I
QUANTITATIVE ANALYSIS OF THE STABILITY
OF JAPAN'S ENERGY SYSTEM
by
Tatsuo Oyama
MIT-EL-017WP
August 1986
Quantitative Analysis of the Stability
of Japan's Energy System
Tatsuo Oyama
Graduate School of Policy Sciences
Saitama University, Urawa, Sai tama
Japan
338
Tel: 0488-52-2111
ext. 2748
Acknowledgements
This research was initiated
Researcher at the Economic
when the author
Institute
Research
was
of
Visiting
a
the
Economic
Planning Agency in Jap an in 1983, then was finished while he
was
a
the
Postdoctoral Fello w
at
the
Laboratory
Energy
Massachusetts Institute of Technology
in
in
June,
from
the U.S.A.
1985 to July, 1986.
T he author would like to express his sincere
thanks to Professors
D avid O.
Jeremy F.
the
Shapiro of
Massachusetts
Institu te
Wood of the Energy Laboratory
Re search
Operations
of
Technology
for
Center
their
very
Institute
Research
of
gratefu 1
Tokyo
Institute
and
in
to
Doctors
M.
Japan
S.
Ishikawa
for
their
computational aspects of this research.
Mori
of
of
the
invaluable
suggestions, comments and their sincere co operation.
is also
at
and
The
the
Science
Electronics
assistance
author
Device
with
the
ABSTRACT
In order to measure Japan's energy system's stability under
an uncertain future availability of energy resources, we built
mathematical programming / economic equilibrium model based
linear programming techniques.
Future uncertainty
as random variables with a given
the
economic
equilibrium
probability
point
is
is
upon
expressed
distribution,
obtained
by
a
and
iterative
convergent computation.
Numerical experiments show an optimal
energy
supply-demand
structure with equilibrium prices of primary energy resources
the future target year,
then
instability probabilities of
price analysis of
an
optimal
we
obtain
supply
stability
our
energy
system.
From
solution
our
energy
at
and
shadow
policy
is
quantitatively evaluated.
Key words: Mathematical Programming, Economic Equilibrium
Energy System Stability
Japan
imports
Major imported
almost 90%
energy
resources
oil, coal, and liquefied
her
of
total
are fossil
natural gas
fuels
region
countries
and
politically
and
Considering
the
the oil embargo of 1973-'74,
it
economically unstable
which arose during
many
countries.
that we will not always
be able to obtain
crude
energy
(LNG). These primary
contain
which
as
such
resources are mostly imported from Middle Eastern
the South Pacific
energy.
primary
difficulties
probable
is
amount
an adequate
of
primary energy resources. The oil embargo by the Arab nations had
and
a very serious impact on our country's economic
system. Therefore, whether or not we can
engineering
enough
obtain
primary
energy resources to meet future demand concerns us greatly.
Once we can meet our
demand requirements
stable.
If
we
expected
we will consider
cannot,
it
is
primary
and
that our
unstable.
final
energy
Under
energy
system
is
supply
given
constraints, we can determine whether our energy system is stable
or unstable at some future 'target' period. This
to measure
the stability
of our energy system.
of
Our model analysis expresses the structure
our
energy
[11.
Then
supply and demand system as a network, as in Hoffman
we formulate a linear programming economic
in which the supply availabilities of
(crude
oil
and
coal)
attempts
paper
are
defined
iterative convergent procedure is then
equilibrium
primary
as
energy
random
proposed
economic equilibrium point. The equilibrium
point
optimal energy supply and demand structure in the
1
problem,
resources
variables.
to
An
compute
an
indicates
an
future
target
year. Our mathematical programming / economic equilibrium (MP/EE)
model
is a mathematical
an economic
equilibrium
programming
point
optimization
as
an
optimal
model
that
solution
finds
of
the
linear programming problem.
We define the stability Probability
the probability
that
the
given
of our energy system as
linear
programming
feasible under the 'randomized' supply availability
model
is
constraints.
In addition to these stability probabilities of our energy system
at the target year, distributions
of prices
and demand
of two major primary energy resources (crude oil
and
obtained from the economic equilibrium solutions.
quantities
coal)
are
can
also
We
evaluate our energy conservation policy in various demand sectors
by
combining
shadow
price
analysis
with
the
probabilistic
approach.
Energy
Situation in Japan
Japan consumed 440x101
3
kcal (454x106 kl oil equivalent) of
primary energy in 1983. Annual growth rates of our total
energy supply were above 11% before the
1973,
Japan,
but since
1973,
then they have decreased.
coal had the
replaced
largest
share
until
by oil. The oil share attained
shortly
first
before
decreasing since then;
the
an
first
oil
especially
crisis'
in
Among fuels supplied
to
1962,
but
its highest
embargo.
rapid
after the second 'oil crisis' in 1979. The coal
2
'oil
primary
was
level
It
decrease
share
then
78%, in
has
been
occurred
has
been
increasing
slightly.
Use
of
natural
gas
and
LNG
has
increasing since we started importing LNG in 1968, and
been
presently
their share of the market is about 6.5%. Hydro power's share
about
21% in the 1950's, but
it decreased
to around
6%
in
was
1981.
Nuclear power, which appeared commercially in 1960's, now obtains
almost
the same share as hydro power, which
both coal and nuclear power
consumption
is 6%. We expect that
will
increase
in
the
future, while oil use will decrease.
Japanese industrial energy
high, between 62% and 69% of
industry
consuming
70 -
demand
total
has
energy
historically
demand,
80% of the industrial
first
demand
'oil crisis'.
In
contrast,
has increased steadily
with
share.
energy demand, however, has leveled off at about
60x
to
1950's, although we have seen a slight
13% to 15%). In
decrease
its
the
share
future,
of
increase
industrial
total
since
while
in
1981.
since
recently
energy
demand,
the
energy
25%
Transportation energy demand has been almost constant
heavy
Industrial
residential-commercial
from 17% in 1960
been
demand
the
(from
should
residential
and
commercial demand will increase.
The
domestic
constant at around
energy
5 0xlO
3
supply
in
Japan
kcal (53x106 kl
has
oil equivalent)
the 1950's, and it has consisted
mainly of hydro
power
and domestic
coal.
Our domestic
energy
of the total primary energy supply, a lot
countries'
for Great
domestic ratios,
Britain,
almost
been
power,
supply
less
than
since
nuclear
is about
other
western
e.g. 85X for the United States,
49% for West
Germany,
3
and
25%
for
10%
48%
France.
Crude oil comprises
around
65%
consumption, which is totally
of
our
imported
total
from
primary
foreign
energy
countries
mostly in the Middle East and the South Pacific.
The oil embargo
in 1973 induced a large price
crude oil from $2.51/bbl
for
serious effects
in 1972 to $10.79/bbl
both
the
world
supply-demand system. After this
economy
first
'oil
energy consumption has stopped increasing
decreased
by 4.4% from 1973
the second
Furthermore,
'oil crisis',
Crude oil prices
$32.97/bbl in 1980.
from
1979
decreased
to
rose
Total
by 3.4% during
Considering
events
the crude oil supply
crisis',
a
energy
our
primary
large
$13.77/bbl
the
price
1978
to
by
10.7%
consumption
also
Through
prevailed
in
decreased
energy
1975.
from
resulted
second
imports
primary
in
these two 'oil
our
industrial
on crude oil.
in the last 10 years
or so, we know that
to our country can be greatly
international political
countries.
which
from
has
had
Our energy system has structurally changed and no
longer depends as heavily
the
our
crisis',
the same period.
crises', energy conservation
demand sector.
and
It
of
as rapidly. Oil imports
caused
Our crude oil
1980.
1974.
to 1974, and 4.8% from 1974 to
Iranian revolution in 1978, also
increase.
in
increase
It is probable
that
we
will
in
induced by a disruption
unexpected happening in these
of
situations
oil
have
yet
supplies
countries. The
influenced
oil
by
exporting
another
'oil
due
some
crisis
may
to
effect
our energy system both physically and economically. Therefore, it
would be very
important
for
us
to
4
quantitatively
evaluate
our
energy system's stability under various levels of primary
energy
supply constraints. The 'stability' of our energy system is fully
dependent
on the possibility
of importing
Our energy system can be determined
crude
to
be
oil
stable
corresponding to whether or not we can have a
and
coal.
unstable
or
sufficient
supply
of primary energy resources to meet our future energy demand.
By
defining the supply stability probability of our energy system as
the probability that our mathematical
has a feasible solution,
the
programming
supply
stability
energy
of
model
our
energy
system can be quantitatively investigated.
MP/EE Energy Model
Energy System and Linear Programming Model
Our energy system, which involves energy flow
supply
regions
to final
demand
sectors,
is
from
illustrated
network system in Figure 1. The supply sector consists
divisions, including five supply
and
stockpile-transfer.
hydro-nuclear,
crude
Four
oil,
regions,
kinds
coal,
and
transformed into petroleum products,
energy of electricity
and
consists of four
categories:
city
of
various
as
of
a
seven
domestic
production,
primary
energy
LNG-natural
gas
of
are
coal products, and secondary
gas.
The
industry,
final
demand
sector
residential-commercial,
transportation, and stockpile-transfer.
A feasible
future
energy
energy
demand
flow in the network
under
various
5
has
supply
to
satisfy
constraints,
the
and
physical and engineering constraints. The
arcs of the network correspond
energy
flows
to unknown variables
and network constraints are linear
equalities
on
the
of the model,
and
inequalities
using those variables. Thus the problem of finding an equilibrium
energy flow can be formulated as a linear programming problem.
The goal is to
corresponding
to
obtain
an
a
desirable
economic
feasible
equilibrium
energy
point
flow
in
our
mathematical programming energy model. A desirable energy flow in
the network
energy system
of Figure 1 can be determined
as a flow
attaining a maximum economic surplus criterion; i.e., minimizing
supply
cost
less demand cost. One can obtain
flow by solving linear programming
problems
an
optimal
iteratively,
energy
until
satisfying a convergence criterion.
Structure of the MP/EE Energy Model
In the MP/EE energy
variables (xi, yj,
correspond
the remaining
there are five kinds of endogenous
wi, dij). Four kinds of them (xi,yj,Zk,WI)
Zk,
to energy
model
flows, as in the network of Figure
endogenous
variables
(dij)
indicate
1,
while
"flexible"
demands for imported primary energies of crude oil and coal.
xi,
iM
= (1,...,17) : primary
energy
transported
from
the
supply regions and the stockpile-transfer
node
yj, jeN = {1,...,14)
: primary
energy
transformed
into petroleum
products, coal products, or secondary
6
energy; primary energy directly consumed
in demand sector
kK
Zk,
= {1,...,28) : petroleum and coal products transformed
into secondary
energy
or consumed
in
demand sector
:
wi, laL = {1,...,5}
energy consumed
secondary
in final demand
sector
dj,
= {R,L}, jJ
iI
= { ±1, ±2,..., ±6)
variables indicating the perturbation of
primary energy resources' (R:crude oil
L:coal) demand from their standard demands
Using
the
programming
above
variables,
model are expressed
constraints
in
the
linear
as follows:
(1) Availability Constraints of Primary Energy Resources
In the energy network of Figure 1, primary energy
enter the system through supply
nodes.
amount
The
resources
of
primary
energy at each supply node has an upper bound determined
by
the
physical, economical or, sometimes,
in
the
political situations
supply regions. The physical availability of each primary
energy
resource from each supply region is given as follows:
xi
(1)
isM.
b,
(2) Flow Conservation Constraints
The set of nodes
in the network
is divided
into three groups
supply nodes corresponding to supply regions, demand nodes
indicating demand
sectors
intermediate nodes.
At
from
each
p=13
to
p=16,
intermediate
7
node
and
remaining
pe{1,...,12},
in-flow has to be equal
Z
x
Z
yj =
z
yj +
iEMp
jENp
jENp
=
jENp
kaKp
kEKp
to out-flow. Therefore,
yj
p=1,2,3
(2)
Zk
p=4,...,10
(3)
p=11,12
(4)
Zk = Z
leLp
WI
(3) Upper and Lower Bounding Capacity Constraints
Upper
and
lower
bounds
are
given
for
the
variables
indicating production of petroleum and coal products
{Zk,
and consumption
laL).
constraints
of electricity
are written
and
city
gas
These
as follows:
LBZk
Zk
UBZk,
kKI'CK
(5)
LBWIl
Wi
UBW1,
1EL
(6)
where LBZk, UBZk, LBWi and UBWi are lower
{Zk)
({w,
kK'CK},
and
upper
and ({w), respectively. K' is a proper subset
constraints (5) are given for some variables of
of
bounds
of
K,
hence
{Zk).
(4) Yield Constraints of Petroleum Products
Refinery systems have their
own
restrictions regarding the yields
of
physical
petroleum
and
engineering
products.
Each
petroleum product has both lower and upper production bounds.
yLjC
yj
YUjC
3s;jS6
C = xI + X2 + X4 + X7 + X
+ X12 + X 5 - Y2
where yLj
and yuj are the lower and upper
petroleum
product
indicated
(7)
bounds
of the yield
by yj, and C is the total
entering the refineries.
8
crude
of
oil
(5) Demand Requirement Constraints for
Imported
Primary
Energy
Resources
Imports of primary energy resources are
restricted
by
the
following constraint:
Z
xi
Dk
i6Mk
where J
and J-
+
Z
jaJs
dkj -
Z
jaJ-
dkj
ke(R,L)
(8)
indicate sets of positive and negative indices of
J=(±l, ±2,..., ±6),
respectively.
The
inequality expresses the flow of
left
each
side
primary
of
the
energy
above
resource
from each supply region to Japan, while the right side
expresses
the perturbed demand of each primary
from
standard demand. The
variable
energy
(dkj),
with
resource
the
the
superscript
k
deleted, is illustrated in Figure 2, where the demand quantity De
corresponds
to the standard
Each variable
interval
dkj
has
an
demand
upper
Dke
in
bound
the
constraint
corresponding
(8).
to
the
in Figure 2.
0
dk j
k6(R,L),jJ.
k jA,
(9)
(6) Final Energy Demand Requirement Constraints
In
the
four
demand
residential-commercial,
nodes
corresponding
transportation
and
to
industry,
stockpile-transfer,
the following final energy demand requirement constraints have to
be satisfied:
Z
jeNp
yj
+
C
keKp
Zk +
C
leLp
wi
9
Dp
p=13,...,16
(10)
(7) Objective Function
The total supply
defined
cost of the energy system
as the sum of fuel costs
region and transporting
in each supply
the CIF cost. The transformation
i.e.,
Figure
transformation
the fuel cost to be the cost
this model, we define
the resource
and
in
1
is
costs.
In
of
obtaining
it to
cost is defined
Japan,
the
to be
cost for transforming primary energy resources into petroleum and
of
coal products, electricity, and city gas, and consists mainly
the capital
fuel
cost necessary
for
energy
transformation.
cost per thermal unit (1010 kcal) be ci, iM,
transformation
be dj, jN.
The transformation
is given by e,
written
cost per unit kcal of petroleum
the
let
the
and
and coal
products
cost per kcal of secondary
IeL. Then the total energy
supply
cost
energy
can
be
as follows:
Z cixi + Z djyj
isM
jeN
However,
the demand
total cost for meeting
+
cost of the energy system,
energy demand,
a forecast
a step function
g(q)dq =
jEJQ
(11)
Z elw.
lsL
approximation to the area under the nonincreasing
By using
Let
in Figure
defined
is given
demand
as
as
+
jeJQ
an
curve.
2, the cost is
.Pjdj
(12)
dj,
(13)
where
Q = Q
a
10
and
J
is a set
contained
of
indices
to the demand
from (12) the constant
of
corresponding
from 0 to Q,
in the range
corresponding
{j}
in
demand
and Pj is a
the
j-th
in
the
Figure
intervals
commodity
interval.
cost corresponding
the demand curve from 0 to Q
following
to
Subtracting
to the
2,
price
integral
obtain
we
the
sum:
f(q)dq =
*Go
where sgn(j)
indicates
sgn(j)
Z
jJoJ
sgn(j)Pjdj
(14)
the sign of index j (j*O), i.e.,
= 1
if j>0
if j<O.
-1
Thus adding (12) for each iI={R,L),
our objective
function
can
be given as follows:
Minimize
Z cixi
ieM
+
jeN
djyj
+
leL
ewi -
Z . sgn(j)Pjdij.
i IjJ
The negative of the above objective function can
be
(15)
interpreted
as maximizing the demand cost less supply cost, while meeting the
future energy demand requirements. Hence the optimization problem
corresponds to finding the economic equilibrium point
economic surplus.
11
maximizing
The probabilistic
aspects
of
our
energy
model are
as
follows: many energy supplying countries are somewhat politically
and
economically
availabilities
to the
unstable.
Hence,
right
hand side values bi in
denoted by b
q,
distributions
bin,
and
which
correspond
random
variables.
availability
of
some
and
parameters
the
random
lower
bounds
are
are
integers
b
variable
has
p
the
densityfunction:
(b-bm)
f(b)
upper
whose
Then
supply
overseas supply region to Japan
whose
and
respectively.
followingprobability
are
bound of the total
primary energy resources from the
follow beta
(1)
that
assume
of the primary energy resources
Suppose that the upper
and
we
P
=
-
(b-b)
q-
1
bm
,
K
where K is a constant. When parameters
b
p and q are
b
integers,
(16)
K is
given by
K(p))r(q)
r(p+q)
=(bM-bm)P+q- t
(p+q-1)
t
12
(17)
where r(-)
is a Gamma function
co
r(p)
Suppose
e-xxP
=
parameters
-
ldx.
p>l and q>1, then
p and q satisfy
variable b has the following mean ,
variance
2,
the
and mode m.
bmq-+bp
A =
random
(18a)
p + q
C2
pq(bM - bm)
(p+q)
2
2 (p+q-1)
2
(18c)
bm(q-l) + b(p-l)
m=
(18c)
p+q-2
Computational Method
Our energy
formulated
model described
in a vector-matrix
to
form as follows:
(19)
A2x = b2
(20b)
A3X 2 b3 + Kd
(20c)
d
(20d)
5 b4
>
(20e)
0
where the unknown variable
yj,
be
(20a)
x,
(xi,
can
Aix : bi
d
variables
section
cx - pd
Minimize
subject
in the previous
vectors
Zk, Wi) and (dij),
13
x and d consist
respectively.
Here,
of
the
p and d
are price and commodity vectors, whose elements are given by
and sgn(j)d'j
(20c)
are
in (14), respectively.
the
resource
Constraints
availability
(20a),
constraints,
Pj
(20b) and
balancing
equations and demand requirement constraints, respectively. (20d)
indicates
the
variables
(dk j ).
bounding
constraints
in
(9)
for
the
demand
We define our resource supply cost minimization submodel
as
follows:
csxs
Minimize
subject
to
A s ix
(21)
s
bsi
(22a)
AS2Xs = bS2
(2 2b)
AS3Xs
(2 2c)
L bS3
(2 2d)
Xs a 0
A 5 i,
where
bSi
submatrices and
cS
i=1,2,3,
subvectors
of
the
c and x in the original
i=1,2,3,
by (19)-(20).
imply
for
that
The
our
submatrices
resource
and
xs
are,
corresponding
respecti vely,
Ai,
b;
for
linear programming problem given
and
supply
subvectors
submodel
described
consists
components related to crude oil and coal only,
above
of
the
than
the
soluti on,
for
rather
whole supply model.
Let
the
th e shadow price,
demand
resource
requirement
ie I be
i.e., the optimal dual
constraint
r'i. Then the
(22c)
equil i brating
of
primary
condition
energy
for
our
MP/EE energ:y model can be written as follows:
w'i = gi(Di'
+
-
Z d'j
jEJ+
14
Z d i 'j)
jeJ -
(23)
where gi indicates
the i-th
component
of
the
demand
g(q), corresponding to the primary energy resource iI.
function
Die
and
di'j indicate the standard energy demand for the resource iIl and
an optimal solution for the demand variable dj,
respec tively.
The optimization problem given by (19)-(20)
surplus
maximization
problem.
A
general
problem cannot always be transformed
into
maximization problem.
the
In order
possible, the demand function
e.g.
Hurwicz [153).
demand function has
for
to
be
the
an
economic
equilibrium
economic
surplus
transformation
be
i.e.,
to
integrable
Jacobian
symmetric,
an
market
g(q) needs to
Therefore,
is
matrix
the
be
(see
of
cross
the
price
elasticities between two different commodities must be symmetric.
In our energy model, cross price elasticities
commodities
were assumed
between
different
to be zero, and thus our Jacobian
matrix
is symmetric.
Let us look at a computational
economic equilibrium
point
in
procedure
our
energy
for
obtaining
model.
Firstly,
sequence of random numbers with a beta distribution (beta
assigned
and
to the corresponding
by (1). Then the linear
each
pair
of
these
numbers
right side in the constraints
programming
solved to obtain an optimal energy
economic
flow
equilibrium
meeting
demand. The computational method in our MP/EE model
a
random
numbers) are generated. Two sequences of beta random numbers
generated simultaneously,
an
are
is
given
model is
future
energy
analysis
is
presented in the flow chart of Figure 3. Solving our MP/EE energy
15
a
is
model iteratively
part
principal
algorithm are given
details of the solution
our
of
in Figure 4.
The correcting process at the t-th iteration, given
energy prices pt
and supply costs c,
1
pt+I
2
= C
Cst+l
+
('t
+
The
analysis.
primary
as follows:
is written
t)
(24)
Cst
(25)
where
AcSt = A(*Ot
- cS),
0
A
1.
(26)
of this iterativ e computation
The convergence
is
attained
when
the shadow price of each primary energy resource of crude oil and
coal equals that
corresponding
obtained
to optimal
the
from
approximate
demand
curve
commodi ty demand.
Numerical Results
Assumptions and Input Data
We define
the year 1983 as the base year, and then
the year 1990 as our
future
target.
average annual growth
rates of final energy
we
Firstly,
demand
look
assume
between
base year and the target year are 2.0%, 3.0%, 2.0% and
industry, residential commercial, transportation
transfer, respectively.
Final energy demands
given in Table 1. Supplies of primary energy
16
and
2.0%
at
that
the
for
stockpile-
in 1983 and 1990 are
resources
in
the
base year and the target year are
shown
Table, Other Middle East Region denotes
Table
in
2.
In
Middle
o il-exporting
the
the
East countries excluding Saudi Arabia, i.the. U,Iran, Iraq, Bahrein,
U nited
Kuwait, Neutral Region, Qatar, Oman and the
The South Region consists mainly of
such as Indonesia,
crude
oil
includes
and Nigeria.
Algeria
The
The
Region
Other
Other
coal- exporting
countries
for
such
countries
oil-exporting
African
Pacific
Southern
Brunei and A ustralia.
Arab Emirates.
as
incl udes
Region
South Africa, China, and Soviet Union.
Upper bound availabilities of primary
1990 are estimated
from the Middle East
import
In
estimating
upper
bound
upper
based
is
of 4.0% between
increase
1990.
as follows. The
resources
energy
upon
for
an
for
coal
1
year
target
from
import
oil
an nua
average
se year 1983 and the
bounds
crude
in
the
Southern Region, North-South America and Other Region, an ave:rage
annual
increase
Estimates
4.0% is assumed.
from the Other Middle East Region,
crude oil
and
LNG
North-South America Region, crude
oil
the
Other
all based
on the average annual
LNG
from
the
Reg ion,
gas and stockpile-transfer
crude oil, coal, natural
domestic
from
of
supplies
for
increase
1983 to 1990. The upper bound availability
are
2.0 - 4.0%
from
of hydro-nuclear
p ower
rates
is estimated according to an average annual increase of 4.0 - 5.0
% from the base year.
CIF prices of primary energy resources from
regions
in 1983
in Table 3.
and their estimates
Crude oil prices
for the year 1990
in 1983 indicate
17
various
are
'average'
su pply
g iven
pr ices
in oil-exporting countries in the region. For example, the
oil price
in Saudi Arabia
is that of Arabian
light,
price in Other Middle East Region is based
Emirates Murban.
Prices
in
the
on
Southern
the
Blend,
from 1983 data by assuming an average annual
except that the increase
oil
Arab
North-South
Indonesian
Sumatra
respectively.
Crude oil price estimates for the target year 1990
4.0%,
the
United
Region,
America Region and Other Region are those of
Light, Mexican Isthmus and Algerian Sahara
and
crude
are
price
obtained
increase
rate is 5.0% for Other
as
Middle East
Region.
Coal prices
material
in 1983 are the weighted
mean of steam
coal from each supply region. Coal prices
Region and North-South America
Australian
and
the
Region's coal price
United
is
Region
States,
the
weighted
are
based
annual
Natural gas and LNG prices
mean
of
increase
South Region, and Alaskan for
on
coal
South
of
Other
African,
prices
in
of 5.0% from 1983.
and
North-South
and
those
The
South
in 1983 are those
LNG for the Middle East Region, Brunei
the
respectively.
Chinese and Russian coals. Estimates for future
1990 are based on an average
in
coal
of
Abu
Indonesian
America
Dhabi
for
the
Region.
LNG
price estimates for the target year 1990 are obtained
from
1983
data by assuming an average annual increase of 4.0x.
Transformation costs for petroleum products, coal
and secondary energy (electricity and
Table
4.
Costs
for fuel oil,
city
kerosine-gas
naphtha are weighted means minus fuel costs.
18
products,
gas)
are
given
in
oil,
and
gasoline-
The other petroleum
products' transformation cost is basically the LPG price, and the
coke transformation cost is the coke
price
minus
coal price. Electricity transformation costs
for
the
material
both
industry
and transportation are the capital costs in the electricity
for industry, and those
for residential-commercial
capital costs in the rate for residences.
costs are the capital
costs
in the
are
rate
also
the
City gas transformation
industrial
and
residential-
commercial city gas rates.
