# The Diffusion of Photovoltaics: Background, Modeling,

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The Diffusion of Photovoltaics: Background, Modeling,
Calibration and Implications for Government Policy
Gary L. Lilien
May 1978
MIT ENERGY LABORATORY REPORT - MIT-EL-78-019
PREPARED FOR THE UNITED STATES
DEPARTMENT OF ENERGY
Under Contract No. EX-76-A-01-2295
Note:
Charles Allen, Arnold Barnett, Pat Burns, Tom McCormick, Bruce
Schweitzer, and Leif Soderberg contributed significantly to the
material described in this paper.
APPENDIX 1.1
A.l.1
PV PROGRAM
A Users Guide to the Photovoltaics Simulation System
The photovoltaics simulation system is a self-contained interactive
set of computer programs on the Multics system at MIT;
It allows the
user to exercise the PV model for different sets of parameters by providing general facilities for creating, saving, editing, using, displaying and deleting sets of parameters.
parameters reflects the model:
The organization of these sets of
the full set of input data is kept in a
model, which contains parameters which apply across sectors, such as the
cost decline factor, as well as the names of its constituent sectors.
A
sector, in turn, contains the data that are relevant only to itself, such
as average PV installation size.
Finally, when a model and its associated
sectors are used to perform a simulation, the results are saved in a run,
which records all the model and sector input data as well.
The commands
in the system are used to manipulate these three types of sets of data,
or records.
The actual model is defined by the following key variables and
equations:
Xit
=
number of kwatts of government PV installed in sector
i during
time t
Yit = number of Kwatts of private PV installed in sector i
during time t
Ct
= cost of one Kwatt of PV at the end of period t
-195-
*0
Zi
= average number of Kwatts per PV installation in sector i
Nit
cummulative Kwatts of PV installed in sector i up to
time t
Nt
= cumulative production of Kwatts of PV up to time t
No
= Kwatts of PV installed before model begins
CO
= cost per Kwatt of PV before model begins
X
=
decimal factor by which cost declines when PV production doubles
F
()
= cumulative normal distribution for cost acceptability,
ci
and aoc
parameterized by
Ci
F
(.) = cumulative normal distribution for successful installa
SI
and a
tions acceptability, parameterized by pi
i
hi
si
= probability of purchase by randomly selected individual
in sector i given acceptability
Pio
=
Pt
= total potential market for PV in sector i at time t
Qi
initial potential market for PV in sector i
= potential original equipment PV installation that becoifies
available per time period
= growth rate of existing market potential in sector i
The equations relating these variables are:
t
N
- I (xi+Y
T=
-196-
Nt
=
Pit
=
1
F
+
N
Nit
(Pit-l-Qi)(l+gi) + Qi
l(C 1) = probability that cost is acceptable at
the beginning of period in sector i.
F
s.
(N
it-
/Zi) = probability that installed PV is accept-
i
able in sector i at the beginning of
period
Yit
=
Ct
= Co(Nt/No)
t.
it- Nit-1/Zi)(1-Fci(Ctt- )) Fsi(Nit-1/Zi)
hi
Zi
g 2X
These equations describe the model that is embodied in the simulation
system.
Tables I, II and II (for models, sectors and runs, respectively)
show the correspondence between the variable names used here and the names
as they appear
in the computer
programs
and on the output.
In use, the computer and the user exchange information through a computer terminal.
There is a wide variety of computer terminals, which have
various switches that must be properly set to communicate with Multics.
After turning the terminal on, a COM/LCL switch should be on COM, a
REMOTE/LOCAL switch on REMOTE, a FULL/HALF switch on HALF, a CONT MSG
switch ON, and a speed switch on HIGH.
Once the terminal is ready, you
must &quot;dial-up&quot; the computer, that is, establish the telephone link between
-197-
it and your terminal.
The phone number is 258-8215 for IBM 2741-like ter-
minals and 258-8313 for ASCII-type terminals.
The general procedure is
to dial the number; you should get a high-pitched tone in response.
Then, depending on the communication system used, push the DATA button
or pull up the white button in the handset cradle or insert the handset
into the acoustic coupler.
with a two-line message.
On 2741-like terminals, Multics will respond
For ASCII-type terminals, you must type a speed
identification character first, followed by a line-feed.
Use &quot;s&quot; for
110-baud terminals (11 characters/second), &quot;y&quot; for 150-baud (15 cps) or
&quot;o&quot; for 330-band (30 cps).
message.
Multics will then respond with its two-line
Then type
and hit carriage return for 2741, line-feed for ASCII.
You will then be
prompted for the password; enter it, again followed by carriage return for
2741, line-feed for ASCII.
After that, Multics will print some login
messages and the simulation system's opening message will print.
From
this point on, everything you type on either kind of terminal should be
terminated with a carriage return.
The system will ask if you need the system explained (see attached
sample terminal session).
otherwise type &quot;yes&quot;.
If you are familiar with the system, type &quot;no&quot;,
The system expects the full word &quot;yes&quot; or &quot;no&quot; when-
ever it asks you a question, though it generally assumes a &quot;no&quot; answer if
anything other than &quot;yes&quot; is typed.
If you answered &quot;yes&quot;, a brief
description of the commands will be printed.
In either case, the system
then prints &quot;READY&quot; as a signal that it is awaiting your first command.
-198-
If you mistype the
You then type in the command that you want performed.
command or type a non-existent command, the system will respond:
The command XXX does not exist - try again
with XXX replaced by the command you typed.
valid command.
You should then type in a
After the command has finished processing (which may
require that you type in information), the system again prints &quot;READY&quot;
and awaits your next command.
This cycle is terminated when you type
the command:
return
which will stop processing and automatically log you off the computer.
You will be informed of this by the message:
You will now be disconnected from the computer.
Goodbye.
There may be circumstances when you don't want a command to finish
an action it has started.
For instance, you may start displaying some-
thing that is longer than you thought.
In any such situation, press the
&quot;quit&quot; button on the terminal (which may be labeled &quot;attn&quot; or &quot;break&quot; on
some terminals).
Whatever the command is doing will stop, &quot;QUIT&quot; will be
printed, and then &quot;READY&quot; will be printed to indicate that the next
command may be entered.
In certain situations an additional informational
message in parentheses will be printed before &quot;QUIT&quot; to inform you of the
status of the records that were being processed.
Also, should an unfor-
seen error arise, an error message will be printed containing information
so the system maintainer can diagnose the error, followed by an artificially created quit signal to return you to command level.
sages should be reported promptly to the system maintainer.
Any error mes-
There are general conventions that apply to all commands.
The
format in which they are typed is the command name followed by other
information, separated by blanks.
For instance,
create model example
would be used to create a model named example.
The commands are flexi-
ble though; in this example, &quot;create&quot; could be abbreviated to &quot;cr&quot;, and
any word beginning with &quot;m&quot;
could be substituted for &quot;model.&quot;
Thus
cr m example
create mood example
cr money example
would all have the same effect as the previous example.
In general, most
commands have one or two letter abbreviations, and whenever the next
field entered is a type (&quot;model,&quot; &quot;sector&quot; or &quot;run&quot;), only the first
letter is significant.
allowed.
All names are used as is with no abbreviation
Different types of records are allowed to have the same name
(i.e., you can have a model and a sector both named test), but two records
of the same type must have different names.
The programs check for this
and give you a chance to re-enter a different name or overwrite the already existing record if you try (inadvertently or not) to use a duplicate name.
You can also choose to &quot;cancel&quot; in such circumstances, which
means that you are returned to command level as if you never entered the
original command.
record type,
The programs check to see if you have entered a valid
and prompt you if you haven't.
cr x wrong
-200-
For example:
i
by:
First argument must be model or sector.
Reenter:
You can then type a word beginning with &quot;m&quot; or &quot;s&quot; and create the proper
record.
If you realize that you don't want to create anything, you can
hit &quot;quit&quot; to return to command level.
In addition, if you leave out
arguments, the programs will prompt you for them.
For example:
cr model
by:
Enter model name:
Also several commands allow you to process several records of the same
type at once.
For instance:
cr mood testl test2 test3
will create three models.
