6 D6 ; 2 6 D6 2 =Η > 4 D D6 22.( 2 D6 2.( 2 D6 2 2 D6 2 =6;>2 D6 26 1Χ =57)2;57) >2 D6 24 ) =6;> 24 D6 2: ) =6;> 2: D6 2; =6;>2; D6 2 2 D6 22 D6 2 =6;>222 D: 2 Γ)Χ%=; > 2

8 2 # /%7 4 /%7 # ; # + ; # / ) 6 2 % # ; 6 ( 2 : 6 D 2 : 6 D ( 24

9 + # ), #) ) / < ) # 7 ), + + D ), / < )) # % ) %) +). + # ) ) ) # 2

10 2> % #?D 2 1Χ #Ι Ι 1Χ, +) 6+)1Χ ) 7 & ϑ &. Η 1 > < ( ) Η & # ) ). & # > < ). &2 # Η ) Χ 2 ) 3 #) &. ) #, +< 2 +:2

11 & + > < Κ Λ ) ) 7 + ) 7. ) ) < +. ). < ) 7. ) ) % + ) ( &&. + ),,

12 %8 =8(/.ΕΦ> = > 8 ϑ ) # %& #% # D ) & Μ7 )Μ + 6 #+ 4, 4 +, +

13 D ) & 1 ( ) D &) ) < ) #, ϑ < Κ, Λ 7 7 ) ) Χ. 8< + % Β # ) 6 2 Β 6 # 8 # Κ% D Λ # # 1 ( # + 6

14 8 6 Κ Λ ; = Ι > 1. ) & 6 ) =8.%Ε;Φ>% # 8 0 Κ Λ # / & ) # & # +=(. Ε22Φ> TWh of raw fuel Percentage of UK fuel use Petrol, DERV % Aviation Fuel % Electricity, domestic % Gas, domestic % Electricity, industrial % Gas, industrial % % )4& #, 4

UK raw fuel usage (before conversion) TWh/a 316.17 418.68 Petrol, DERV 139.

15 UK raw fuel usage (before conversion) TWh/a Petrol, DERV Aviation Fuel Electricity, domestic Gas, domestic Electricity, industrial Gas, industrial )4& #, (. #.Ε Φ 7 )4) :

16 < Ε22Φ Raw Fuel Use TWh / a Percentage of UK raw fuel use Coal % Oil % Gas % Nuclear % Hydro % Other renewables % Other fuels % Imports % % )4), :7 2Ι + # Ε22Φ Α; Β: =Α, =57> =7> 7? 6 2 4: 22;6 4Ι < : : 2;Ι + 2 ;:4 64 Ι ;: ::Ι ;2 : Ι 5< 26 2; Ι 3= > 26 2; ;4 Ι 3=> ; Ι < Ι. 2:6 ; 2 ;Ι ( /5( 5 +(Χ )4/ < ) # : Ι ). % <). ) Ι 6Ι2Ι 7 +: : :2 2ΝΟ4Ι ;

17 + : + # #)? Μ % =1)Π;:4 Ε757ΦΠ?D> Ι Θ 7 (1Χ 57 ΡΙ 8 67 =457 > ΡΙ 8 =267>? 57 Ι < 57 Ι Ι 8 :7 27 6Ι 1 67 ( 7 3 Θ Θ 7 0=2Ι # Σ> :Ι Θ Ν2 57 Ι 1 Ν 47 =Ν57 8 =Ν6:7 > > )4( : ) < 1Χ). & % = > + & #

18 :7 :7 &, #. & 0 =:Ι> & = >2Ι # % & 7 # & ) & = >. # ) Κ Λ3 50 ΚΛ 8 ) 5 7 )& :6)Χ )Χ=3Χ> ) / =/ > ) 8 = 8 ) > 0 =0 > ) 51 =51>Ε2 Φ 0)1 =01>5 ) 5 = > ) ) ) ) 2

19 # ) 2)Χ =3ΧΧ?Χ> ) (8< =( 8 ) <> Χ?Χ ) 3 ) ) ) ) ) ) ( =( > / (8< ( / =(/> & 67 #Χ% #7D ) ( ) ++ / & ) (. ( (8<ΚΛ / / ) +. =/ > )/ 8% (8< ( 8< + Κ Λ < 1 0% 8% 22

20 % +. < /1. Τ / Ε2 Φ/1. Τ +=/1.> ΤΙ <+ ) < + / +=> = > ) (8</ Β& ) % /. <. ) Χ&) ) # % )+.Τ& Α Τ2 1 Τ2 & Α Τ 1 Τ + ΟΙ ΤΙ Τ E = Q Q X P P X X X Q 1 = 2 P 1 2 X 2 Q ( Q + Q ) X1 P X 2 X 1 ( ) P + P X1 X 1 X 2 X 2 D & ; 22 E = =

21 & 2 ( 1+ a) ( 1 a) 2a PX 2 PX 1 QX 2 = QX 1 QX 2 QX 1 = QX 1 a = E 1 a P + P ( ) ( ) ( ) X 2 X 1 D ) 4 5 =Ρ 2> & / 7) = 2ΡΡ> ++ / 1, ) + # %& ( ) +,. %&, #,,, #,,, #,,, # /,, / /, /#,, )4+ 2 A classic example is the Parker 25 fountain pen, whose sales were poor as it was perceived as too cheap and tacky, until the price was increased and sales rocketed. Also, the beer Stella will always be reassuringly expensive. 2

22 2. % ) + + ) Τ Θ D + ) ) + ΟΙ ΤΙ Θ ( ) ( ) + + = = Y Y Y Y X X X X Y Y X X P P P P Q Q Q Q P P Q Q E D / 4 % Τ Θ => = = Y X Y X Y X Y X Y X Y X P P P P Q Q Q Q P P Q Q E Ο Ι Τ ΘΙ Τ Θ D ( # # < #)Κ :Λ Κ Λ ) & + )

23 0 0 ). 7 8% 0% 8%) / ) Κ0 ΛΕ:Φ) ). Β < + +, = ) > Υ26 7. ) +. ) + )( )Μ 5 + +)?ΕΦ ΚΛ %= % > /.=/. +>/. + Τ./. Τ 2 => Τ )% ). ) ) ) ) 26

24 1 D Κ Λ ). 2Ι) 6Ι 7 )4/ 34?, #)00&Ε)+Φ 70 = (8<> ) 7 )4( 7 Γ 24

25 ) & ϑ )) Χ, % 1 /. % + 7 ) ; 6 )2 6 Peak demand forecast , England and Wales Peak demand (GW) Week number ( ) 7 )4+ )00(4)00+#?, ( ) ) )2 % Margin forecast , England and Wales Margin (GW) Week number ( ) 7 )41 )00(4)00+#?, 2:

26 + ) + ( ) D 571 ) )6 Σ, + Β ) Peak demand and capacity forecast , England and Wales Margin (GW) Week number ( ) Demand Capacity 7 )45 )00(4)00+#?, ),. #? / & Ι= )> # 2ΙΙ. Ι # Ι 2;

27 0 23)43,,,,/ 1 + # )) + Ε2 Φ 8%Ε:Φ & #Β+ ) ) )2Ι + ) ΕΦ 97 : ; Γ 4 ; Γ 7 +0 Ε:Φ % + Γ England and Wales electricity price volatility, January 2003 Demand (GW) Buy and sell price ( /MWh) Jan 14 00:00 Jan 13 12:00 Jan 13 00:00 Jan 12 12:00 Jan 12 00:00 Jan 11 12:00 Jan 11 00:00 Jan 10 12:00 Jan 10 00:00 Jan 09 12:00 Jan 09 00:00 Jan 08 12:00 Jan 08 00:00 Jan 07 12:00 Jan 07 00:00 Jan 06 12:00 Jan 06 00:00 Jan 05 12:00 Jan 05 00:00 Date and time England and Wales demand Sell price Buy price 7 )48?, # )00/. Γ % Ι ) D 57 + #+ # ) ) % ) )

28 Γ Υ67 Υ26 7Υ67 % ) ) England and Wales electricity price volatility, January 2004 Demand (GW) Buy and sell price ( /MWh) Jan 26 00:00 Jan 25 12:00 Jan 25 00:00 Jan 27 12:00 Jan 27 00:00 Jan 26 12:00 Jan 28 12:00 Jan 28 00:00 Jan 31 00:00 Jan 30 12:00 Jan 30 00:00 Jan 29 12:00 Jan 29 00:00 Date and time England and Wales demand Sell price Buy price 7 )4.?, # )00(. Γ ) Γ 6657 )Υ267 7 ; 1& ). 0 1 Γ ΚΛ +Β < &