Upper and lower bounds
respect
to variables
for
constraints
(5)
and wi, are presented
Zk
Lower bounds for petroleum
those whose consumption
and coal products
is relatively
small,
and
(6)
in Tables 5 and 6.
are either
the consumption
0.0
for
or the amount of the
base year's consumption when they are large. The upper
either
with
of the base year or a 50%
bound
increase
added
when they are relatively small. The average annual increase
is assumed from 1983 to
large.
Lower
bounds
1990
for
for
those
electricity
whose
and
4.0%
consumption
city
gas
is
is
are
the
consumption in the base year. Upper bounds are obtained from
the
base year's consumption by assuming an average annual increase of
5.0%.
The upper and lower bounds for
given
in Table 7
are based
petroleum
on the assumptions
products'
that
yields
demand
for
light petroleum products such as kerosine, gas oil, gasoline
naphtha will increase
in
the
future,
while
demand
for
and
heavy
petroleum products such as heavy fuel oil will decrease.
The main sources
of Japanese
19
energy data used
in
our
model
analysis are Energy Statistics [23], Handbook of
Industry [7], Industrial Statistics
Statistics
and
bm and b
the
for
maximum,
estimates for the future
Parameters
[24],
and
Power
Petroleum
[8].
Parameters
minimum
Table
Electric
the
availability
future
distribution
respectively,
p and q are determined
equal to the expected
beta
indicating
of
crude
oil
extreme
and
coal.
so that mean values are
availability
of
these
the
are
nearly
resources.
These parameters are presented in Table 8.
Beta random numbers are generated by
transformation method to uniformly
applying
distributed
the
inverse
random
numbers.
Random numbers following uniform distribution between 0 and 1 are
generated by using the square method.
random number generation,
see e.g.
(For more
Fishman
information
[93, Bratley,
et
on
al
[3].)
Approximate demand curves for imported crude
are based upon their
price elasticity
own and cross
price
oil
elastic ity
ij, i,je{R:crude oil, L:coal),
and
coal
The
data.
represents
the
decrease (%) of commodity i's demand corresponding to a unit
commodity j's price increase. According
analysis
in Oyama
[22],
own
and
cross
primary energy resources in 1980 are
0.69,
LL=-0. 7 4 .
Hence
that in Japan crude
of
RL
price
ERR=-0.07,
oil is rather price
and
the
translog
e
are substitutes
ERL=O.04 ,
LR.
20
can
insensitiv e compared
each
mlode1
lasticitie:
from the above data on s;j 's we
coal, and these resources
positivity
to
of
o ther
from
of
L R=
say
with
the
An Optimal Energy Supply
and Demand Structure
We wrote a FORTRAN computer program
model.
to
our
analyze
The program consists of nearly 3800 statements,
which (around 80X) comprise the product form simplex
solving the linear programming problem.
histgrams
and
(including 17 xi's, 14 yj's, 28 Zk'S,
variables)
and
51
a second
output
5
contains
of CPU time on
requiring about 140 pivots if
for
random
formatting
121
we
variables
dij's
and
33
(excluding
bounding
is
obtained
for each iteration
IBM
3033
compu ter
system,
start
from
phase
of
the
simplex technique. In order to
24
wi'S,
constraints
constraints). An optimal solution
within
method
of
and so on.
The linear programming MP/EE model
slack
most
Others relate to
number generation, iterative procedures,
for figures,
MP/EE
obtain
an
1
the
eq ui 1 ibrium
economic
point for each pair of resource availability constraints
,
necessary to solve 4 - 6 linear programming problems
ternating
al
between the MP/EE energy model and the supply submode 1
Since
the latter model
Hence,
solving
is rather
simple
the MP/EE energy model
time. We generated
it
problems.
it can be sol ved quickly.
takes up most
250 pairs of beta random numbers.
of
the
So,
a
CPU
total
of 250 cases of these economic equilibrium problems are solved
about
is
in
12 CPU minutes by the IBM 3033 system.
An optimal
Tables
solution
9 and 10. Total
and 72.016x101
3
for one of these 250 cases is
import availabilities
kcal for crude oil and
coal,
shown
are 266.137x10
respectively.
1 3
in
kcal
The
obtained economic equilibrium point implies an optimal supply and
21
resource
supply
total
primary
energy
be
483.6 7 x10'
demand structure under the given primary energy
constraints.
As presented
supply to Japan
in
in Tables
1990
9 and 10, the
is
expected
I
equivalent),
(5 14.54 xl 0 6 kl oil
from 1983 to 1990.
in crease
a 22%
1990
ar e
while those shares
coal and hy dro-nuclear should increase by around 3%,
crude
consumes
oi 1 will be unchanged.
al 1 the available
t hat
are
role in meeting
Equilibrium pric es of crude oil
and
Yen/10 4 kcal
and
iterations.
The se equilibrium
Yen/104 kcal,
and
coal
the
Japan
I
n
m
odel 's
and
energy
are
given
respectively,
for
that
th at
first,
chosen
LNG-natural gas plays a marginal
24.99
wh ile
implies
from cost criteria
crude oil and coal
1983
in
crude oil and coal supplies.
MP/EE model we can recognize
structure
The solution
gas
59.4%,
The sh are s
were 59.2%, 18.9%, 8.2% and 13.7%, respectively.
of
kcal
LNG-natura 1
in Japan in the year
3.1% and 15.5%, respectively,
3
during seven years
coal,
Shares of crude oil,
and hydro-n uclear supplies
22.0%,
to
our
then
demand.
by
69.38
six
after
prices show that crude oil
prices
rise around 44% from 1983 to 1990, while coal prices go up by 26%
during
the same period.
In the year 1990,
production
ratios of fuel
oil,
kerosine-
gas oil, naphtha-gasoline and other petroleum products are 43.6%,
19.7%,
24.1% and 12.6%, respectively.
that principal
heavy
volume
of our refinery
fuel oil to light gasoline
that our refinery
From the solution
output
and kerosine.
system will be adjusted
22
is
know
changing
Thus,
to meet
we
a
we
from
expect
demand
for
more
light petroleum
products
and less heavy
in
ones
the
near
future.
We can
also
see
consumption
of
secondary energy of electricity and city gas increases about
40%
from 1983 to 1990.
especially
in
These
our
shadow
constraints (10), we
has
a
and
that
prices
for
that
shadow
price
and
will play major
energy
the
demand
industrial
transportation
shadow prices
transportation
than the
demand
industrial
former demand sectors consume
in
the
the
constraints
The
expensive
demand
and
demand
This
(electricity and city gas) than the latter.
tell us that energy conservation
energy
residential-
constraints
one.
more
requirement
Yen/lO04 kcal
59.51
4 kcal.
both have shadow prices of 90.45 Yen/lO0
commercial and
roles
commercial demand sectors.
note
residential- commercial
results
energy resources
residential
Looking at
constraint
from
have
is
higher
because
energy
the
resources
These shadow
prices
residential-commercial
demand sector is almost 50% more effective than that in industry,
from the point of total energy system cost reduction.
Let us look at shadow prices
constraints
given by (1).
constraint of the
decrease
corresponding to a unit
supply availability.
the
The shadow price
imported
interpreted as the
for
resource
of
the
increase
We know
that
from
formula:
23
7r
(104
region
kcal)
shadow
the
ieM
supply
can
function
of
price
supply
availability
for
objective
resource availability constraint for the
given by the following
supply
the
7i
region
be
value
resource
from
the
iM
is
xi =
+ (EP - CSi) + CT,
isM.
In the above formula, which holds for
resource, 7a indicates
the
shadow
(27)
each
primary
energy
for
domestic
energy
price
production constraint (i.e., the price for the stockpile-transfer
constraint
is
equal
to
The
,W).
value
Yen/10 4 kcal for crude oil and 59.505 Yen/10
an equilibrium price of the
primary
4
of
energy
of primary energy
constant
respectively.
resource
term given
CSi
57.495
resources
EP is
obtained
Yen/104 kcal
24.992
is the respective
from each supply
region
for each primary energy
1.216 Yen/l0 4 kcal and 0.0 Yen/10 4 kcal for
is
kcal for coal.
from the model, i.e., 69.379 Yen/104 kcal and
for crude oil and coal,
xn
ieM.
resource,
crude
cost
CT is
that
oil
and
a
is,
coal,
respectively.
As mentioned
as
a
"benefit"
before,
the shadow price
obtained
from
increasing
resource availability by a unit amount (104
can conclude that a unit amount (104
oil or coal availability
i can be
the
kcal).
interpreted
corresponding
Therefore
kcal) of increase
can basically
produce
The
why coal is a little more beneficial than crude oil is
directly
latter needs to be transformed
to the demand
to other types
petroleum products, electricity and city gas.
24
crude
a benefit of
Yen or 59.5 Yen, respectively, to our energy system.
former can be transfered
of
we
57.5
reason
that
the
sector, while
the
of
energy,
i.e.,
Stability Probability and EquiliIbrium Prices
As shown
in Figure
3, we
N(=250)
so lved our MP/EE model for
cases. For some combinations of beta random numbers the model may
be infeasible
insufficient
since either crude oil
or
coal
supplies
be
pair
of
to meet our future final energy demand.
Figure 5 shows model feasibility results for
beta
may
random
numbers.
In
Figure
5,
the
each
vertical
coordi nate
indicates the availability of imported coal, while the horizo ntal
coordinate indicates the availability of imported crude oil.
For
each combination of these energy resources' availability, an F or
I indicates
whether
Let the number
Ni. Then we define
energy
the model is feasible
of infeasible
or infeasible.
cases among total
the "supply stability
N
probability"
system by the ratio of the feasible
cases
cases
be
(Ps) of our
to
the
t otal
number of cases.
Ps =
N -
Ni
(28)
N
Our energy
probability
system
can be
understood
Ps and "unstable" with
Ps and 1-Ps as stability
and
to
be
"stable"
the probability
instability
with
1-Ps. We
probabilities
the
call
of
our
energy system, respectively.
Our
numerical
experiments
show
that
the
stability
instability probabilities of Japan's energy system in the
year 1990 can be presented
as follows:
25
and
target
205
45
1-Ps =
= 0.82,
=
Ps
250
We should consider
that
an
extremely
probability
the instability
difficult
of 0.18 unless
- = 0.18.
250
probability
situation
our
(29)
energy
implication
as an
may
occur
system
with
the
structurally
is
changed. An infeasibility result may be changed into a "feasible"
case by adding more infrastructure to our energy
sy stem,
transforming our energy system into a more flexible one
it
can
meet
variable
final
energy
by
demands
or
so
by
that
promoting
substitution among primary and secondary energy reso urces.
Let us examine
objective
function
the
'stable'
values
for
cases
these
in
more
detail.
cas es
(N-NI)
The
have
the
frequency distribution shown in Figure 6. This distr ibution shows
rather higher
This
frequencies
in the lower part of the
cost
is because the objective function value is dominated
cost of crude.oil,
so if crude oil is abundant
low, the objective function value
imported
crude oil availability
is
rather
and its
range.
by
price
'stable',
of crude oil prices
The above
availability
upper
the
argument
has a distribution
range, so its price
lower
range,
as
also
be
with its peak
Figure
26
8.
Coal
is
a
cases.
applied
will be distributed
in
if
is low, its price goes up rapidly
in the feasible
can
is
but
and objective function value becomes much higher. Figure 7
histgram
the
to
coal.
frequency
with
has
its
in
peak
higher
Coal
the
in
own
price elasticities than crude oil, that
availability
solution,
of
imported
the amount
of
crude
coal
oil
used
is,
dominates
is
Since
IELLI>IRRI.
the
subject
to
optimal
crude
availability. Furthermore since the own price elasticity of
is rather
large, the price
extremity,
thus splitting
prices,
of coal can
the
frequency
quickly
to
distribution
coal
either
of
coal
as in Figure 8.
Figure 9 is obtained
in Figure
5 with the
crude oil and coal.
by combining
equilibrium
into nine parts
equilibrium
the
prices'
feasibility
results
The figure first divides
into feasible and infeasible
divided
move
oil
areas.
depending
prices PR and PL.
Then
on
of
crude
this
imported
whole
the feasible
the
Note that
the
results
region
region
oil
and
division
is
coal
of
the
feasible region is not very accurate, since the equilibrium price
of each primary energy resource can vary depending on the
availability
of the other.
However,
this partitioning
supply
helps
us
to know approximately how much each energy resource's equilibrium
price
will be changed by the degrees of supply availability.
Summary
During
the years
following
the Arab
oil
embargo
of
1973,
there have been many energy policy debates throughout the world,
including Japan. Energy policy debates concern various technical,
environmental, social, economical, political
problems. Energy policy modeling efforts have
27
and
even
increased
military
due
to
not only the necessity
of such
interdisciplinary
also the greater availability of
Hoffman
[11
supply-demand
proposed
energy
high
energy
network
problems,
approaches have been developed.
[19, 20, 21], Modiano
computers.
systems
various
(See e.g.
Manne, et al [17] for energy models.
Oyama
speed
Also
and Shapiro
research,
Since
analyses
systems
see Shapiro
and
for
analysis
Charpentier
[18],
but
[4]
and
[26,
28],
Shapiro
and
White [29].) We investigated the Japanese electric
power
(see Energy Study Group
to
see
what
our energy supply and demand situation will be like in
the
year
[6], Saito and Oyama
[25])
system
2000.
Many economic equilibrium models have
also
developed
been
which use linear and nonl inear programming techniques.
Kennedy
[16], Hogan
Daniel and Goldberg
[12], Griffin
[5].
[10], Hogan
See also Takayama
price and resource allocation
models.)
and
and
Furthermore, Shapiro [277]
Kuhn-Tucker optimality
presents
conditions
Shapiro
[141,
[30]
for
disc usses
[27]
between
1 inear
of energy planning moidels.
the
as
Weyant
Judge
decomposition techniques to show the relationship
programming and economet ric components
(See e.g.
interpretation
an
economic
of
the
equili brium
point for certain matheniatical programming models.
However,
in most
of
these modeling
studies,
demands and the availabi lity
of
given exogenously.
that our future energy
We believe
primary
energy
future
e nergy
resources
demand
are
and
the availability of primary energy resources should be uncertain.
In this
analysis,
we
have
28
investigated
the
effects
of
constraints on Japan's
primary energy resources' supply
system.
It
is
generally
resource 's availability
that
true
when
a
primary
its commodity
is high,
price
and vice versa. However, since we have considered
primary energy resources simultan eously,
el asticities
price
resources
are
supply
analyse.
a re
constraints
Through
our
MP/EE
a
mo del
energy
goes
two
down
kinds
their
furthermore
the
price
effects
of
more
complex
to
econ
omic
little
we
obtain
an
equilibr ium point for each energy resource as shown in Figure
However, the
splitting
based
regions
complica ted
on
because
the comp utational
of
feasible
the
commodity
pr,ices,
as
region
in
into
the
the equilibri um points may
figu
be
re,
depen,
dent
Hogan
[12], and
its
main
framework,
procedur'es and some assumptions
are different.
assumed
to be
supply mapping
our model
mappings.
existence
MP/EE
Hogan,
et
al
[133;
and Ahn and Hogan [2] for co nvergence arguments for
in
In
the
PIES
continuo us ly
except
on
that
model,
the
special
function
and
the
while we
existence
solution.
29
for
functions were po int-to-set
and the uniqu enless of an equilibrium
the
was
(inverse)
as sumed
The assumptionis of the PIES model guarantee
only
also Ahn
i terative
the
demand
differentiable,
was a po in t-to-set mapping,
assumes
the
a bout supply and demand functions
that both supzpl y and demand
model
is
method and its convergence criteria.
PIES mod.el (see e.g.
cases)
9.
sma 11er
Our computational technique is fundamentally similar to
[1],
of
and
dif ferent,
very
energy
point,
of
an
both
the
while
our
equilibrium
Our iterative computational method worked very well, and
could obtain an equilibrium point after
all
feasible
cases.
computational
convergence
Although
the
method is not given
is guaranteed
several
convergence
in this paper,
by showing
the
fact
iterations
for
proof
our
for
we
believe
that
the
the
shadow
price of the demand requirement constraint (25c) is expressed
the
approximate
demand
function
value
we
corresponding
to
by
the
optimal resource demand. We are presently working on this proof.
Approximating a demand function is another problem. In
paper we assumed the existence of nonzero own price
for
primary
energy
resources
only,
neglecting
this
elasticities
cross
price
elasticities between two distinct primary energy resources.
Both
own and cross price elasticities can be simultaneously considered
in our model analysis
K
matrix
price
of
by incorporating
(20c).
elasticities
The
does
this information
consideration
not
make
of
solving
into
nonzero
the
the
cross
problem
more
difficult, but rather changes the problem formulation slightly by
adding more nonzero elements in the coefficient matrix.
Applying
decomposition techniques should also be very effective in solving
our MP/EE model
in this case.
The stability probability was defined to be the
that the MP/EE model was feasible.
random
obtained
numbers
the
as
our
stability
import
and
We tried a single
supply
instability
probability
sequence
availability, and
probabilities
of
of
then
our
energy system. We know that if the substitutability between crude
oil and coal increases,
then the stability probability
30
will
also
increase,
since there are more ways to meet
the
forecast
energy
demand.
We can conclude
that the Japanese
energy system
more flexible, so that it can structurally
primary energy supply availability.
For
adjust
example,
needs to
variations
the
(from 1979 to 1980).
In another good example,
of
Japanese
cement industry changed almost totally from fuel oil to
one year
be
coal
our
in
power
industry is introducing mixed fuel thermal power plants consuming
fuel oil, coal and LNG.
If our energy system were well
coal,
it will greatly
heighten
organized
the stability
primary energy supply, and also lower
cost. We would not have
the
to
consume
probability
total
to depend so heavily
more
of
energy
system
on crude oil,
which
has higher supply uncertainty and instability. We must note
coal transportation and storage infrastructure and
countermeasures for SOx and NOx emissions and
very important
in the case where we consume
a
the
that
environmental
burned
ashes
are
great
amount
of
coal.
Thus, the substitutability among primary energy resources is
a very important factor in
our
energy
system's
stability.
In
order to further elucidate the relationship between the stability
probability
and the energy
resources
substitutability,
we
need
more numerical experiments, trying different values for lower and
upper
bounds of certain
energy
flows, and varying
the yields
and
efficiencies of petroleum and coal products.
The model described
in Section 2 is a single
31
period
static
optimization model.
a part of the right hand side in
Letting
apply
could
we
variable,
linear programming model be a random
the
probabilistic and stochastic analyses, obtaining supply stability
and instability probabilities. We can
and
increasing the number of periods
probability
in
the
more
distant
add
the
estimating
future.
In
this
stability
case,
following difficulties occur: uncertainty with respect to
primary
energy
variations
of
prices
and
final energy
supply
forecast
availability
and
of
the
reliability of data. Obtaining the large scale structure
linear programming model and computational
to obtain an
economic
equilibrium point
another difficulty. Therefore we believe
representing
techniques
necessary
will
efficiently
two
future
solutions,
optimal
justification of probability distribution, and
the
subsequent
availability,
and
by
analysis
dynamic
or
three
be
stages,
the next 10 - 15 years will be the largest time span
we can deal with reasonably.
We believe that the approach introduced in this paper can be
useful to quantitatively analyse the energy system
countries like Japan which depend
heavily
energy resources. We are considering further
on
stability
imported primary
modification of
energy systems approach by incorporating dynamic terms
modeling
of national economic structures.
