A Continuous Time Model of Electricity Consumption
Which Incorporates Time of Day Pricing, a Demand
Charge, and Other Commodities
by
A. Ronald Gallant
March 10, 1980
A Working Paper Supporting North Carolina Agricultural
Experiment Station Project No. NC0364l,
"Statistical
Methods for Continuous Time Demand Systems with Application to Electricity Pricing."
1
e
x.
A Continuous Time Model of Electricity Consumption Which Incorporates
Time of Day Pricing, a Demand Charge, and Other Commodities
Over an interval of time
[O,T],
a path of electricity consumption
c(t)
say, a month, a consumer chooses
with units in kilowatts per hour
and, under conventional pricing, is billed proportionally to the total kilowatt
hours consumed over the period, computed as
usea, instead, the billing formula
J:
J:
c(t)dt.
ret) c(t) dt
Time of day pricing
where
ret)
is the
rate schedule with units in dollars per kilowatt hour.
The method of computing a demand charge varies from utility to utility.
The method described here is that used in the North Carolina Time of Day Rate
Demonstration Project.
At selected times during
[O,T]
denoted as
the average kilowatts per hour that a customer consumes over the next
fifteen minutes is computed,
(1/6)
t.+6
~
Jt i
d( t)
dt.
2
The maximum of these as
i
ranges from 1 to K is,. say,
d • D is added to the customer's bill where
per kilowatt per month.
~
An amount
has units in dollars
Fo·r no·tational. convenience, set
_
_ [101D.
I. (t)
d
D.
- t.
t. <
_. t
<.
~
~
+ D.
otherwise
whence
Jo
T I.(t)c(t)dt
Let
q
~
= (l/l!)
t.+D.
Jt'i
~
c(t)dt.
denote an N-vector of conventional commodities, let
an N-vector of corresponding prices, and let
e
expenditures over the period
(O,T].
of electricity consumption paths
q
c,(t)
p
be
Y denote a consumer's total
The consumer is constrained to choices
and quantities of other commoditi'es
which satisfy
qO·( c,q,D)
q.(c,D)
~
J~
= J~
=
=
i
r(t)c(t)dt +p'q+dD-Y < 0
I. (t)c(t)dt - D
~
< 0
l,2, ••• ,K.
These equations define the income constraint set.
The consumer is presumed to choose an electricity consumption path
(t)
and other commodities
the income constraint set.
function
c(t)
8
L2 (0, T]
q
by maximizing a functional
The functional
u(c,q)
and a commodity vector
u(c,q)
over
maps a square integrable
q
8
R N into the real
3
e
line.
One has in mind, possibly, an analogy to the Generalized Leontief
functional form
u(c,q)
= LN
1 a.
i
=
In q. +
1
fT
1
0
N
aCt) In [c(t)]dt
1
\'
(3 •• (q . q . ) ~
+
~i=l ~i=l 1J 1 J
+ \'N
JT fT s(s , t)[ c ( t) c (s) 1~
0 0
1
ds dt
Electricity is an input into household production so that this objective
function is to be regarded as a composition of household production functions
and a utility function.
There is no mathematical impediment to the construction
of a household production type of model but since electricity consumption is
not measured by end use in the available data there is little point to the
exercise.
To solve this optimization problem, a notion of differentiation of a
functional such as
is required.
u(c,q)
To do this, suppose that
(~2/ax2)K(t,x)
are continuous with
that a functional
f(c)
s
fCc)
f:
K(t,x),
bounded over
(a/ax)K(t,x), and
0 < t < T,
_00
< x <
00
is defined by
c,h
f(c+h) - f(c)
=
where
E
f:
L [O,T],
2
(a/ax)K[t,c(t)]h(t)dt +
0(11 hill
This equation suggests a definition of a
'iJ f (c)
c
and a
d~finition
of a differential
'iJ
c
f(c)h.
c(t)
(a 2 /ax 2 )K(t,x)
K[t,c(t)]dt.
It follows that, for
derivative
with respect to a square integrable function
and
4
The square integrable function
7 f(c)(t) = (d/dx)K[t,c(t)]
c
may be regarded as the derivative of
Vf(c)h
c
f(c)
evaluated at
c
and
= fT0 7c f(c)(t)h(t)dt
may be regarded as the differential approximation to the difference
f(c+h) - f(c)
•
In general, i f one can find a square integrable function
F(t)
which satisfies
the equation
£(C+h) - ftc) •
for square integrable
7 f(c)(t)
c
f:
c,h
F(t)o(t)dt
q
o(
= F(t).
(3aq f(q), aaq
1
q
E
N
R retains its conventional meaning
a)
f(q), ••• , aQNf(q)·.
2
•
Thus
f(q+h) - f(q)
where
II h II
II h II )
then
Differentiation with respect to
v ~ f(q) =
+
= V~f(q)h
+ a (11h
II)
5
As shown in detail later il.n this section, the Kuhn-Tucker first order
conditions for the consumers optimization problem are
iT u(c,q}
q
- AOP
= 0
- AOd
- I~=l
= 0
A. (-1)
J.