29 England and Wales electricity price volatility, June 2004 Demand (GW) Buy and sell price ( /MWh) Jun 09 00:00 Jun 08 12:00 Jun 08 00:00 Jun 07 12:00 Jun 07 00:00 Jun 10 00:00 Jun 09 12:00 Jun 11 00:00 Jun 10 12:00 Jun 12 00:00 Jun 11 12:00 Jun 14 00:00 Jun 13 12:00 Jun 13 00:00 Jun 12 12:00 Date and time England and Wales demand Sell price Buy price 7 )4&0?, # )00( + ) ) 7 Γ 8) 57) % DΓ 22 Γ 2 Γ 2 % Γ ; )Υ67Β Γ 1) Γ : 657 ;, 5722 %1 ; ;% +. ) )ΚΛ 57( & 57 ) 57, & ) 1 ) Κ Λ + System buy price, Jan 2003 w eather event System buy price, Jan 2004 w eather event Systen buy price ( /MWh) Demand (GW) Systen buy price ( /MWh) Dem and (GW) 2

30 7 )4&& Η )00/)00( D 22 Γ % )+ +7 D ) ;57Υ27 System buy price, Jan 2003 and Jan 2004 weather events Systen buy price ( /MWh) Demand (GW) 7 )4&) Η )00/)00( % & ) System buy price, June 2004 fine weather week Systen buy price ( /MWh) Demand (GW) 7 )4&/ Η )00(% % & D 2 )Υ67

31 =D 2> ) ) ) Κ Λ & )+. Χ +) ϑ ϑ % + + ) D Κ Λ + 1 ) ( ) )% ) = > D 2 D 2. ) Ι2Ι & Υ27=2)7> + Κ Λ Υ67 =6)7> Ι2Ι Ι ;Π6Ο 57. &

32 System buy price vs demand, simple winter model Systen buy price ( /MWh) Demand (GW) 7 )4&( #?, # Κ Λ & + = : > # )+ ) + = > ( + =D > 7 + D 2 ) ( )

33 ( 7 D ;D 2 ) ) =+> = > ) 0 D ;D 2 ) / 32 / =/ > 3 # + Κ :Λ Κ Λ ) / + )7 3 / 0 / )7) / 0. / + ) Υ26 7=26 )7> ( +:6)7 =Υ:67> ( + : ;)7 =Υ: ;7> 2 24)7 =2 24Υ7> : Ε2 Φ ( ) 6Ι ) % Κ)Λ ( / % + /. / ) % + / 6

34 < )7 / ϑ Υ 47= 4)7>( Γ Υ;7+ (8< Υ:67 / Β+ )7< / / ))= >) Σ 3 / / + / ) / 5.. ) & Χ # ) )+ +,) ) 3 + ) ) ) Υ6Υ2Υ 7 Υ267 ) ) 4

35 8%ΕΦ D+ 7 Υ267 Υ4 + :Ι );74 7) Υ27 Υ;8 + Υ<Υ; <6Υ ), ) )%, / / )& + ) / 5 & ) 5 Η, 2 & 2 3 & / 3 In 2003, the NETA buy/sell differential was large enough to dissuade pumped hydro operators from trading. Instead, they sometimes chose to leave the network be and let NETA balance the system with CCGT generation. This adds to market volatility. :

36 / + =) Κ) Λ > + ) Κ Λ/ & 2Ι < 2Ι / 06 1 #+):;;7Ε2Φ,)Ε2Φ ( 2:;7 27: :;Ι Ε2Φ D 472 7:Ι Ε2Φ D :Ι = > # =:7 > & # )).+ ) # ) ))( ) &+6 4 D ) ) + ΟΧς Ο ;2+Ο ΓΝΟ;)7 ςο2) Ο ;2 ;

37 % ) + ;; Ο ; )7 ;+; Ο 4 )7 + ; Ι :Ι = > )7 # # Ε;Φ ϑ 2 ) / 1. ) ΚΛ% ( 2 ) 27ϑΚΛ 2Ι2Ι. ΚΛ W2Ι D+ ) ΚΛ & )

38 // 5 % +: + Pumped Hydro Bulk storage capacity (proven) 14GWh (UK) 1.1 million household-days Storage capacity density, per 10m x 10m x 10m cube Domestic storage feasibility and capacity Lead-acid Small scale for a 5-10 day reserve Heated water Small scale 98 (heat) 7800 Very cheap. Potential 50% loss per day in summer per day, for a small system. Round-trip efficiency Cost Plant life MWh House holddays % NO years 85% /kwh 100% (to heat). 10 years 25 years Feasibility Extremely limited by suitable locations Widespread uptake limited by global lead availability unless electrolyte stores used? May become marginally financially viable. Cheap, simple. BUT, energy must be used as low-grade heat for washing, heating, bathing etc. Hydrogen Not yet proven, but potential on a large scale (heat) Possible 20% (to electricity). 20% (to heat). Flywheels Unproven ? ~60-80% Less for storage times of days (bearing losses?) Compressed air None 2.3 Efficiency drops to ~17% after one day 180 NO ~64% Less for storage over hours or days.? In development 3500 /kwh 25 years? Problems with bearing losses, mechanical stresses, risk of catastrophic failure. Large pressure vessel required. Temperature losses rapidly reduce efficiency.over time. Superconductivity None ??? Long lengths of wire immersed in liquid helium. Tolerable magnetic field strengths? Demand response???? 100% to infinity?? To be discussed )41 1 ) # %

39 & # 1 D ) D +? ) ) & 2 ) & 2 3 & = > Ι 3. ) =2>7 &+ Β.=> Η &) Η ) 5 ) # ) ) + 2

40 / ) ΚΛ ) ) ) / (8< ) & % ) : D 8 8 =D > < 7 7 ) & & ( # ) 7 )=) > ) )= > ) ) = > D %

41 . ) # # = > = > # ) # & ) 6

42 1 ( / % Ε:Φ& ) ) ) Ξ4)7 Ξ2)7, # ) Κ1 Λ Κ :Λ Κ7 Λ (8<= ϑ+ )> + = )> % %. )7 + )? 0 + & (8</ 1 = 1> D & Κ )Λ + > )

43 > +. 1 ΚΛ?) &ΚΛ, / D 2 7? & % ) 7 /4& 7 #)0014)005#,Ε)+Φ 8(/. Ε4Φ < 6

44 )Ξ67 Ξ67 % Ξ6 & 27Η 8(/. Ξ6 Η 2)7 8(/.Ε4Φ 1)Ι = 0 #> ) :Ι ) :Ι 1 D % 8(/. + Η 27 &. Β D ) & ) ) & ) ). ) &). 3 ) ) ). = 1%> ) 1(8< 4

45 :. 3Χ (8< / < (8</ Χ ), ) ) 8 ( ) ) ) ΚΛ/ ) ) ). ) ). & + => ( + ) = > ) / ) + ) ) =. )/ > % ) & + ; % &. Κ Λ ). ) => % + 7 ) :

46 ) 3 ) & % ) ΚΛ Κ Λ ) ) Χ ) / < 1 5 2)7 % ) & ) ( Η & & Κ Λ 8 ( /. =8(/.>Ε4Φ + ;

47 Η =%>& / ( = % % =%> ϑ 3 ( >. => >. =?%8.(8 > > 1 = > >. =?Χ > > ) & ) > > & + + ϑ =>, ϑ % %++Ε26Φ. Κ Λ % 3Η + )2 ) &, ). )

48 . # Γ/ =Γ / >Ε2Φ. # 2;6 2; 63Η & 8 & (./ & ) = > & + & 3, ) (/% Ε:Φ =%> Ξ6 Ξ6,Ξ2 %) D & = > + 3 % & % =)> %< %) % 7 % ) 1 Χ < 7 Η % 9

49 / > >. < =< > = >Κ+Λ ) + : 97 + < ) =; 2> :6)7 6)7=2;>Κ7Λ ( Κ7 Λ ), & < 2Ι, ) <. & )< / 8 % < & ϑ. ) ) Κ ) Λ =%Η. 7 > ) % ). Κ )Λ(8< / )7 2

50 ) & / ϑ) ) # ), ) + ) ) /? 2. ) ) (8</ 1 1& 7 ) % 1 ) ) < )) )+). # ) % ) Κ Λ 8 ) ) ) Β ) ) Κ Λ ) // > 8 < ) +) ) < / )). ) ) )

