1
Good Consumer or Bad Consumer: Economic
Information Revealed from Demand Profiles
Yang Yu, Student Member, IEEE, Guangyi Liu, Senior Member, IEEE, Wendong Zhu, Fei Wang, Bin Shu,
Kai Zhang, Nicolas Astier, and Ram Rajagopal, Member, IEEE,.
Abstract—In this paper, we demonstrate that a consumer’s
marginal system impact is determined only by their demand profile, rather than by their demand level. Demand-profile clustering
is identical to clustering consumers according to their marginal
impacts on system costs. A profile-based, uniform-rate price is
economically efficient as real-time pricing. We develop a criteria
system to evaluate the economic efficiency of an implemented
retail price scheme in a distribution system by comparing profile
clustering and daily-average clustering. Our criteria system can
examine the extent of a retail price scheme’s inefficiency, even
without information about the distribution system’s daily cost
structure. For this work, we analyze data from a real distribution
system in China. In this system, we find that targeting each
consumer’s high-impact days is more efficient than targeting
high-impact consumers.
Index Terms—Demand profile, marginal system impacts, retail
price, clustering, distribution system
I. I NTRODUCTION
A. Information for retail market design
Retail-price design and demand-repose (DR) development
need two types of information. One is the marginal system
impacts caused by a consumer’s demand and variation. The
other is every consumer’s demand function. The former reflects how sensitive the costs of a distribution system are
when a consumer’s demand changes. The latter determines
how costly it is for a consumer to respond to a retail-price
change or a DR project. In this paper, we examine the linkage
between the marginal system impacts of serving a consumer’s
demand and this consumer’s demand profile, which we define
as the hourly proportion of a consumer’s daily demand.
The profile of the system load in a DS determines the
DS’s daily system cost of purchasing energy, denoted by
the system cost in this paper for short [1]. For example, a
distribution system will face to a higher daily cost for serving
the same demand level when the aggregate demand’s peak
level is higher. If two consumers have heterogeneous demand
profiles on the same day, the impacts of their demands on
Yang Yu is with the Precourt Energy Efficiency Center and Department of Management Science and Engineering, Stanford University, CA,
94305,[email protected].
Guangyi Liu, Wendong Zhu, and Fei Wang are with GEIRI North
America, 5451 Great America Parkway, Santa Clara, CA 95054,
[email protected].
Bin Shu and Kai Wang are with Beijing Electric Power Economic Technology Research Institute, Beijing, China, 100055.
Ram Rajagopal is with the Department of Civil and Environmental Engineering and Department of Electronic Engineering, Stanford University, CA,
94305,[email protected].
Nicolas Astier is with the Toulouse School of Economics, Toulouse, France,
31500,[email protected].
the system aggregate demand profile can be different. Thus,
it is important to analyze how a consumer’s demand level
and profile influence the system load profile and the marginal
cost of serving him/her, which fundamentally determines the
economic optimal rate for this consumer.
The profile of the system load in a DS determines the
daily system cost of purchasing energy, denoted in this paper
as the “system cost” [1]. For example, a DS’s system cost
will increase when the daily system load has a higher peak
level even if the total system consumption keeps the same.
At the same time, the system-load profile can change when
consumers change their demand levels or profiles. Thus, it
is important to analyze how a consumer’s demand level and
profile influence the system-load profile as well as the system
cost so that we can clarify the linkage between a consumer’s
demand profile and the marginal cost of serving him/her.
The information about the individual user’s daily demand
profile used to be unavailable or incomplete. Consequently,
research on retail price design is based on the assumption that
all consumers have very similar electricity usage patterns [2].
Thus, often, implemented retail prices and DR projects are
designed independent of the consumers’ demand profiles. For
example, retail electricity bills in most current U.S. markets
include two parts: the tiered rates for electricity energy consumption, and the fixed charge to balance the utility’s budget
for fixed costs and other service costs [3]. Both parts are
independent of a consumer’s demand profile. However, even
if some price designs take a consumer’s demand profile into
consideration, they do not differentiate the consumers by their
profiles. For example, the demand charge prices consumers
only according to their peak demand levels in a month, no
matter when the peak demand occurs.
Currently, comprehensive data of individual daily demand
profiles are available, and researchers find remarkable heterogeneity in consumers’ daily demand profiles [3]. For example,
annual consumers’ demand profiles for PG&E can be clustered
into more than 200 types [4]. Thus, it is necessary to reexamine
the effectiveness of the current retail price schemes and DR
projects, which originally are designed on the homogeneousprofile assumption.
B. Literature on utilizing the information of consumers’ demand profiles
The available literature discusses how the demand-profile
information can be used to deepen our understanding of
consumers’ behaviors and improve demand management in
2
DSs. Researchers explore how consumers vary their demand
profiles to respond to fluctuations in the outside environment,
such as temperature change [5]. Additionally, consumers’
employment statuses and other demographic characteristics are
also correlated with their demand profiles [6]. However, the absence of research on the economic essence of demand profiles
prevents the discussion of how consumers’ profile variations
influence the system costs of serving them. For example, it is
still unknown whether a consumer’s demand-profile variation
causes an increase in the marginal system cost to serve him/her
when the outside environment or the occupancy pattern of
the building change, such as a temperature increases or a job
change of a consumer.
Some studies use demand profiles and their clustering
results to improve the system-load forecast. For example,
Guilumba et al. use the clustering results directly as a predictor
for load forecasting [7]. Additionally, certain investigators
suggest that forecasting the aggregate loads of consumers
belonging to the same profile cluster has fewer errors than
forecasting the load of any other consumer aggregation [8].
Methodologies have also been developed to predict individual
consumer profiles according to demographic characteristics
[9]. However, we still do not know whether a consumer’s
deviation from their forecasted demand profile will cause
a significant error for the system-demand forecast, as it is
unclear how the variation of individual demand profiles impact
the system-load profile. Consequently, we cannot target the key
consumers to improve the system-load forecast.
Many other analyses use demand-profile information to
target consumers for DR projects. For instance, Kavousian
et al. rank consumers according to their energy efficiency
by controlling the effects of demand profiles [10]. Variations
in a consumer’s demand profiles over days is considered a
key feature to determine whether this consumer should be
targeted for a DR project [11]. Gulbinas et al. suggest a
method to target building occupants for an energy efficiency
program by balancing their profile randomness, efficiency,
and intensity [12]. Certain researchers note that a consumer’s
demand profile significantly impacts the value of targeting
him/her in a DR project. Kouaelis et al. argue that it is
important to calculate the net benefit caused by the profile
change of a flexible consumer’s demand [13]. However, the
lack of an explanation about the link between the system
costs of serving a consumer and their demand profile is a
barrier to understanding the benefits of a DR project [14].
For example, we cannot conclude whether a consumer with
varying demand profiles over days is less valuable than another
consumer whose daily consumption pattern remains stable.
Understanding the relation between consumer demand profile and the system cost is the foundation for measuring the
profitability of serving a consumer, assessing the efficiency of
a retail price design or demand response project, and examining the quality of demand-response resources in a DS. In this
paper, we explain the economic nexus between consumer’s
demand profile, marginal system cost, and economically optimal daily average retail rate. This economic explanation is also
used to develop criteria to assess the economic inefficiency and
association impacts of an implemented retail price scheme in
a DS. The remainder of this paper is organized as follows.
Section II provides the theoretical discussion about consumerdemand profiles and marginal system impacts. Section III
explains the nexus between demand-profile clustering and
daily-average-rate clustering. Section IV discusses how we
develop criteria to assess a retail-price design. Section V
summarizes our empirical analysis on a China DS. Finally,
Section VI offers a discussion on the empirical study and
concludes the paper.
II. L OAD PROFILE CLUSTERING , MARGINAL SYSTEM
IMPACT AND PRICING DESIGN
A. Consumer’s marginal system impacts
We examine a DS that serves N consumers whose hourly
demands are positive. Consumer i’s daily demand vector is
−
→
denoted by Li = (li,1 , ..., li,h , ...li,24 ) ∈ R24 , where li,h is
consumer i’s hourly demand in hour h.
Definition Consumer i’s daily demand profile as
−
→
li,1
li,24
Li
, ..., P24
).
P fi = −
→ = ( P24
|Li |1
h=1 li,h
h=1 li,h
(1)
Here, k • k1 is the l1 norm of R24 .