T
JO
=y
r(t)c(t)dt + p'q + dD
=0
for
e
i = O,l, ••• ,K.
Assume that exactly one constraint
"i
is binding;
this is approximately the same as assuming that the consumers chosen consumption path
c(t)
has a single maximum.
In this case the first order
conditions are
iJcu(c,q)(t) - AOr(t) -
iJ
u(c,q)
q
-
J.
J.
+ A'i"
r(t)c(t)dt + p'q + dD
JT
°
= 0
= 0
AOP
AOd
f:
A~I~(t)
I~(t)c(t)dt
J.
Algebraic manipulation yields
= 0
= y
= D
6
A =
o
[J: ~
cu(c,q}(t)c(t)dt
+
~~U(C,q)~ ly
A"i· = A0d
whence the first order conditions become
= AOP'
vq u(c,q)
T "
o r(t)c(t)dt + p'q = Y
J
r(t)
= r(t) +
dI~(t),
~
Marginal utility is proportional to price at each instant of time.
result generalizes;
The
conditional on
T
fo
c(t)I. (t)dt
~
the first order conditions are as above but with
r(t) = r ( t) + d max [I" (t) , I~ (t) ] •
i
i
Another implication of the first order conditions is that consumption
is homogeneous in prices and income.
That is, the electricity consumption
path is of the form
,..
c(t)
= c[(r,p)/y](t)
and the consumption of other commodities is given by a function of the form
q = q[(r,p)/y].
7
Typically, the price path
!(t)
is periodic over
-~
<
t <
~;
that is
r(t+kT)
= r(t),
k
= 0,
=1, :2,
It also seems natural to impose periodicity on marginal utility
v c a(c,q)(t+kT) = Vc u(c,q)(t).
The optimal consumption path
A
c[(r,p)/y](t)
will, as a consequence, be periodic.
A stocastic specification of an observed demand path
where
~
i
°
- f(T),
the time of day, and
T,
< T <
ci(t)
is
represents small errors in knowledge of
e.~ - g(e)
represents an additive error of observation.
The expected demand path is, then,
c[(r,p)/y](t)
= E[c.(t)]
~
=
T
fo c[(~,p)/y](t-T)
-f:
where
f(T)
f(t-s)
~(t.p)/y](s)ds
has been extended to a periodic function.
and continuous then
lim
f(T) dT
t+t
o
c [(~,p)/y](t)
If
f(T)
is bounded
8
T
=
fa
[lim +
t
ta
f(t-s)]c[(t,p)!y](s}ds
=
whence the expected demand path is continuous regardless of whether or not
the optimal demand path is continuous.
The path
c[(r,p)ly]
is a convolution.
Thus, if
fer)
and
c[(r,p)!Y'](t)
have Fourier expansions with absolutely summable coefficients
c[(r,p)!y](t) = "':'
~J=-""
S. (1!IT)e i (2'IT!T)jt
J
then
= ,,""
~j=-""
(a.S.) (1!IT)e i (2'IT!T)jt.
J J
Thus, the Fourier coefficients of the expected demand path are attenuated
by those of the timing error distribution
f(L);
they decrease to zero
faster than those of the optimal demand path.
When fitting this demand system to data, i
'" i)
~
or (i,
may be is found by inspecting the consumption path
in the data.
c(t)
as the case
which obtains
This introduces an errors in variables problem as, due to
9
·e
error, the
,.
i
,.
which obtains may differ from the optimal i
This problem
is addressed in a later section.
Another problem in empirical work with the demand system is that consumption is defined as an implicit function of price.
as an explicit function of price is preferable.
is available to address this problem.
v(~,x)
Consumption expressed
A form of Roy's identity
An indirect utility functional
has income normalized prices as its arguments
~(t)
=
x =
~(t)ty
p/ y.
One has in mind as a choice of
v(~,x)
a generalization of a flexible
functional form along the lines described earlier.
ROy's identity is
Next, a detailed verification of the first order conditions is presented.
The reference for notation, definitions, manipulative results such as the chain
rule, and the Kuhn-Tucker first order conditions is Wouk (1979, Ch. 12).
The fact that both
L [0,T]
2
in the development.
and
N
R are self dual is exp10itled repeatedly
Thus, a bounded linear operator
<·,r'>
on
1 [0,T]
2
must be of the form
<c,.'> where
r'
J:
.(t).'(t)dt
itself is in
L [O,T].
2
This being the case, the notation
r'
to
10
indicate membership in the dual space is dropped in favor of the notation
since
r'
is, in fact,
in
1 [0,T].
2
r
This gives rise to another fact which
is exploited repeatedly
<c,r'
«
~ c(t)~(t)dt
- <r,c',
=
The Frechet partial
of a mapping
N
R ,
derivative with respect to
f(c,q)
operator denoted as
its range,
from
'i/
c
c
evaluated at
to, say,
f(c,q)
which maps
h
E
(c,q)
is a bounded linear
1 [0,T],
2
its domain, into
and satisfies
f(c+h,q) - f(c,q) =
The notation
<p,q>.
'i/ f
c
'i/
c
f(c,q)h + o(llhll).
is used frequently to mean
.