51 + ), ) => => + /7 1 7 ) D+ & &. + <+ D Η = >) Η = >. 3, ) %. 0 % ) & Κ& ΤΛ Κ ΤΛ Κ Λ

52 Κ Λ Κ Λ /# 5 ) ϑ, 1. # ΚΛ + 0(/% = Ε:Φ> 0Ε Φ+ + %+ < < + Χ & < < ) < Χ < 0 < + < % # +

53 /#? ) 1 = > =7> =3> =)7> Ψ:) )) 66 ; 2 ) ΧΧ ; 4 ΧΧ 2 2 ; ΧΧ Ζ ΖΧ( ( : : 6 : ; 4 # : ? Π 2 Π 2 2 ) Π 2; Π 7 Π 2 2 Π Π 2 26 : 26 Π #2 (.& 2 /4& % < Η 3 )7 7 6

54 +6+ )7 ) 1 ) ). +. ) 22 / ( / / / ) + )) )/ / / ) 1 ) ) /. < / <. Κ Λ/ = + > )/ < ) + / 4

55 / 1. ) ) / > % 1 7 (4& Ι :

56 +. ) Κ Λ ΚΧΛ % % + % + D.,) = [ > ΚΛ % Κ Λ /? 33. 8< % Η 8 ) Η. Η8 & ) )?) ) ) )+ = > = > / # ) ) D ; D 2 ) &. ;

57 ( =Χ> ( ( =D> ( =7 > () ( ( = >. = > 8 ) D + & ) ) Χ ) 7) ) : 7 :, 2 //D2 72 D/D W /D2 /D 7 (4) 3

58 + 7 & % D % % + =Χϑ >. = Ο22 2 Ο2 2>& D 7 (4/ %54 # % + 6

59 // ((.Ε;2Φ ) # ) 2 (. (. # # % 31 ) Lighting Cold Cooking Brown Wet Space Heat Water Total Commercial Industrial Elec Stats % Houses 100.0% 100.0% 60.0% 100.0% 100.0% 15.0% 15.0% TWh/annum (UK total) TWh/a kwh/house/day (ALL) avg kwh/house/day kwh/house/day (installed) avg kwh/house/day Gas stats % Houses 40.0% 70.0% 70.0% TWh/annum (UK total) TWh/a kwh/house/day (ALL) avg kwh/house/day kwh/house/day (installed) avg kwh/house/day UKHouses 25,017,000 TWh/annum to kwh/house/day toe to MWh (4& 62

60 ) = 7 > & 2 2=% & 7. 2 ) 8 Θ>.., ) ), D =0 77 ). > D? 3 + => // : ) ( 2 ;1 /) =% 2 8 Θ & >?, (D?=(D?> = 7 > + 0 2Ι 8 (D?27?D =?D> (D?8 = >. 1? + L I ( ) = + t Pt MLF 1 MLF DFML t 1. +(D?=(D?> 6

61 ) => ). = > % 2Ι = > & & ? = > + % ) 8 2 Κ Λ Κ Λ ) &Κ. Λ ? 8+ + % 26 3? 7 = > & 2 & Κ Λ + i ( k DDi SI i ) T P E = f. =+>Ο++W 3 = 7 >(( Κ Λ = >. 7= > 1 ) = 7 > & & (. ( =+>= 6

62 Σ>& ) )8 / + =)((. > + & // 5 3?Α ) ) D Ι26Ι ) 5 Ι 4Ι (.. = 2> ) ) & ) 7 % = >. )) 31 = 4> ) ΕΦ W Ο46 31 W Ο4 31 W Ο = > 31Ι. Κ)Λ = ) > = > ) &) = > 6

63 ) 2 =24 Ζ22Ο :)7 :Ζ26 Ο6;)7 > // )28 & (.+ Γ 2 Γ 2 Γ (4( )( :&4&=. = 7> 26Ι ) 7, +726Ο2)7 66

64 )= > Γ 4 Ο6 ) Ο? : 7 (4+? % :4 = # + 6 # ) ) Γ % 6 # 7 (41 % :4 = ) = )> 26Ι # % # + 64

65 & Κ+Λ 8+ ) Γ ) = 7> D ; D 7 (45 % :4 = 0 # ) = > # ) ) D 2 97 D 2= > #=6 > 6:

66 7 (48 % :4 = 7 ) 2Ι= > ) (. D 2Ι D. Σ / ) 0) 7 1Χ 7 6;

67 ) ) = > % ) 7 ) D ) ) = 1)7> % 7 = 2> % Β3 Ζ /7. 7 <7 1Χ 1Χ) % Β3 Ζ3 3 1Χ /7= 2> = 26> 1Χ 8 ) Β)1Χ ) ) 6

68 ) ). /7 1 0;33?60. (1Χ ( ( 31 1Χ ) = 6> 1Χ+ ) +:6. 3 ) 7 (4. # 31 ) (. + %+ D 6 D : 31= > 6Ι 26Ι :Ι= 2> 6Ι =:Π6Ο6Ι> 31D ) Γ 31 # )2578 ) 31 # ) 4

69 26Ι 7 (4&0 <;?% :4 = 31 # D : ) =6 Ο257>= 4Ο2457>) (4&& <;% :4 = ) ) 42

70 7 (4&) <;% :4 = 31)D ; 31 +:Ι 6Ι 31 # Σ7 ) 57 + ) % 31 6Ι:6Ι #) 257 ) :57 / 31 :Ι. 31 Κ Λ # 31 Ι6Ι=6Ι :Ι>0 31 ) #= > ) % ) 26Ι= >. # = > 31 /# 0 ) 4

71 System buy price, Jan 2003 and Jan 2004 weather events Systen buy price ( /MWh) Demand (GW) System buy price Model (High) Model (Low) Model (Mid) 7 (4&/, Pr ice = Ae Bx + C Υ7 + Demand x = Capacity L arg estgeneratorcapacity %0. ) +, ( = ;Υ7Ε2 Φ> & & % Υ67 =6)7>% +( % :6)7= ( ;)7 + :6 +; Ο 6 )7 1 + & + + )7 8 < + 4

72 7 (4&( ; ) ) 3 & ) & 3 ϑ =+ > += > / < 8 % ). & [ = > ). & < & 4

73 8% # 2. ) 3 Χ ) / &,., ) ) +))8 + ) )) = 26> ) = > ) = > & )Β. ) ) 7 (4&+ 7 46

?# 7?)D (D]?# 7 + 3 ) ) 7 = > +?# 7 +?)D (D]?# 7 Η.

74 ?# 7?)D (D]?# ) ) 7 = > +?# 7 +?)D (D]?# 7 Η. 7 (4&1 + Ο26 + = > 2 2 +=/1.>) /1. Ρ6W26 )./1. 44

75 ;2 ) = 22> + 2Ι 67 ;:57 6 # ) 4Ι =D 2> < ). =D 24> =>, + 2 (/?# 7 (/?)D (D]?# 7. = > & = > % +: ) = > ) = > 7+ ) = > = > 4:

76 8Τ ). + =>+ /Β 1 < & ) ) & & % < ) )? + ) 3(/ 2. #Χ7# 7# + &ΕΦ. ) 2 )6 2? ( 8 6 ( +( + ) ( +( ). +?? )? ) ;

77 ) ) ) Η2 Η ) ) ) Κ Λ ) ) Κ Λ ) 1 & ). 1Ο =7> ΑΟ ) Ο= >3 % ). 23 ) 7# dt dt W P Q = mc V 7 T W b c c( t t 0 ) = + Ae b P + HT A = mc V H c = mc V % 7 + ) t = t b TW ln c A + c 0 + ) ) =2 > ;)7 = 4 >? ( = 6 > ) 2; )78 ) )? ) ) & ) ) &. ) = ) Σ> 1 ) ΚΛ 4

78 7 (4&5 % :

79 7 (4&8 % ) = > /Β 5 ) & =8< > 3 = > ) 7 ) ;6 ΚΛ + )1 22)7 = 2> 22+;6Ο;4)7 26)7 ) )?? D )??&? ) :2

80 = > = 4 >. ) )= > ) (, 7 ) & ) 7 =2> )66 7 )66 =6 > ) ) 7 ) & ) ( /Β 1 =(/> ) Κ Λ (/ ) & 1? 3 = > ) & ) 3 (/ )& & )= Σ> ) (/ ) )?? = > = 4 > (/ ) (. D (/ ) ) ( = :>7 )?? ) :

81 ) ) ( (/ ) > > + ) /Β 5 < % 7 (4&. %= > % ) ΕΦ Χ ;7 #Η 4ΕΦ 1/?1 D 1/?1 D ). ( 2 27 ; 2. ) (/ ) ). ( :