32
of
and
our
more
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36
Price
and
Table 1.
Final Energy Demand
(1010 kcal)
Sectors
1983
1990
211858
243358
Residential-Commercial
99465
122329
Transportation
56869
65324
Stockpile-Transfer
45844
52660
Industry
Table 3.
Energy Prices by Supply Region
(106Yen/10'0kcal)
Energy
Supply
Region
1983 Data
1990 Price
Saudi Arabia
45.996
60.488
Other Middle East
47.437
66.749
South Region
North-South America
48.964
47.398
64.433
62.373
Other Region
50.729
66.756
South Region
22.712
31.958
North-South America
19.184
26.994
Other Region
17.761
24.992
Middle East
48.072
63.259
Gas
South Region
45.219
59.505
LNG
North-South America
46.042
60.588
Crude
Oil
Coal
Natural
1
Table
2.
Primary Energy Resources by Supply Region
(1018 kcal)
Energy
Crude
Oil
Supply Region
1983 Data
Natural
Suiply
Saudi Arabia
59904
74100
Other Middle East
91649
113368
South Region
North-South America
38882
9196
48096
11375
Other Region
13213
16344
447
588
Stockpile-Transfer
28531
37545
South Region
North-South America
26890
21312
35385
28045
Other Region
Domestic
8834
11173
11625
12834
Stockpile-Transfer
7197
8851
Middle East
2411
2965
23655
29093
Domestic
Coal
1990
Avai1abil ty
South Region
Gas
North-South America
1389
1708
LNG
Domestic
2154
2474
Stockpile-Transfer
1335
1642
2
Table
4.
Costs of Secondary Energy Resources
Transformation
(10'0 Yen/101 0kcal)
Energy
Transformation costs
Resources
Fuel Oil
2.01
Kerosine-Gas Oil
32.96
Naphtha-Gasoline
49.85
Other Petroleum Products
24.85
Coke
Coke Gas-Blast
19.29
31.39
Furnace Gas
Electricity(Industry,
Electricity
Transport)
215.52
(Residential)
284.42
City Gas (Industry)
City Gas (Residential)
Table 6.
71.95
106.57
Upper and Lower Bounds
for
Secondary
Energy
(101' kcal)
Secondary Energy
Energy
Lower Bound
Use
Industry
Electricity
Upper Bound
101,500
120,500
60,000
70,000
Transportation
Industry
4,000
2,400
5,500
3,500
Residential
9,500
13,500
Residential
3
Table 5.
Upper and Lower Bounds for Petroleum and Coal Products
(1010 kcal)
Products
Energy
Crude Oil
Fuel Oil
Kerosine
Gas Oil
Naphtha
Gasoline
Lower Bound
Use
Electricity
Stockpile
0.0
32,000
18,600
41,000
Electricity
32,000
41,200
Industry
37,000
49,400
Residential
0.0
12,600
Transportation
0.0
7,000
Industry
0.0
12,000
Residential
25,000
33,000
Transportation
13,500
17,600
Industry
City Gas
22,000
0.0
29,000
1,400
Transportation
35,000
41,500
Electricity
Petroleum
Products
Coke
Coke Gas
Upper Bound
Industry
Residential
2,600
3,500
12,500
16,500
0.0
9,500
Transportation
2,000
3,500
City Gas
2,000
2,900
Industry
32,000
42,500
0.0
Residential
50
Stockpile
1,300
1,800
City Gas
2,000
2,900
Electricity
4,200
5,600
0,300
14,500
Industry
1
4
Table 7.
Upper and Lower Bounds
for Petroleum Products Yields
Upper Bounds
Lower Bounds
Petroleum Products
Fuel Oil
0.40
0.55
Kerosine-Gas Oil
0.10
0.30
Naphtha-Gasoline
0.20
0.0
0.30
0.15
Other Petroleum Products
Table 8.
Parameters for Beta Distribution
Parameters
Energy Resource
bm
Crude
Coal
Oil
bM
p
_q
200,000
300,000
4.0
2.0
45,000
135,000
2.0
4.0
5
Table 9.
Optimal Primary Energy Supply
(1010 kcal)
Energy
Supply Region
Saudi Arabia
Crude Oil
Coal
Natural Gas
LNG
72,683
*
*
111,200
47,176
*
*
38,966
North-South America
11,157
30,884
*
Other Region
16,031
12,802
*
600
12,900
2,550
37,600
8,950
1,750
296,447
104,502
15,283
Other Middle East
South Region
Domestic
Stockpile-Transfer
Total
6
10,983
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d-2
d-i
De di
d2
Approximate demand function
d3
and supply functions
START
I
I1
Generate two sequences of beta random numbers
Bl(i), B2(i), i=l,...,N.
If
For each pair of Bl(i), B2(i),
i=l,...,N,
solve the MP/EE model and obtain an economic
equilibrium solution.
w
I
r
Obtain the probability of supply stability
and distributions of total energy system
cost, primary energy prices and their quantities.
-
Nr
END
Figure
3. Computational
procedure
for
the model analysis
START
|
II
Initial
p,
q,
p=Q(qo).
t=O. I
I
IJE
l
l
Solve the MP/EE model
Optimal solution
X't, Y't,
and supply submodel.
t.
Yes
i
--
"Converged".
Equilibrium
point obtained
t1No
t+l - )t.
Cst=cS+ACst
Figure 4.
pt=('
t+pt -)/2.
ll
END
Algorithm for solving the MP/EEmodel
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Figure 6.
**
HISTGRAM
9.664583E+08 -
Distribution of objective function values
0.667239E+08 -
0.665911E+08
0 .667239E+08
0.668567E+08
0.668567E+B -
0. 669895E+08
0.665911E+38 0.669895E+03
0.671223E+08
0.672551E+08
0.673880E+08
0.675208E+08
0.676536E+0O
0.677864E+D8
0.679192E+08
0.680520E+08
0.681848E+08
0.683176E+08
0.684504E+08
0.685832E+08
0.687160E+08
0.683488E+08
0.689817E+08
-
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0.671223E+08
0.672551E+08
0.673879E+08
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8)
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0.679192E+08
0.680520E+08
0.681848E+08
0.683176E+08
0.684504E+08
0.685832E+08
3)
0. 687160E+08
0. 688488E+08
3)
0.689816E+08
0.691145E+08
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Figure
Distribution of crude oil prices.
** HISTGRAM
49.5565
-
55 .9771
-
52.7668
59. 1874
62.3977
65.6080
68. 8183
72.0286
75.2389
78. 4492
Figure
52.7668
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29.4750
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