'i/ f (c ,q) ;
c
to avoid any confusion caused by this abbreviation.
Table 1 is provided
11
1_
Freehet Partial Derivative
Freehet Partial Derivatives
Evaluated at
'i/ u
e
'i/ u
q
=
<. ,'i/ 1.1>
e
(e ,q)
=
<', 'i/u>
q
'i/D u
=
<,,0>
Domain
Range
(e,q)
L
2
N
R
R
(e,q)
R
R
R
R
'i/ egO
=
<', r>
(e,q,D)
'i/ g
q 0
=
<', p>
(e,q,D)
L
2
N
R
'i/DgO
=
<-, d>
(e,q,D)
R
R
'i/ egi
=
<-,I.>
1.
(e,D)
L
2
R
'i/ qgi
-:=
<-,0>
(e,D)
N
R
R
'i/Dgi
=
<-,-1>
(e,D)
R
R
<·,'i/...v>
r
<', 'i/ v>
(r,p)
L
2
N
R
R
L
2
N
R
L~
'i/..v
r
'i/ v
P
'i/...e
r
'i/ e
p
'i/... q
r
'i/ pq
=
=
P
(r,p)
(r,p)
(r,p)
(r,p)
(r,p)
L2
N
R
R
R
L2
N
R
N
R
12
The consumers optimization problem is to
maximize:
u(c,q)
subject to:
= <c,r> +
gO(c,q,D)
<q,p> + <D,d>
- Y: 0
= 1,2, ••• ,K.
i
A unique solution
satisfies
and
r(t) > 0
= 0,
gO(c,q,D)
qi > 0
for
all
i
(c,q)
g. (c,D) = 0
J.
= 1,2, ••• N
t e: [O,T],
Frechet derivative of
of the consumers optimization problem which
i,
c(t) > O.
is presumed to exist for each
Pi > 0
u(c,q)
for some
for i
= 1,2, ••• N,
and
is presumed to exist at
all ;·.t e: [0, T],
(r,p,d)
d > O.
The
(c,q).
The Kuhn-Tucher first order conditions for this problem are
'i/
'i/
C
q
u(c,q)
L~=l
A <., r>
O
u(c,q)
-
A.<·,1.>
J.
J.
AO<"P>
L~-l
A <. ,d>
O
A.<·,-l>
J.
<c,r> + <q,p> + <D,d>
A.]. [<c,r.>
+ <D,-l>]
].
If exactly one constraint
'"
i
=
0
=
0
=
0
=
y
=
0
is active, then
and the first order conditions may be written as
- AO <., d> - Ai <',-1>
=
o
=
o
= o
A.].
.
=
0
for
'"
i of i,O
with
13
<c,r> + <q,p> + <D,d>
<c, Ii> + <D, -1>
=
Y
= o
then
[<c,'iJ u> + <:tq,'iJ u> + 0] - AO[<c,r> + <q,p> + <D,d>]
q
C
-
A~[<c,I~>
~
~
+ <D,-l>]
=
0
which simplifies to
[<c,'iJ u> + <q,'iJ u>] - A Y = 0
c
q
0
<C,I7> + <D,-l>
using the income constraint and
~
Thus,
Now the equation
may be rewritten as
A d<· -i> - A~<·,-l> = 0
0'
~
whence
The first order conditions become
...
<., r>
=
0
= O.
14
....
<c,r> + <q,p>
=
Y
with
....
=r
r
o=
A
+
dI~
~
[<c,V u> + <q,V u>]/y
q
C
Suppose that there is a differentiable mapping
q[(~,p)/y]}
of
(r,p)/y
optimization problem.
v[(~,p)/y]
into the point
(c,q)
(c,q)
= {c[(~,p)/y]~
which solves the consumers
The indirect utility function is, then,
= u{c[(~,p)/y]
q[(~,p)Iy]}.
By the chain rule
V uV"'c
v,.."
r = c r +
V
v
p
= Vc uV p c +
V
q
uV.. . q ,
r'
V uV pq
q
Substitution of the first order conditions derived earlier yields
V,..v
r
V v == A
p
0
....
<', r>
V c + A <',p> V q
P
O
P
Differentiation of the income constraint
<c[(r,p)/y],~> + <q[(~,p)/y],p>
yields the equations
=y
15
'"
'" 'ilAc + <. t p> 'ilAq
< • t c[ (r t p) j"y] > + <. t r>
r
r
A
<--tq[(rtp)/'y]> + <. t r> 'il
p
C
=
0
+ <. t p> 'ilp q = 0
Substitution into the equations for
Then
<rt'ilAv'> + <pt'il v> = -AoGtc[{rtp)/y]> + <ptq[(rtp)/y]>]
r
P
= -Ao[3c[(rtp)/yLr> + <q[(~tP)/y.]tP~
Substituting for
If one writes
x=
A yields Roy's identity
O
v·(~,x)
r/y
x = ply
then Roy'S identity becomes
<. ,c>
<. ,q>
for the indirect utility function where
16
Reference
Wouk, Arthur (1979) A Course of Applied Functional Analysis.
John Wiley and Sons.
New York: