2. Theoretical Formulation of AIM/Enduse
This section explains the linear programming formulation of AIM/Enduse model.
The code written in GAMS is included in Appendix B. The formulation described
in this section is valid for a given year. Since the formulation allows transfer of
relevant information like stock of devices and improvement in performance of devices across successive years, the model can be run for any number of years. Thus,
if the simulation period selected by the user is 30 years, the model makes the calculations successively for each of the 30 years.
2.1 Indices and Sets
i
j
k
l
m
p
Wj
z
Rz
Gi
e
Re
Vj,k
Sector
Service kind
Energy kind
Device
Gas (emission) kind
Gas (emission) removal process
Set of combinations of device and removal process (l, p) that can satisfy
service kind j
Index denoting group of sectors categorized for the purpose of emission
control
Set of sectors belonging to group z
Group of sectors selected for representing energy constraint
Index denoting group of sectors categorized for purpose of energy supply
Set of sectors belonging to group e
Set of internal service kinds j corresponding to internal energy kinds k
(i.e. all k belonging to this set must be supplied by all j belonging to this
set)
2.2 Expression for Emission Quantity Estimation
Emission quantity of a gas is estimated adding up quantity of emissions from all
devices. Emission from a device is estimated by multiplying operating quantity of
the device with emission quantity per unit of device.
Qim  
 X
j ( l , p )W j
l , p ,i
 elm, p ,i



elm, p ,i   f 0m,l   f km,l  1   k ,l ,i   Ek ,l , p ,i  U k ,l   d lm, p ,i
k


(2.1)
(2.2)
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Where,
Qim:
Emission of gas m in sector i
elm, p,i : Emission of gas m from an operating unit of combination of device l with
removal process p in sector i
Operating quantity of combination of device l with removal process p in
sector i
Energy use of energy kind k per operating unit of combination of device l
with removal process p in sector i (same as specific energy input)
Emission of gas m from operations other than energy combustion of a
unit of device l (same as gas m’s emission coefficient of device l)
Emission of gas m from combustion of energy kind k by a unit energy
use of device l
Energy saving ratio due to efficiency improvement in use of energy kind
k by device l in sector i
Proportion of energy kind k used in device l for combustion operations,
or burning rate (Note: 1- Uk,l or proportion of k used for non-combustion
operations in device l is taken as input in database system)
Emission rate (1- removal ratio) of gas m from combination of device l
with removal process p in sector i
Xl,p,i:
Ek,l,p,i:
f0,lm:
fk,lm:
ξk,l,i:
Uk,l:
dl,p,im:
2.3 Expressions for Constraints
2.3.1 Emission constraints
Emission of gas m in sector i must not exceed allowable maximum emission limit
in sector set Rz .
m
m
Qi  Qˆ z
(2.3)
iRz
Where,
Q̂ zm :
Allowable maximum limit on emission of gas m in group z
2.3.2 Service demand constraints
For a given service, its demand must be met by the quantity of service output supplied by all devices.
D j ,i  1   j ,i  
Where,
A
( l , p )W j
l , j ,i
 X l , p ,i
(2.4)
2. Theoretical Formulation of AIM/Enduse
Al,j,i :
Ψ,j,i :
Dj,i :
257
Supply output of service j per operating unit of device l in sector i (same
as specific service output)
A measure of service efficiency of service type j in sector i (Note: Negative of Ψj,i , a measure of loss of service j, is taken as input in database
system; Negative of Ψj,i is the loss incurred during delivery of service j,
for example transmission and distribution loss of electricity supply)
Service demand quantity of service type j in sector i
2.3.3 Device share ratio constraints
For a given service, ratio of service output of a device to total service output of all
devices must not exceed its upper limit or maximum share.
θl , j ,i   Al , j ,i  X l , p,i  Al , j ,i  X l , p ,i
( l , p )W j
(2.5)
p
Where,
 l , j : Maximum share of device l in service j
2.3.4 Operating capacity constraints
Operating quantity of a combination of device l with removal process p must not
exceed its stock net of operating rate.
X l , p ,i  (1   l ,i )  S l , p ,i
(2.6)
Where,
1+Λl,i: Operating rate of device l in sector i (Note: 1+Λl,i is taken as input in database system)
Sl,p,i : Stock of combination of device l with removal process p in sector i
2.3.5 Stock exchange constraints
Every device has a life. Stock of a device recruited in a given year will retire at the
end of its life, with its quantity reducing linearly during its lifetime. Thus, out of
total stock of a combination of device l with removal process p that was available
in the previous year, a certain fraction (inverse of life of device) retires and the
balance stock is passed on to the current year. Certain stock of previous year’s
combination of device l with removal process p can be replaced (or exchanged) in
the current year by its combination with another removal process p1. However, the
stock that is replaced in the current year cannot exceed the stock that is passed on
from the previous year.
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
1
S l , p ,i 1
 T
l ,i


   M l , p  p ,i
1
 p1

(2.7)
Where,
S l , p ,i : Stock of combination of device l with removal process p in sector i in the
previous year
M l , p p1 ,i :Previous year’s stock of combination of device l with removal process p
that is replaced in the current year by its combination with removal process p1
Life of device l in sector i (this is the average life of stock of device l in
previous year)
T l ,i :
2.3.6 Energy supply constraints
Total quantity of supply of energy kind k cannot exceed its allowable maximum
supply quantity in a sector.

   1     E
iGi
j

k ,l ,i
( l , p )W j
k ,l , p ,i

 X l , p ,i   Eˆ k ,Gi


(2.8)
Where,
Eˆ k ,Gi : Allowable maximum supply quantity of energy kind k
2.4 Internal Service and Internal Energy Balance
All internal energy kinds must be supplied as internal services from within the
model (from energy conversion and supply sectors).



    1     A
iRe
 jV j , k 
j ,i
( l , p )W j
l , j ,i
 


 X l , p,i        1   k ,l ,i   E k ,l , p,i  X l , p,i 
 iR kV  (l , p )W

e
j ,k 
j


(2.9)
2.5 Stock Balance
Stock of combination of device l and removal process p in the current year is
equal to the sum of the stock of that combination transferred from previous year,
2. Theoretical Formulation of AIM/Enduse
259
the quantity of that combination recruited in current year, and the net stock of other combinations of device l that are exchanged by its combination with removal
process p in current year.

1
S l , p ,i  S l , p ,i 1
 T l ,i

  rl , p ,i   M l , p1  p ,i  M l , p p1 ,i
p1



(2.10)
Where,
rl,p,i :
Quantity of combination of device l with the removal equipment p recruited in current year in sector i
The performance of a device can also change over time. Average performance
of combination of device l with removal process p on a given parameter in current
year is estimated from the weighted average of performances of its stock passed
on from previous year, its quantity recruited in current year, and the net stock of
this combination that is obtained from exchanges with other combinations of device l in current year. Expressions (2.11), (2.12), (2.13), and (2.14) estimate the
average performance of combination of device l with removal process p on different performance-parameters. Note: expression (2.14) is not used in current version
on the model.
d
m
l , p ,i
 S l , p ,i  d
m
l , p ,i

 m 
 1 

   M l , p p1 ,i   d l , p ,i  rl , p ,i   M l , p1  p ,i 
S l , p ,i 1
p1


 T l ,i  p1


(2.11)


1
E k ,l , p ,i  S l , p ,i  E k ,l , p ,i  S l , p ,i 1
 T l ,i





   M l , p  p1 ,i 
 p1


 E k ,l , p ,i rl , p ,i   E k ,l , p1 ,i  ΔEk ,l , p1  p  M l , p1  p ,i
(2.12)
p1


1
Al , j ,i   S l , p ,i  A l , j ,i   S l , p ,i 1
p
 T l ,i
p


   M l , p  p1 ,i 
 p1


 Al , j ,i  rl , p ,i  Al , j ,i  M l , p1  p ,i
p
(2.13)
p1

1  
Tl ,i   Sl , p ,i  T l ,i   S l , p ,i 1
  T l ,i   rl , p ,i
p
p
p
 T l ,i 
(2.14)
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Part IV. Manual: A Guide to AIM/Enduse Model
Where,
m
d l , p ,i : Emission rate (1- removal ratio) of gas m from combination of device l
with removal process p in sector i in the previous year
m
d l , p ,i :
E k ,l , p ,i
Emission rate (1- removal ratio) of gas m from combination of device l
with removal process p in sector i, for stock of that combination obtained
in the current year from either recruitment or exchange with other combinations.
: Energy use of energy kind k per operating unit (or specific energy input)
of combination of device l with removal process p in sector i in the previous year