82 . ) ) ) D Η. +D & Q S = GAτ α C P [ UA( T T )] C A Α 5 = > +:6 C is the transmissivity P is the absorptivity T C is the plate temperature T A is the ambient temperature (taken from the climate data). 3 D7 = ) > & 7 T C T S ( T + T ) FW S = 2 P = TFW + mc V Χ m & & T S = T FW A + mc V T Gτ Cα P U 2 AU 1+ 2mc V FW T A /, :

83 0) = >, = 1Χ> 3 # 31 1Χ 3 ) = 2>, % & 8< ( = ) > % =(/> ) 8 3 = 6 #>6 Η Κ Λ8. & ϑ2ϑ % 26Ι &Κ,Λ 8 3 :6

84 /, (8 % 8 3 () =Θ8>Μ ( =Θ8>Μ (, =Θ8>Μ ( =Θ8>Μ 8< ( 31=Θ8>Μ ( =Θ8>Μ (1Χ =Θ8>Μ Κ Λ = [ > Β3 Ζ3 1Χ & 0<3 1Χ Κ Λ & 1Χ => Η 1Χ /, Χ 8 7 +, = :2> 0? ) ) :4

7& ) & & 1 P ( k) ( np) k np e = 1=)>) k + D D+ΚΛ 267 6 + # 2 )7 +6

85 7& ) & & 1 P ( k) ( np) k np e = 1=)>) k + D D+ΚΛ # 2 )7 +6 &[ D )7 2 # 2 )7 + 2 ) < 0 + Γ & = > =)7> + ) ) )= 2> / 7 (4)0 D ::

86 ) ) ) 7 ) & ) ) 2 ) 7 (4)& D % )8 Ο2 ) 7 (4)) D % #&00 3 ) =Ο6 :6> =Ο26 2:6> 8 Ο2 26Ι 26 )& % Ο26 7( Η Η ). ) &) + :;

87 )Η 7( ) + /, 8 < & % 6 / 1 ) & + ΚΛ ) ) = ) > Β?Β 3Β )3 Β (?Β =0 Ζ7Ζ ) Ζ Ζ?Ζ3Ζ )3 > # ( Β 3 </ =(/> ) )373Ο3Ζ37 # ( ) =(/> ) ) ) 7Β ) ) ) Β ) 7 )( Ο7Ζ37 ( Β => ) #( Β ) Β 7 )( ) ) :

88 31535 Β & 31=5 > Β & 31=5 > (7 5 Β =5 > (1Χ5 Β1Χ =5 > (5 Β =5 > (5 Ο Ζ Ζ(7 5 Ζ(1Χ5 0Β D Β (Β (Ο0ΖD (( Β (( Ο(?Ζ(5 Ζ( Β. Β ( Β ( Ο(( Ζ Ζ. 0)5 Β) 0)7 5 Β) 0)1Χ5 Β)1Χ 0)5 Β) 0)5 Ο0)5 Ζ0)7 5 Ζ0)1Χ5 Α Β 5 Ο==( 0)5 > > D0)5 Β) D( Β (Β? Β ) = )( )3? )3 7( > = 7( 5 > 7 Η ;

89 7Κ Λ + 1Β=Υ7>= (< > ) D1Β=Υ7>= (< > D1%Β=Υ7> D/1. +Β + D/1. +ΟD1D1% D 1) Β 2 D/1. + = > 08 Β 2=>D/1. +WW2= > D/1. +22=>D/1. +ΡΡ2= 22> 7 ( 7 ( Β = > = 4 > 7 )Β = > ) )??Β )=> )( Β )= > ) Β )=7> ) = 6 > 0 Β =7> D Β =7> Β=7> Ο0 ΖD Β =7> Ο Ζ ) D = :6)7 Υ:67> D ;2

90 = ;>?DΟ= 1 1)1 > Η = > = > ΚΛ = > = 1 1)1 >3 2. = > / 1 ( = ) > % =(/> )= > / 0 8 : ;. Κ/1. +01? Λ Κ08 Λ Ζ2 D/1. + Ρ /1. +01? + + ) ) 2 D/1. + W 2/1. +01? + + ) + ) = > )) 22. D/1. + ;

91 D1% =Υ7>. /1. +01?. ) + = > ϑκ08 Λ D 1) + + ) / ) :2 & # D += ) Χ + Χ >0 D + %& = 2>) 7 ) D) += Σ> D + 8 => 27 + (<+ = >). = > D + = ϑ>7+ ;

92 . # /? + 2 D ) 3 ) ) Η D 7). / /1. +01? = 6 6> ( =Θ8>Μ ( =(/> ) ) =Θ8>Μ. =Θ8>Μ. ) =Θ8>Μ. ) =Θ8>Μ 8. =Θ8>Μ. ) =Θ8>Μ. =Θ8>Μ. ) =Θ8>Μ (=Θ8>Μ ( =Θ8>Μ ;

93 / 1 D % ) 08 2 = 22> D Κ Λ // ) & 2 = :6)7> & + / D = + 2> D +) &., = 1 1 > & / : + & 2D=7 3 7>. + = 22> = 2> = > + D+ 2 ;6

94 . < & = > + & ) /7? ) 08 +/1. + = 22> = /1. + 1) >7 ) (8</. ) ;4

95 % +) 7 (4)/ %. +) Ο4 ;% = 57 # > = > ) + % 57 = +> 3 ) = Σ> ) Κ) Λ Κ Λ ) Σ 8 ) Ο4666 ) ) /# ; + + ) + % 46 ) 2 ;:

96 Η ;;

97 (. = > # D ) # # 31 D 8 & % D Ε2:2;Φ & 4 ; =) > 6: = ) > D ϑ ) + 2 (/% = Ε:Φ> 64 +4& +1 ;

98 7 +4& +1 ΕΦ :6 2; ) # ) + ) % < => &. ) ) 7 3 ) 22 + ) 2 + D +&+ 6 D + = 2> D) + = 2> D

99 D +& 6 D + = 2> ) D %, D + + < 26Ι=> 5 :Ι=>6Ι 31 </ 31 =Σ> ) 4 ) 4Ι ) Ι=> %Ι)1Χ=47) > %Ι) 26Ι = 6)7)> 2Ι ) :6Ι =6Π26ΙΟ:6> < 657=)> ) 57=)> = > < 2

100 7 +4) 4:=. 1Χ 5 % 2 = :6)7> 7 +4/ & ) # 57 =>) )= D6 4 D6 :>.

101 =) > ). ) % + ). ) + =/ +:6)7> 7 +4( &% % 6 :2.) = > % = 6 6;+> :6)7 )

102 7 +4+ # #4 & =Ο 6 Υ:67> ) + D6 ; D6

7 +41 &<;:&)Α,%= :Ι 6Ι=6Ι> 31 31 < 31 Υ226 ) 7 +45 &;6:) +Α,%# & +Α,%

103 7 +41 &<;:&)Α,%= :Ι 6Ι=6Ι> < 31 Υ226 ) &;6:) +Α,%# & +Α,% = Ι 1Χ 6Ι=Ι26Ι>, ) ) 31Χ <Ι1Χ 4Ι 1Χ Υ & 6Ι Υ66&= 6Ι > 1Χ ) 1Χ & 6

104 3 (= > 657 ) :6)7 3 ( 657 )? ) 7? Η?? Η ) & :ϑ= D6 ; +=/1.> ϑ = >7 /1.). ) 6:3 = + Ο6;2 Ο42> + ) ΚΛ Κ Λ =>= /1.> Σ < 0 :6)7 ) 2 4

105 %,ΚΚ %,ΚΚ Χ ; ΖΙ, 2 26 Ζ:Ι %: 2; 2; #10Χ= ΖΙ : ;; Ζ6Ι #&+Χ=,: 2 4 Ζ6Ι #&+Χ= : = 26: 2:; Ζ2Ι +4) & Υ2 4Υ2; )7 :6)7 4 )7 + )7 6 )7 :2 )7 4)7 % :5 +Κ%,= : = 7 :Κ%,= ;: :2 :Κ%,= : 6: +4/ 7 & 3, = > / :6)7 / ϑ :

106 : )7 :6)7< / ϑ ) 31 &. = >:2)7=46)7 > 6:)7 / ). < % + ) (/%Ε:Φ +Ξ4)7 = 2> Ξ2% ) )+ % 4572ΠΞ4ΟΞ4. (/ 6 +Ξ2ΟΞ6 (/%Ε:Φ Ξ2)7=) > Ξ2 ) 2)745764Ο 7< ) 3 &? 6Ι= > 31 ) ) ) 31 1Χ < ;

107 + )? ) # 57 0 Κ Λ ) ) = Ζ257> ΚΛ + / ),. ) => / ). +Υ Υ #. :6 )7 2. ;6Ι& 26Ι = > 2 2Ι ) 2 < 31 2: 2 )7 2; 26)7 = 2> ( )4Ι ; =6)7 )>

108 (1Χ)4Ι 24 = )7)> 4Ι & )7 + = >) &. ) + 2Ι ) %8( & ) ) = ) 2> 2 ) ) ) + 3 & ;57 = > 25757= >Ζ257= >Ζ2657 =>Ζ257=>Ζ57= >ΖΤ ΤΟ4;577 Σ. # +4;57Σ 2 2 & 2 =D 6 > ) = ; > % )7 & 3 4)7 +6 )7 :657 4) Σ1 )7 &) ) 2

109 )D 3.53 =D 2 > ) 6; & < ) =.(> 7 +4&0 ; ) D )7 7 =4Ι+6 +7Ο57> 2= > = ) > ) 5..( Σ 22

110 7 +4&& )# 0 Η &) )# ) =D6 > 24571) 2;57=257;57 > 2

111 7 +4&/ ) ) ) =ΖΖ > Α ) 2Ι=)7 )7 >. 4Ι 6; 2 2

112 7 +4&( ):+080= 0 1Χ )57 ) 2;57 6; : 1Χ ) 657) 2 657)<) 7 +4&+ );6:)/Α,%# &8Α,% = D :5 +Κ%,= : = 7 :Κ%,= 2;: ; :Κ%,= 2;2 26 2

113 +4( 7 ) % 2. ). D 2. 3 :6)7 ) D 2 :6)7 =/ > )= 2 )7> ),. + =Ξ4)7> ):6)7 )7 2 )7 6)7 0 :6)7 Σ %,ΚΚ %,ΚΚ Χ Ι, 2 2 2Ι %:&00Χ = 2; 2Ι : Ι :&00Χ 4 2Ι 26

=,:&00Χ = : = +4+ 6 64 ;; 2Ι ) Υ;; )2 )7 Υ :2 ) = 2 )7> = 31> % = > )

114 =,:&00Χ = : = ;; 2Ι ) Υ;; )2 )7 Υ :2 ) = 2 )7> = 31> % = > ) 4Ι =)7> 0 = + > 0 ) ) ) )= >. ) 7 +4&1 )#% :+080= ϑ +=/1.>7 Κ 7( Λ ) 24

= &>Η Η& ) Β ) ). Κ 7( Λ)7 )7. 4)7 3 7 +4&5 )#% :+080= )= > + = +/1.Ρ2>( ) +7 =/1.

115 = &>Η Η& ) Β ) ). Κ 7( Λ)7 )7. 4) &5 )#% :+080= )= > + = +/1.Ρ2>( ) +7 =/1.W2> ) )7 ) ) = > 2 6; : % + ) ) 8 )D Υ;22 =Υ:4ΖΥ2 ΟΥ:;2> Υ2 Υ;22 Υ:;2Υ <Υ2 <6Υ4 ) 2:

116 7 +4&8 )# :+080= 7 + =4Ι )7 + 7> 7 )7 7 =)77Ο > =/1.> < 6: /1. ) 0 4: :2/1. 4: :2 ) ) + <4Ι ;67 & ΚΛ Ι% Υ&Υ; 7 ΝΟΥ4 Υ2)7= +:2> :6) Σ % % =/1.> 6 6% 6 %6.+ 2;

117 Electrical storage revenue vs RPI buy price threshold Revenue per day Buy threshold (RPI) 7 +4&. )#. ΚΛ + ; 6 ) ) % ) ) & % = )Τ Τ> ) ;% + % + 7 ) D & &) D Ξ6 =6 +Ξ2 62>D & +) 657%Ξ4)7 2

118 ) Ξ2 + + ) 57=: ) >. 57= > ΚΛ ) ) 5D % ) 2 = > Χ D6 2 D ) Ι 7 +4)0 /# 22

119 7 +4)& /# :+080=? % &. ) # ) 2Ι 2457 (/Χ 7 =;57> Χ ) ) % ) ( ) 3 ) ( ) + D & + ) ) =ΝΟ > ) 222

120 / )2 # # ) ) + ) + < ) D ) 01=0) > 01 )& ) ) ) ) ) =+> ) ) & + / (8< ) 8%. ) 22

121 7? ) +). # Υ Υ2 ) ) = > #. + ΚΛ. 3 = > ) D Κ Λ + # ) ( = + > 22

122 . + & < ) (=> & Η & < ) ) & + +? ). % ) ) ) ) % ), &) ) < + ) / Χ ) + 22

123 ) 0 ) &Β %+ ). # ) ) & 226

124 7 < ( ) D ( & & ) + & & 0& & ΚΛ ) ϑ Β Η.. ( ) ) + +(. + + ) +& & ) 224

125 # (2 # Ε % % % (/ ( / Χϑ ) 1 8 ) ) 0 0 Η (8< ( 8 )< (/ ( / (/ ( / (. (. ( ( 3Χ + 3Χ5 )Χ8 #:6)Χ ) % % 3Χ 3Χ44)Χ2)Χ:6)Χ )Χ # Γ/ Γ /?D % + ). )27 7 ϑ2οι?d?d 2Ι?? D )) %?D?Χ? Χ. # Χ= > 26Χ=> Χ Χ. # 44)Χ22)Χ )Χ # < < ) 1Χ 1 22:

126 / /1. /1. Τ /. 1% % < 8 7 / (8< ) (8< ) (8< / 1. + = > 1 < / + =/1.> + ) =Τ> / % % /. ) < = > 8 ) 7 22;

127 # # # Ν27 46Ε4Φ. Ν24Ι =0 Ν2 Ι> 32Ι # 0=2Ι ;: Ο ;2 Ε4Φ> % Ο46Π Π ;2Π2Π=242 > ΟΝ7Ψ24Ι =+> ΟΝ257Ψ24ΙΠΙ ΟΝ;7Ψ2 Ι => # ΝΟ 7Ε22Φ ; Ο:Ι Ι & # 0 0; #=(.Ε;Φ> % + # Ν27 46Ε4Φ 1) # = ) & ΝΟ2)7 > 27 1Χ 26Ι Π2Π26Ο47) % # ΝΟ64Π Π2Π Π46Π26ΝΟ27 1) # # +64Π Π6Π26Ο: =47> ; =;7)> 57=67>24 = )7)> 1Χ # ΟΙ # #=(.Ε;Φ> % + # Ν27 46Ε4Φ 22

128 1) # = ) & ΝΟ2)7 > 1) 27 6)7 +4Ι % # ΝΟ64Π Π2Π Π46Π4ΝΟ67 1) # # 64Π Π6Π4Ο57 457; =6)7)> #/ 0 8 7) 27 8 # + 2 2) ) Π2Π22Π27Ο Π27Ο 577 ) Ι # 0 < 57 Ι % ) # # % ) :6)7 )27 1 ) =) 2) 4> ) 2)2Π Π27Ο2 57 Ι= 1>% ) ## # ) 27? 6Ι 2

129 # % Κ0ΛΕ6Φ ) #Β 0 % Ε6Φ# +, )./0 12/. 34/ (. 22; & 47Π22; Ο2;57 47Π22;26Ο457 7 # :2 2Ι # = ;:> 257 % ) # 257 ) 22

130 # # 5 ( & ) ) % ;+2;+2 )Ε2Φ 2%Ψ2ΧΥ;:ΖΧ% :Ι+2+2Ο; )7 Υ6 )&. & % ( ( % : ;Ι Ι & ; : Ο 4)7 ; + :Ο;2)7 24)7 2 ϑ +24;2ΝΟ24Υ; 62 ;24 ) & & Υ Υ;. ) + & % :)7 24)7 = 4)7 > Υ Υ ; ) 2 D ) Κ Λ 4 The incentive for these people to install domestic storage is that current grid connection of domestic renewables provides vastly different (but fixed) rates for import and export. The export rate paid to the user is very low (below the lowest wholesale price). Therefore, these people currently justify their investment in battery storage because it makes them almost self-sufficient from the grid, so that they do not get punished by the imbalanced import and export rates. 2

131 3 ),. 24 Υ; = > < 24+ 4Ο264)7Υ67= )> Υ; ( 24+;2Ο2 4)7)Υ67 = )> Υ4 ; + Υ4 < Υ 4<2 26 Υ;= > & = >, 3 ) ) & # # & & Η & & ) 1 Σ% & % =2 & > 8Ο2+2+2Π=;+2;+2 >Ο6 Ο6+;2)7Ο;7 # W W &? & 2

132 2Ι ) ) 6 ΟΧ+ς+ + ςο2) Ο 2:)Γ)# Ο = 6>+ 2:ΓΟ6 65ΓΝΟ ;7 & ;24Ο:; % ) 2=2> ) =)Ο 7#> 6 = > ΑΟ%+ ) +) ΑΟ4+2+2+= 6>2+ Ο )7 3 + = :>+4Ο2; )= ) = 6>Ο6Ι > ) & < 2 )2 ; 46 Ο2+2+=46 ;>+ 2:Ο;6Γ ΝΟ: )7 3 )7 )4 )=(Ο 2 3Ο > ) =)Ο 7#> ΑΟ%+ ) +) ΑΟ=_=(> Ζ_(3>+=;6 >+ Ο2;

133 2;4=2+2+ 2:>+4Ο2 Ο =ΝΟ6Ι > Η + 3 ) & =) Σ> : )7= ) > +24)7 # #Ε;Φ% + & = ) > # 6 3 => 5 + Η = > & =+ Σ> W3 W ΙΕΦ ) 5 4 ) 6 3/// 4// &) 3 ) 3 ) 1/0 3 =3 > 67 2 Π2Ο; )3 2 =6 2>; ) ; +=Ζ24>Ο;4) =3 <Ζ24Ο2;> ;4) ;42;+2 Ο D ;6;)Γ 3 ;6; Ο2:45ΓΝΟ6 7 +:Ι 26

134 ;Ι Ι Ι 3. = >. = > &. & #. & :2) ) & :2) & 57 ;7 ;7.+ & & ) #/ < 1 E = Jω 2 Γ 2 Γ + D+ J = m r i i 2 i. J R = 2. π. r. ρ. H. r. dr = π. ρ. H. R 0 2 R J = 2. π. r. ρ. H. r. dr = π. ρ. H.( R2 2 R1 4 R1 ) ) + Γ J ρ. V. R 2 Χ / + + T ρ R ω 24

135 = >, 3 1 & 0 ( / Ε Φ & :62 22) + 68 = > ω max 1 R T ρ J ρ. V. R 2 70) ( ) R ( ) ( ) V 2. π E = Jω 45MJ 12kWh 2 % :6) Η 7 Κ Λ4 Κ Λ & / Ε Φ Ξ4)7 #? % ) 2:

136 2+2+26= + Σ> ΟαΟ2 + Work = m. c T T 2 1 p = p 2 1 V γ 1 γ.( T 2 T1 ) pv = RT 2 Ο2 6 8 Ο Ο # Ο; 4# D Ο:2;Γ)# /Ο;:Γ)#. Ο/ Ο62 ) ) ; 4# 2 :2;=; 4 >Ο; 5ΓΝΟ7 2Ι + ) ; / ) : 2 ;)7 ); 4# =4 > 6Ι Ι ) = >. = > Ο Ο 1 Ο/ Ο;:2+ 62 Ο =24> % + Ο2#= 2 2 > 7)Ο Χ = >Ο2 :2;= 2>Ο5ΓΝΟ47 6;+;Ο4 Ι % ;+;+=4>Ο2:Ι 2;

137 #7 5 Η Η Ο[?. 1, Κ, Λ +Κ Λ Κ Λ % & +2ΕΦ = > & ϑ ) =,, > => 4Γ:6)Χ% =ΝΟ 26)7> # Ε Φ Ε2Φ ΕΦ Η ), = 2 > )= )>, 7 54& )#)00%6:8 4= 2

138 #/ 58 = > = > & = > = > +. & + 8? % Baseline demand + - Actual demand G(z) Price H(z). 5=Η>] & 3=Η> ) & = :> Η + ( 1 + ( 1 2

139 Baseline Demand - D Actual Demand - D 0 G(z) (Price - P o )/P 0 H(z) 3 ( 1 DW /1. + = 22 22> ϑ 5=Η> 1 Bx Ae + C + DuOS Pr ice = ( 1 P 0 Demand x = Capacity L arg estgeneratorcapacity 5=Η> P BAe Bx ( Capacity L arg estgeneratorcapacity) 0 3=Η>3=Η> =>] ] = 7> ) Η ;. Η ) Χ ) )

140 +.),)Β )ΣΗ Η 2 8 3=Η> D + H R1Buy P N PS P R2Buy 8 3 P S PR Buy PR 2Buy ) PR 1 Buy 6 PR 2 Buy W2 +Β ( / 1. ) 1. & 5=Η> 3=Η> G( z) 1+ G( z) H ( z) Η Η& 1+ G( z) H ( z). & 1+ Kg( z) # & =Η>. =Η>Η 2 #7 Η& ΗΟ # 2

141 % +Η. # 7 # )) + +, Ο6 Ο657? 5 Ο57ΟW? 5 Ο657 1 Ο26 )7=26Υ7> 150 C DUoS ln A 3 x = B =D 2>%Ο20Ο26 Ο2 (;Υ7 ( +Π657Ο :57 7 & + + K K Storage Elasticity < 1 < 1 & + P P 0 0 BAe ( Capacity Larg estgeneratorcapacity ) BAe Bx Bx P N H P S P R1Buy ( ) [ E ] arg 1 0D0 < Capacity L estgeneratorcapacity R2Buy < 1 2

142 15* e *10 9 ( ) 15* e *10 15* * ( ) 6 25*10 P S < [ E * 46.5*10 ] < * * *10 P S < [ E * 47.0 *10 ] < 1 0 P S Ρ:7 Ρ; 7 7 ΒΒ6 Β; # # ; ::7 0 Η + = > D 2

143 # 14 The following algorithm finds the angles of the sun for a given location on the planet and time, and computes these angles relative to a solar panel orientation. The collectable "power factor" of the solar collector relative to the total direct normal solar irradiance available is calculated. Inputs:- Latitude, Longitude Time (in GMT, relative to midday on 21 June - at time t=0 a point at longitude 0 in the N hemisphere has it's highest sun angle possible) Azimuth (towards East) and declination rotation offsets for the solar panel from nominal flat. Outputs:- Solar declination, in degrees from directly overhead Solar bearing, a compass bearing of the sun position relative to true North The AimFactor power, which is the power available relative to direct normal solar The bearing, declination, and overall angle errors of the panel orientation The earth centre is considered stationary at [0,0,0] with the sun stationary at [infinity,0,0] We begin by considering the point closest to the sun at t=0, on the equator.this is a vector of [1,0,0]. This vector towards the sun points directly overhead in a direction of [1,0,0]. We also know that views East and North from this point at this time are [0,1,0] and [0,0,1] respectively. Know we apply the following rotations in order, to find these three vector directions for the correct location and time:- Rotation of latitude Rotation of longitude Rotation due to rotation of earth about N-S (the z) axis (every 23 hours 56 minutes) Rotation of the N-S axis of earth about the y axis (23.45 degrees off axis) Rotation earth and its N-S axis about the z axis every days The three transformed (overhead, East and West view) vectors for are now compared to the sun-view vector which remains [1,0,0] The three vectors EastVector, NorthVector and OverheadPointer form a right-handed orthogonal set of axes. We use these three vectors to form the actual panel pointer vector, by using a rotation towards East and a declination from overhead. This is an azimuth-declination transform similar to radar antenna pointing. > restart; > RzAnnual:=linalg[matrix](3,3,[[cos(epsilon),- sin(epsilon),0],[sin(epsilon),cos(epsilon),0],[0,0,1]]); cos( ε ) sin( ε ) 0 RzAnnual := sin( ε ) cos( ε ) > RyAxisOffset:=linalg[matrix](3,3,[[cos(phi),0,sin(phi)],[0,1,0 ],[-sin(phi),0,cos(phi)]]); 26

144 RyAxisOffset cos( φ ) 0 sin( φ ) := sin( φ ) 0 cos( φ ) > RzDaily:=linalg[matrix](3,3,[[cos(theta),- sin(theta),0],[sin(theta),cos(theta),0],[0,0,1]]); cos( θ ) sin( θ ) 0 RzDaily := sin( θ ) cos( θ ) > RzLongitude:=linalg[matrix](3,3,[[cos(alpha),- sin(alpha),0],[sin(alpha),cos(alpha),0],[0,0,1]]); cos( α ) sin( α ) 0 RzLongitude := sin( α ) cos( α ) > RyLatitude:=linalg[matrix](3,3,[[cos(beta),0,sin(beta)],[0,1,0 ],[-sin(beta),0,cos(beta)]]); RyLatitude cos( β ) 0 sin( β ) := sin( β ) 0 cos( β ) > OverheadRotationCombo:=linalg[multiply](RzAnnual,RyAxisOffset, RzDaily,RzLongitude,RyLatitude); OverheadRotationCombo := [ ( ( cos( ε ) cos( φ ) cos( θ ) sin( ε ) sin( θ ) ) cos( α ) + ( cos( ε ) cos( φ ) sin( θ ) sin( ε ) cos( θ ) ) sin( α ) ) cos( β ) cos( ε ) sin( φ ) sin( β ), ( cos( ε ) cos( φ ) cos( θ ) sin( ε ) sin( θ ) ) sin( α ) + ( cos( ε ) cos( φ ) sin( θ ) sin( ε ) cos( θ ) ) cos( α ), ( ( cos( ε ) cos( φ ) cos( θ ) sin( ε ) sin( θ ) ) cos( α ) + ( cos( ε ) cos( φ ) sin( θ ) sin( ε ) cos( θ ) ) sin( α ) ) sin( β ) + cos( ε ) sin( φ ) cos( β ) ] [ ( ( sin( ε ) cos( φ ) cos( θ ) + cos( ε ) sin( θ ) ) cos( α ) + ( sin( ε ) cos( φ ) sin( θ ) + cos( ε ) cos( θ ) ) sin( α ) ) cos( β ) sin( ε ) sin( φ ) sin( β ), ( sin( ε ) cos( φ ) cos( θ ) + cos( ε ) sin( θ ) ) sin( α ) + ( sin( ε ) cos( φ ) sin( θ ) + cos( ε ) cos( θ ) ) cos( α ), ( ( sin( ε ) cos( φ ) cos( θ ) + cos( ε ) sin( θ ) ) cos( α ) + ( sin( ε ) cos( φ ) sin( θ ) + cos( ε ) cos( θ ) ) sin( α ) ) sin( β ) + sin( ε ) sin( φ ) cos( β ) ] [ ( sin( φ ) cos( θ ) cos( α ) + sin( φ ) sin( θ ) sin( α ) ) cos( β ) cos( φ ) sin( β ), sin( φ ) cos( θ ) sin( α ) + sin( φ ) sin( θ ) cos( α ), ( sin( φ ) cos( θ ) cos( α ) + sin( φ ) sin( θ ) sin( α ) ) sin( β ) + cos( φ ) cos( β ) ] > OverheadPointer:=linalg[multiply](OverheadRotationCombo,linalg [vector]([1,0,0])); 24

145 OverheadPointer := [ ( ( cos( ε ) cos( φ ) cos( θ ) sin( ε ) sin( θ ) ) cos( α ) + ( cos( ε ) cos( φ ) sin( θ ) sin( ε ) cos( θ ) ) sin( α ) ) cos( β ) cos( ε ) sin( φ ) sin( β ), ( ( sin( ε ) cos( φ ) cos( θ ) + cos( ε ) sin( θ ) ) cos( α ) + ( sin( ε ) cos( φ ) sin( θ ) + cos( ε ) cos( θ ) ) sin( α ) ) cos( β ) sin( ε ) sin( φ ) sin( β ), ( sin( φ ) cos( θ ) cos( α ) + sin( φ ) sin( θ ) sin( α ) ) cos( β ) cos( φ ) sin( β ) ] > NorthVector:=linalg[multiply](OverheadRotationCombo,linalg[vec tor]([0,0,1])); NorthVector := [ ( ( cos( ε ) cos( φ ) cos( θ ) sin( ε ) sin( θ ) ) cos( α ) + ( cos( ε ) cos( φ ) sin( θ ) sin( ε ) cos( θ ) ) sin( α ) ) sin( β ) + cos( ε ) sin( φ ) cos( β ), ( ( sin( ε ) cos( φ ) cos( θ ) + cos( ε ) sin( θ ) ) cos( α ) + ( sin( ε ) cos( φ ) sin( θ ) + cos( ε ) cos( θ ) ) sin( α ) ) sin( β ) + sin( ε ) sin( φ ) cos( β ), ( sin( φ ) cos( θ ) cos( α ) + sin( φ ) sin( θ ) sin( α ) ) sin( β ) + cos( φ ) cos( β ) ] > EastVector:=linalg[multiply](OverheadRotationCombo,linalg[vect or]([0,1,0])); EastVector := [ ( cos( ε ) cos( φ ) cos( θ ) sin( ε ) sin( θ ) ) sin( α ) + ( cos( ε ) cos( φ ) sin( θ ) sin( ε ) cos( θ ) ) cos( α ), ( sin( ε ) cos( φ ) cos( θ ) + cos( ε ) sin( θ ) ) sin( α ) + ( sin( ε ) cos( φ ) sin( θ ) + cos( ε ) cos( θ ) ) cos( α ), sin( φ ) cos( θ ) sin( α ) + sin( φ ) sin( θ ) cos( α ) ] > PanelPointer:=evalm(linalg[scalarmul](OverheadPointer,cos(SPD) )+linalg[scalarmul](eastvector,sin(spd)*sin(sre))- linalg[scalarmul](northvector,sin(spd)*cos(sre))); PanelPointer := [ cos( SPD ) ( ( ( cos( ε ) cos( φ ) cos( θ ) sin( ε ) sin( θ ) ) cos( α ) + ( cos( ε ) cos( φ ) sin( θ ) sin( ε ) cos( θ ) ) sin( α ) ) cos( β ) cos( ε ) sin( φ ) sin( β ) ) + sin( SPD ) sin( SRE ) ( ( cos( ε ) cos( φ ) cos( θ ) sin( ε ) sin( θ ) ) sin( α ) + ( cos( ε ) cos( φ ) sin( θ ) sin( ε ) cos( θ ) ) cos( α ) ) sin( SPD ) cos( SRE ) ( ( ( cos( ε ) cos( φ ) cos( θ ) sin( ε ) sin( θ ) ) cos( α ) + ( cos( ε ) cos( φ ) sin( θ ) sin( ε ) cos( θ ) ) sin( α ) ) sin( β ) + cos( ε ) sin( φ ) cos( β ) ), cos( SPD ) ( ( ( sin( ε ) cos( φ ) cos( θ ) + cos( ε ) sin( θ ) ) cos( α ) + ( sin( ε ) cos( φ ) sin( θ ) + cos( ε ) cos( θ ) ) sin( α ) ) cos( β ) 2:

146 sin( ε ) sin( φ ) sin( β ) ) + sin( SPD ) sin( SRE ) ( ( sin( ε ) cos( φ ) cos( θ ) + cos( ε ) sin( θ ) ) sin( α ) + ( sin( ε ) cos( φ ) sin( θ ) + cos( ε ) cos( θ ) ) cos( α ) ) sin( SPD ) cos( SRE ) ( ( ( sin( ε ) cos( φ ) cos( θ ) + cos( ε ) sin( θ ) ) cos( α ) + ( sin( ε ) cos( φ ) sin( θ ) + cos( ε ) cos( θ ) ) sin( α ) ) sin( β ) + sin( ε ) sin( φ ) cos( β ) ), cos( SPD ) ( ( sin( φ ) cos( θ ) cos( α ) + sin( φ ) sin( θ ) sin( α ) ) cos( β ) cos( φ ) sin( β ) ) + sin( SPD ) sin( SRE ) ( sin( φ ) cos( θ ) sin( α ) + sin( φ ) sin( θ ) cos( α ) ) sin( SPD ) cos( SRE ) ( ( sin( φ ) cos( θ ) cos( α ) + sin( φ ) sin( θ ) sin( α ) ) sin( β ) + cos( φ ) cos( β ) ) ] > SunDeclinationFromOverhead_F:=arccos(OverheadPointer[1])*180/P i; SunDeclinationFromOverhead_F := 180 arccos ( ( ( cos( ε ) cos( φ ) cos( θ ) sin( ε ) sin( θ ) ) cos( α ) + ( cos( ε ) cos( φ ) sin( θ ) sin( ε ) cos( θ ) ) sin( α ) ) cos( β ) cos( ε ) ( ) sin φ sin( β ) )/ π > North:=linalg[dotprod](NorthVector,linalg[vector]([1,0,0])); North := sin( β ) cos( α ) cos( ε ) cos( φ ) cos( θ ) sin( β ) cos( α ) sin( ε ) sin( θ ) sin( β ) sin( α ) cos( ε ) cos( φ ) sin( θ ) sin( β ) sin( α ) sin( ε ) cos( θ ) + cos( ε ) sin( φ ) cos( β ) > East:=linalg[dotprod](EastVector,linalg[vector]([1,0,0])); East := sin( α ) cos( ε ) cos( φ ) cos( θ ) + sin( α ) sin( ε ) sin( θ ) cos( α ) cos( ε ) cos( φ ) sin( θ ) cos( α ) sin( ε ) cos( θ ) > SunBearing:=arctan(East,North)*180/Pi; SunBearing := 180 arctan ( sin( α ) cos( ε ) cos( φ ) cos( θ ) + sin( α ) sin( ε ) sin( θ ) cos( α ) cos( ε ) cos( φ ) sin( θ ) cos( α ) sin( ε ) cos( θ ), sin( β ) cos( α ) cos( ε ) cos( φ ) cos( θ ) sin( β ) cos( α ) sin( ε ) sin( θ ) sin( β ) sin( α ) cos( ε ) cos( φ ) sin( θ ) sin( β ) sin( α ) sin( ε ) cos( θ ) + cos( ε ) sin( φ ) cos( β ) )/ π > AimFactor:=evalm(linalg[dotprod](PanelPointer,linalg[vector]([ 1,0,0]))); 2;

147 AimFactor := cos( SPD ) cos( β ) cos( α ) cos( ε ) cos( φ ) cos( θ ) cos( SPD ) cos( β ) cos( α ) sin( ε ) sin( θ ) cos( SPD ) cos( β ) sin( α ) cos( ε ) cos( φ ) sin( θ ) cos( SPD ) cos( β ) sin( α ) sin( ε ) cos( θ ) cos( SPD ) cos( ε ) sin( φ ) sin( β ) sin( SPD ) sin( SRE ) sin( α ) cos( ε ) cos( φ ) cos( θ ) + sin( SPD ) sin( SRE ) sin( α ) sin( ε ) sin( θ ) sin( SPD ) sin( SRE ) cos( α ) cos( ε ) cos( φ ) sin( θ ) sin( SPD ) sin( SRE ) cos( α ) sin( ε ) cos( θ ) sin( SPD ) cos( SRE ) sin( β ) cos( α ) cos( ε ) cos( φ ) cos( θ ) + sin( SPD ) cos( SRE ) sin( β ) cos( α ) sin( ε ) sin( θ ) + sin( SPD ) cos( SRE ) sin( β ) sin( α ) cos( ε ) cos( φ ) sin( θ ) + sin( SPD ) cos( SRE ) sin( β ) sin( α ) sin( ε ) cos( θ ) sin( SPD ) cos( SRE ) cos( ε ) sin( φ ) cos( β ) > SunOffsetFromPanelDegs_F:=arccos(AimFactor)*180/Pi; SunOffsetFromPanelDegs_F := 180 ( π arccos ( cos( SPD ) cos( β ) cos( α ) cos( ε ) cos( φ ) cos( θ ) + cos( SPD ) cos( β ) cos( α ) sin( ε ) sin( θ ) + cos( SPD ) cos( β ) sin( α ) cos( ε ) cos( φ ) sin( θ ) + cos( SPD ) cos( β ) sin( α ) sin( ε ) cos( θ ) + cos( SPD ) cos( ε ) sin( φ ) sin( β ) + sin( SPD ) sin( SRE ) sin( α ) cos( ε ) cos( φ ) cos( θ ) sin( SPD ) sin( SRE ) sin( α ) sin( ε ) sin( θ ) + sin( SPD ) sin( SRE ) cos( α ) cos( ε ) cos( φ ) sin( θ ) + sin( SPD ) sin( SRE ) cos( α ) sin( ε ) cos( θ ) + sin( SPD ) cos( SRE ) sin( β ) cos( α ) cos( ε ) cos( φ ) cos( θ ) sin( SPD ) cos( SRE ) sin( β ) cos( α ) sin( ε ) sin( θ ) sin( SPD ) cos( SRE ) sin( β ) sin( α ) cos( ε ) cos( φ ) sin( θ ) sin( SPD ) cos( SRE ) sin( β ) sin( α ) sin( ε ) cos( θ ) + sin( SPD ) cos( SRE ) cos( ε ) sin( φ ) cos( β ) ) )/ π > DPE_F:=evalm(linalg[dotprod](PanelPointer,EastVector)); > DPN_F:=evalm(linalg[dotprod](PanelPointer,NorthVector)); > PanelBearing:=arctan(DPE,DPN)*180/Pi; arctan ( DPE, DPN ) PanelBearing := 180 π > PanelDeclination_F:=arccos(evalm(linalg[dotprod](PanelPointer, OverheadPointer)))*180/Pi; Above are all the big formulas. Below we put the numbers in 2

148 phi:=23.45*pi/180; φ := π > LongEastDegs:=-3.169; LatNDegs:=55.977; LongEastDegs := LatNDegs := > SolarPanelRotationToEast:=0; SolarPanelDeclination:=30; := SolarPanelRotationToEast 0 > Date:=242.5; SolarPanelDeclination := 30 Date := t is time (in days) relative to midday (GMT) on June 21 of a reference year when Greenwich obtains it's most overhead sun possible at this time. > t:=(date-171.5); t := 71.0 > alpha:=longeastdegs*pi/180; α := π > beta:=-latndegs*pi/180; β := π > SRE:=SolarPanelRotationToEast*Pi/180; SPD:=SolarPanelDeclination*Pi/180; SRE := 0 SPD := 1 6 π > epsilon:=-t/365.25*2*pi; ε := π > theta:=t*(1+1/365.25)*2*pi; θ := π > SB:=evalf(SunBearing); SB2:=round(SB); SunBearingTrue:=SB2 mod SB-SB2; SB := SB2 := 174 SunBearingTrue := > SunDeclinationFromOverhead:=evalf(SunDeclinationFromOverhead_F ); SunDeclinationFromOverhead := > SunOffsetFromPanelDegs:=evalf(SunOffsetFromPanelDegs_F); PowerRelativeToDirectNormal:=evalf(AimFactor); 2

149 SunOffsetFromPanelDegs := PowerRelativeToDirectNormal := > DPE:=evalf(DPE_F); DPN:=evalf(DPN_F); PB:=evalf(PanelBearing); PB2:=round(PB); PanelBearingTrue:=PB2 mod PB-PB2; BE:=SunBearingTrue-PanelBearingTrue; BE2:=round(BE); BearingError:=(BE2 +180) mod BE-BE2; DPE := DPN := PB := PB2 := 180 PanelBearingTrue := BE := BE2 := -6 BearingError := > PanelDeclination:=evalf(PanelDeclination_F); DeclinationError:=SunDeclinationFromOverhead-PanelDeclination; PanelDeclination := > DeclinationError :=

150 2 %Γ = > Ρ β,β β β ββw %D/?%D/?= > Ρ Η 0(2? W %?= > Ρ )3. +2W 02 ;/ / =2 ;>6 6 05% Ρ ) 7β +W 4 0(2 ; 0 3 : (/%( = > Ρ W ; (. #=>= > Ρ ) β +W (. 2Ρ ) +W 2 (. = > Ρ ) β +W 22 (. = > Ρ ) β +W 2 / = > Ρ &).7β()β?W 2 = > Ρ )βββ ββw 2 + = >Ρ + )W 26 %+% = >Ρ W 24 8 )% = >Ρ )W 2: D 2 / 2:=2 6>;2 2; D 2 6 2:8=2 6> 2 : % 2ΚΓ,ΜΛ 2 Γ/ Γ / = >Ρ,)W?2( ) Ρ W / 1 = >Ρ ΗΜ%.(Ο22W 2

151 =>= > Ρ ) β β W 4 8(/.8 ( ( / 0/ 8 ϑ1 ) Ρ W : 8%8 % 0 = > Ρ +β W ; 8.%7 = > Ρ )β2w / = >Ρ W 2 / = > Ρ Μ(.(Ο;29.(Ο 44W = >Ρ ))W 7)= > Ρ ) ) β β βw 2

Homework 6 Solar PV Energy MAE 119 W2017 Professor G.R. Tynan

Homework 6 Solar PV Energy MAE 119 W2017 Professor G.R. Tynan 1. What is the most likely wavelength and frequency of light emitted from the sun which has a black body temperature of about 6600 deg K? What