The available commands are:
create
- creates models or sectors
copy
- copies models or sectors
modify
- modifies models or sectors
delete
- deletes models, sectors or runs
display - displays models, sectors or runs
run
- runs a model creating a run
index
- lists records on file of a given type
help
- prints a brief description of command syntax
return
- terminates processing, logs you off
--201-
I
The
The last two commands are self-explanatory and have no arguments.
other seven commands are described below.
create:
Syntax:
create TYPE NAME1 NAME2
Abbreviation: cr
TYPE:
can be model or sector
are 0 or more names of records to be created.
NAMES:
When this command is invoked it will successively prompt you for
all the data items needed to create a model or sector.
When the last
item has been entered, a message is issued confirming that the record
was created, then the next record (if any) is processed.
copy:
copy TYPE OLDNAME NEWNAME
Syntax:
Abbreviation:
TYPE:
cp
can be model or sector
OLDNAME: name of existing record which is to be copied
NEWNAME: name that the copy is to saved under
This command copies the record OLDNAME of the indicated TYPE under
the name NEWNAME and issues a confirming message.
Since the use of this
command is frequently to have several versions of the same record with
minor changes, you will then be asked if you want to modify the new record.
Answering &quot;yes&quot; is equivalent to issuing the command &quot;modify TYPE NEWNAME.&quot;
-202-
modify:
Syntax:
modify TYPE NAME1 NAME2 ...
Abbreviation:m
TYPE: can be model or sector
NAMES:
are 0 or more names of records to be modified
When invoked, modify asks for your first subcommand.
of subcommands is printed by typing &quot;help.&quot;
names of variables that can be changed.
name.
A dictionary
Most subcommands are the
To do this, enter the variable
The program reports back the old value and asks for the new value.
Then it prompts you for the next subcommand.
There are also subcommands
to display the modified record and to finish processing and return to command level.
delete:
Syntax:
TYPE:
NAMES:
delete TYPE NAME1 NAME2 ...
can be model, run or sector
are 0 or more names of records to be deleted
There is no abbreviation for delete, to avoid accidental deletions.
For models and runs, delete checks the index to see if they have been run
and displayed, respectively, and asks you to confirm whether you really
want to delete these unused records or.not.
printed after each deletion.
-203-
A confirming message is
display:
display TYPE NAME1 NAME2 ...
Syntax:
Abbreviation:
TYPE:
dp
can be model, sector or run
NAMES:
are 0 or more names or records to be displayed
This command prints the requested records on the terminal, one at a
For a run, display asks you if you want model, sector, both or.
time.
neither input variables printed (again, only the first letter is significant, so &quot;m&quot;, &quot;s&quot;, &quot;b&quot; or &quot;n&quot; is sufficient), and give you the same choice
for output variables.
Successively answering &quot;b&quot;, &quot;b&quot; will print every-
thing, &quot;n&quot;, &quot;n&quot; will print nothing, &quot;s&quot;, &quot;s&quot; will print only sectorspecific input and output variables (which depend on model input variables
however), etc.
run:
run MODELNAME RUNNAME
Syntax:
Abbreviation: r
MODELNAME:
RUNNAME:
name of the model to be run
name that the output will be saved under
After the model is run, the program prints a confirming message and
asks if you would like to have the run displayed.
lent
index:
Answering yes is equiva-
to issuing the command &quot;display run RUNNAME&quot;.
index TYPE
Syntax:
Abbreviation:
TYPE:
i
can be model, sector, run or all
-204-
If TYPE is &quot;all&quot; or left blank, index prints a table of all records
created and still on file giving their name, their type, and for models
and runs, under the heading &quot;Used?&quot;, an indication of whether they have
been run and displayed yet.
If TYPE is model, sector or run, a table
restricted to only that type of record is printed with no type column
but retaining the &quot;Used&quot; column.
To clarify these ideas an annotated terminal session is attached,
followed by a complete listing of the program.
-YI
d
,&middot;
.
-205-
TABLE I:
Model Correspondences
Symbolic
Notation
PL/I
Variable Name
Modify
Mnemonic
Name
mname
name
Number of Sectors
n
n
Number of time periods
tmax
tmax
C0
initcost
initcost
Cost decay rate
A
costdecay
costdecay
Initial installed Kwatts of
N0
initwatt
initwatt
sn(i)
secname(i)
govinv(i,t)
govinv(i,t)
English Concept
Initial cost
PV
Name of sector i
Government investment in
sector
i at time t
Xit
-206-
TABLE II:
English Concept
Sector Correspondences
Symbolic
Notation
Name
Average PV installation
size (Kwatts)
Initial existing potential market (Kwatts)
Z.
PiO
Growth rate of existing
potential market
Potential for original equipment installation per time
period (Kwatts)
Purchase probability given
acceptance
Mean of cost distribution
Std. dev. of cost distribution
Mean of successes distribution
Std. dev. of successes
distribution
Qi
h.
1
Pci
PL/I variable name
Modify
mnemonic
sname
name
avgsize
avgsize
initpotinst
initpotin
buildrate
buildrate
newinst
newinst
ppurch
ppurch
cmean
costmean
cdev
costdev
imean
instmean
idev
instdev
Ci
Si1
Si
1
-207-
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-210-
APPENDIX 2
FACTOR STRUCTURE COMPARISON: AGRICULTURAL SECTOR
With Choffray and Lilien's [13] modification of the Chow test, when
we compare the evaluation criteria of A
against C
at the evaluation of the combined group, (A + C).
we must also look
Actually, we are
comparing A and C with (A + C) to determine equality; our null hypothesis is that the groups are, in fact, equal.
N(1-R2)
C
Defining:
X
N-2q
=
nl(l-RP1)+
n 2 (1-R2 2 )
q
where,
2
2
R'
R2 = squared multiple correlations associated with the
p1' p2
estimation of factor p in sample 1 and 2 respectively
2.
R
nl,,n2
= squared multiple correlation associated with factor p
in pooled sample
= number
N
= n
q
= number
When the value of C
of responses
in sample
1 and 2 respectively
+ n2
of items
exceeds the corresponding F statistic at the
appropriate level of significance, the null hypothesis of equality is
rejected.
Group A = Pre-test and post-test survey
Group B = pre-test survey only
Group C = post-test survey only
-211-
APPENDIX 2 (continued)
TABLE
1
For comparison of A and C:
Factor
C
F
p
.90
.
F
95
.F
99
1
0.41
1.72
2.01
2.75
2
0.42
1.72
2.01
2.75
3
1.79
1.72
2.01
2.75
null hypothesis is accepted at the .95 level
TABLE
2
For comparison of B and C:
Factor
C
p
F
.90
F.
FF
95
99
1
2.62
1.72
2.01
2.75
2
2.53
1.72
2.01
2.75
3
0.22
1.72
2.01
2.75
null hypothesis is rejected at.the .95 level
-212-
APPENDIX 3: FACTOR STRUCTURE COMPARISON: RESIDENTIAL SECTOR
TABLE
1
COMPARISON OF GROUP 1 AND GROUP 2
C
Factor
P
F..g..9 0
F.9 5
F
.9*7
1
5.76
1.72
2.01
2.75
2
10.58
1.72
2.01
2.75
3
3.34
1.72
-2.01
2.75
Null Hypothesis of equality is rejected
COMPARISON OF GROUP 3 AND GROUP 4
C
F. 9 0
F.9 5
F.99
0.94
1.72
2.01
2.75
2
2.42
1.72
2.01
2.75
3
1.24
1.72
2.01
2.75
Factor
P
Null Hypothesis is rejected
at the .95 level
-213-
APPENDIX 3 (continued): TABLE 2
MODIFIED CHOW TEST
COMPARISON OF GROUP 1 AND GROUP 4
C
F
F
.90
F,9 5
2.44
1.72
2.01
2.75
2
. 3.72
1.72
2.01
2.75
3
-8.88
1.72
2.01
2.75
Factor
P
I
.99
Null Hypothesis of equality is rej
COMPARISON OF GROUP 2 AND GROUP 3
C
p
F.90
F.9 5
.95
F.99
.99
1
6.57
1.72
2.01
2.75
2
1.92
1.72
2.01
2.75
3
0.88
1.72
2.01
2.75
Factor
Null Hypothesis of equality is rejected
-214-
A.4
Sample Interactive Program Session
-215-
.
E
-
uq
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