E k ,l , p ,i : Energy use of energy kind k per operating unit (or specific energy input)
of combination of device l with removal process p in sector i in the previous year, for stock of that combination recruited in current year.
E
 k ,l , p  p :Energy efficiency change due to exchange of combination of device l
1
with removal process p1 to its combination with removal process p
Al , j ,i : Supply output of service j per operating unit (or specific service output)
of device l in sector i in the previous year

A l , j ,i : Supply output of service j per operating unit (or specific service output)
of device l in sector i, for stock of that combination recruited in the current year.
Tl,i :
Average life of stock of device l in sector i in the current year

T l ,i :
Life span of the recruited equipment l in sector i, for stock of that device
recruited in the current year (Note: This parameter is assumed constant
since equation (2.14) is not used in current version of the model).
Change in average performance of combination of device l with removal process p over time can be calculated by repeatedly calculating expressions (2.11),
(2.12), (2.13), and (2.14) in every year.
2.6 Expressions for Cost
2.6.1 Annualized initial investment cost (or annualized fixed cost or
annualized capital cost)
Annualized initial investment cost as shown in expression (2.15) is used for evaluating recruitment of devices in a given year.
2. Theoretical Formulation of AIM/Enduse
 




x


 Cl , p  rl , p ,i   C l , p1  p  M l , p1  p ,i 
i
j ( l , p )W j 
p1



261
(2.15)



C l , p  Bl , p  (1  SC l , p ) 
 (1   )T l ,i

(2.16)
(1   )T l ,i  1
Where,

Cl , p :
Annualized investment cost of a unit of combination of device l with removal process p

x
C
l , p1  p
:Annualized investment cost of exchanging a unit of combination
( l , p1 ) to ( l, p )

l,p
B
:
Initial investment cost or fixed cost of recruiting one unit of combination
of device l with removal process p
Discount rate
Subsidy rate
α:
SCl,p :
Bl, p is estimated by expressions (2.17) and (2.18).



Bl , p  Bl'  b p''   E k ,l , p ,i
i
(2.17)
k
Ek ,l , p,i  (1  e p )  Ek' ,l ,i
(2.18)
Where,

Bl' :
Initial investment cost or fixed cost of recruiting one unit of energy device l.

''
p
b :
Initial investment cost or fixed cost of removal process p per energy use
of combination of device l with removal process p.
E
'
k ,l ,i
ep :
: Energy use of energy kind k per operating unit of energy device l.
Additional energy use rate of removal process p.
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Part IV. Manual: A Guide to AIM/Enduse Model
2.6.2 Running cost
Running cost in a given year comprises cost of energy used by devices, and cost of
operation and maintenance of devices.
 0

 g l , p ,i   g k ,i  1   k ,l ,i   Ek ,l , p ,i   X l , p ,i
( l , p )W j 
k


(2.19)
Where,
gl0, p,i : Operating cost per unit of combination of device l with removal process p
in sector i
Price of energy kind k in sector i
gk,i :
gl0, p,i is estimated by expressions (2.18) and (2.20)
g l0, p ,i  g l0,i'  g 0p''   Ek ,l , p ,i
(2.20)
k
Where,
g l0,i' :
Operating cost per unit of energy device l in sector i
g 0p'' :
Operating cost per unit of removal process p per energy use of combination of device l with removal process p
2.7 Objective Function
Objective function is the total cost in a given year as shown in expression (2.21).
This comprises total annualized fixed cost (only for recruitments in that year), total running cost, and total cost of emission tax in that year. Decisions for recruitment quantity and operational quantity for all feasible combinations of devices
and removal processes in a given year are made based on the criterion of total
cost.
2. Theoretical Formulation of AIM/Enduse
263





TC     C l , p  rl , p ,i   C x l , p1  p  M l , p1  p ,i
i  ( l , p )W j 
p1






  g l0, p ,i   g k ,i   k ,i   1   k ,l ,i   E k ,l , p ,i   X l , p ,i     im  Qim   min
k


 m

(2.21)
Where,
TC :
Total cost
 k ,i : Tax on energy k in sector i
 im :
Emission tax on gas m in sector i