The summation of all consumers’ demands is the DS’s daily
PN −
→
−
→
system-aggregate load L = i=1 Li , which determines the
→
−
DS’s system cost. The system cost C( L ) is determined not
only by the total amount of electricity consumption but also
→
−
by the various features of L , including peak level, peak time
→
−
and ranges of ramps. For example, C( L ) can be high when
→
−
L has a big evening ramp and requires more fast-ramping
generators, such as California’s duck-curve concerns [15]. We
→
−
assume L has M features that impact the system cost and use
→
−
φj ( L ) to represent the extent of the feature j. For example,
→
−
→
−
φER ( L ) is the range of the system evening ramp. φj ( L ) is
→
−
a function of L . Studying the marginal impacts of consumer
i’s demand on those features helps us analyze the dynamic
of how a consumer’s daily demand impacts the system cost
→
−
C( L ) through affecting the system-load features.
→
−
L and its features change when consumer i increases
their demand level and keeps their daily demand profile the
same, which means that this additional electricity demand is
proportionally allocated into 24 hours according to consumer
i’s daily demand profile. Thus, consumer i’s profile-keeping
demand growth will change the DS’s daily system cost by
→
−
affecting L ’s features , which determine the daily system cost.
We define i’s marginal system-feature impacts (MFIs) and
marginal system-cost impact (MCI) as the marginal changes
→
−
−
→
of L ’s profile features and system costs corresponding to Li ’s
profile-keeping demand growth.
Definition Consumer i’s marginal system-feature impact on
→
−
φj ( L ) is
→
−
φj ( L + ∆
M F Ii,j = lim
∆→0
−
→
Li
−
→ )
kL i k1
∆
→
−
− φj ( L )
.
(2)
3
Correspondingly, the marginal system-cost impact of consumer i’s daily demand is
−
→
→
−
→
−
Li
C( L + ∆ −
→ ) − C( L )
kLi k1
M CIi = lim
.
(3)
∆→0
∆
We emphasize that MFIs and MCI are actually Gâteaux
derivatives in R24 with the l1 norm, rather than the usual
vector derivative defined in the space with the l2 norm. We
define MFIs and MCIs as the Gâteaux derivative because
these definitions fundamentally reveal the linkage between
−
→
the economically optimal retail price with Li , which will be
demonstrated in later sections. Because the chain rule is also
valid for the Gâteaux derivative, thus we conclude that
M CIi =
M
X
∂C
M F Ii,j ).
(
∂φj
j=1
(4)
→
−
∂C
Here, ∂φ
is the marginal impact of the feature φj ( L ) on the
j
system cost.
B. Consumer’s demand profile and marginal system impacts
Both MFIs and MCI are sensitive to the demand profile
rather than demand level. According to the definitions of MFIs
−
→
−
→
and MCI, two daily demands Li and Lk share the same
demand profile if they are linearly correlated. We argue that if
two consumers’ daily demands share the same demand profile,
they have the same MFIs and MCI even if they consume
different amounts of electricity.
−
→
−
→
Theorem 1. If Li and Lk are linearly correlated, consumers
i and k share the same MFIs and MCI.
Proof. Assume consumers i and k have the same demand
profile on a particular day, which indicates that their daily
demands are linearly correlated. Thus, there exists a positive
−
→
−
→
constant α such that αLi = Lk . Therefore, consumer i’s
→
−
marginal impact on φj ( L ) is the same as consumer j’s
marginal impact on the same feature because
−
→
→
−
→
−
Li
φj ( L + ∆ −
→ ) − φj ( L )
kL i k1
M F Ii,j = lim
∆→0
∆
−
→
→
−
→
−
Li
φj ( L + ∆ α−
→ ) − φj ( L )
kαLi k1
= lim
∆→0
∆
−
→
→
−
→
−
Lk
φj ( L + ∆ −→ ) − φj ( L )
kL k k1
= lim
∆→0
∆
= M F Ik,j .
Because M F Ii,j = M F Ik,j for all j, we conclude that
M CIi = M CIk .
The injective relation between a consumer’s profile and
his/her MFIs and MCI allow us to have the following corollary,
which is the economic explanation for consumer-demandprofile clustering.
Corollary 1. Demand-profile clustering actually clusters consumers according to their MFIs and MCI. Consumers in the
same profile cluster share the same MFIs and MCI.
C. Consumers’ demand profiles and economic retail prices
Consumer MCI is fundamentally correlated with the economically optimal retail price. A retail price design that
charges consumers according to their MCIs is economically
optimal. If consumer i pays their daily bill by a rate equal
to M CIi , we refer to this price scheme as the profilebased rate (PBR). We argue that the PBR is equivalent to
charging consumers by real-time hourly prices (RTPs) in two
ways. First, the PBR gives each consumer the same economic
incentive as the RTPs. Second, the PBR and RTPs lead to
the same market equilibrium in the DS, which means that the
consumers’ demands and the system costs are the same.
Theorem 2. Charging consumers who are price takers according to the PBR provides every consumer the same incentive in all hours and consequently leads to the same market
equilibrium as charging consumers the RTPs in a DS system.
Therefore, the PBR is also the economically efficient/first-best
retail price.
Proof. Our demonstration includes three steps. First, we build
the linkage between consumers’ MCIs and the RTPs. Then, we
demonstrate that the PBR and RTP provide the same incentives
to every consumer when they decide their hourly demand level.
Finally, we demonstrate that the same individual economic
incentives lead to the same market equilibrium.
Step 1: We use λh to represent the wholesale RTP in hour
→
−
h and Λ = (λ1 , ..., λ24 ) to represent the RTPs in the whole
→
− →
−
day. The daily system cost is Λ · L | . Because we assume
consumers are price takers, they cannot impact the wholesale
→
−
RTP Λ . A consumer impacts the system cost only through
→
−
varying Ł . Thus, we calculate consumer i’s MCI as follows.
→
−
C( L + ∆
M CIi = lim
∆→0
= lim
−
→
Li
−
→ )
kLi k1
→
−
− C( L )
∆
−
→
→
− →
−|
→
− →
−
Li
|
Λ · (L + ∆ −
→ ) − Λ · L
kLi k1
∆
−
→
24
X
Li |
li,h
λh .
=
P24
−
→
k Li k1
m=1 li,m
h=1
∆→0
→
−
= Λ·
(5)
Here, li,h is consumer i’s consumption during hour h. Thus,
the MCI is a weighted average of the RTPs, where the weight
of each hour’s RTP is the proportion of the hourly demand to
the daily total demand.
Step 2: We use Ui (li,1 , ..., li,24 ) to represent consumer
i’s utility function. Given the PBR, consumer i pays
M CIi (li,1 , ..., li,24 ) for each unit of electricity consumption.
M CIi (li,1 , ..., li,24 ) is a function of li,h . Thus, consumer i
determine his/her hourly demands by solving the following
welfare maximization problem
max Ui (li,1 , ..., li,24 ) − M CIi (li,1 , ..., li,24 ) ×
(li,h ,∀i)
24
X
li,m .
m=1
(6)
4
Therefore, the first-order conditions are
24
X
∂Ui (li,1 , ..., li,24 )
∂M CIi
=
×
li,m + M CIi , ∀h. (7)
∂li,h
∂li,h
m=1
|
{z
} | {z }
A
B
The left-hand side of (7) is consumer i’s marginal utility
of demanding li,h in hour h. The right-hand side of (7)
is the marginal cost of consumer i when this consumer
demand li,h in hour h, which includes two parts. Consumer i’s
consumption in hour h indirectly impacts his/her costs on all
hours by affecting the rate M CIi . Term A is the sum of the
marginal indirect impacts of li,h on consumer i’s costs on all
hours. Consumer i’s consumption on hour h directly determine
his/her cost in the same hour. Term B is the marginal direct
effect of li,h on the cost for hour h. The summation of Term
A and B is the RTP for hour h.
α
24
X
∂M CIi
×
li,m + M CIi
∂li,h
m=1
P
P
24
X
λh m6=h li,m − m6=h λm li,m
×
li,m
=
P24
( m=1 li,m )2
m=1
+
24
X
li,m
λm = λh .
P24
t=1 li,t
m=1
(8)
Therefore, Equation (7) is equivalent to
∂Ui (li,1 , ..., li,24 )
= λh , ∀h.
∂li,h
(9)
When the RTP is implemented in the DS, consumer i’s welfare
maximization problem is
max Ui (li,1 , ..., li,24 ) −
(li,h ,∀i)
24
X
λh li,h .
at reducing the system costs by changing the system-load
features.
Consumer i’s MFIs and MCI are functions of its own
demand profile P fi and the aggregate system-load profile
→
−
L according to the definition of MFIs and MCI. Thus,
→
−
consumer i’s marginal impact on φj ( L ) or the system cost can
→
−
→
−
be denoted by M F Ii,j ( L , P fi ) and M CIi ( L , P fi ). When
−
→
−
→
consumer i reduces their demand from Li to αLi , where
0 ≤ α ≤ 1, and other consumers keep their demand levels
the same, this consumer’s MFIs and MCI will correspondingly
→
− −
→
−
→
→
− −
→
change to M F Ii,j ( L − Li + αLi , P fi ) and M CIi ( L − Li +
−
→
→
−
αLi , P fi ); this is because the system load switches from L
→
−
−
→
→
−
to L + (α − 1)Li . Consequently, the total change of φj ( L )
due to the reduction of consumer i’s consumption is
Z 1
→
−
−
→
M F Ii,j ( L + (x − 1)Li , P fi )dx.
(12)
Bi,j =
(10)
h=1
Equation (9) also provides the first order conditions of the
optimization problem (10). Thus, the PBR and RTP provide
the same economic incentives to consumer i in all hours. This
conclusion is true for any given consumer i.
Step 3: When all consumers are price takers, the RTP
→
−
in hour h λh ( L ) is a function of only the aggregate demand, and it is not affected by individual consumer demands.
→
−
Therefore, the supply cost of the whole DS is C( L ) =
P24
→
− PN
h=1 λh ( L )
i=1 li,h . Consequently, for both the PBR and
RTP scenarios, the market equilibrium of all hours is solved
from
→
−
∂Ui (li,1 , ..., li,24 )
= λh ( L ), ∀i, h.
(11)
∂li,h
Therefore, the PBR and RTPs lead to the same market equilibrium. Because the RTPs are the economically optimal retailprice scheme, the PBR is also economically optimal.
D. Consumer’s values for reducing system cost in DR projects
A consumer’s marginal system impacts reflect how sensitive
the system costs and load features are when this consumer
changes their demand. Thus, calculating MFIs and MCI is
critical for targeting consumers for DR projects, which aim
Equation (12) allows us to deduce the following conclusion
directly as it pertains to the value of reducing a consumer’s
→
−
demand level for changing the feature φj ( L ).
→
−
−
→
→
−
Corollary 2. If M F Ii,j ( L +(x−1)Li , P fi ) > M F Ik,j ( L +
−
→
(x − 1)Lk , P fk ) for all x ∈ [α, 1], reducing consumer i’s
→
−
demand level is more valuable for changing φj ( L ) than
reducing consumer k’s when the range of demand reduction
is in [α, 1]. Furthermore, if consumers i and k have the same
demand profile, reducing their demands by the same amount of
→
−
energy is equally valuable for changing φj ( L ) because they
have the same MFI functions.
We can calculate the system benefit caused by a consumer’s
demand change according to this consumer’s MCI function.
−
→
Let us assume that consumer i changes their demand from Li
−
→0
to Li . The change of consumer i’s demand can be decomposed
into two processes. First, consumer i reduces their demand
−
→
from Li to 0, causing a system benefit. Then, consumer i
−
→
increases their demand from 0 to L0i , leading to a system cost.
Definition The total system benefit from consumer i’s demand
→
−
→ −
change from Li to L0i is
Z 1
→
−
−
→
Bi,cost =
M CIi ( L + (x − 1)Li , P fi )dx−
0
Z 1
−
→
→
− −
→
M CIi ( L − Li + y L0i , P fi0 )dy.
(13)
0
→
−
→ −
If consumer i changes their demand from Li to L0i because
he/she participants a DR project, Bi,cost is the maximum
payment to this consumer that would guarantee a non-negative
benefit for the DS by including consumer i into the DR project.
If the payment to the consumer i is above Bi,cost , including
this consumer causes a economic loss for the DS.
The fact that reducing consumer i’s demand is more valu→
−
able for changing φj ( L ) than reducing consumer k’s demand
does not mean that reducing consumer i’s demand is more cost
effective than reducing k’s. Equation (12) does not include
information about the cost of reducing a consumer’s demand.
In actuality, the cost of reducing a consumer’s demand is
5
determined by the consumer’s demand function [16], which
is not included in our discussion here.
Cauchy-Kovalevskaya Theorem, Equation (15) has a unique
analytic solution near (λ1 , ..., λ24 ). Because M CIi is a solution of (15), γi must be equal to M CIi .
III. P ROFILE CLUSTERING AND DAILY- AVERAGE - RATE
CLUSTERING
The economic optimality of the PBR scheme indicates
the economic essence of consumer-demand-profile clustering.
Theorems 1 and 2 together indicate that consumers who share
the same demand profile pay the same daily-average rate under
the PBR scheme. Therefore, consumers in the same demandprofile cluster pay the same average rate when the PBR is
implemented. Because of the economic linkage between the
PBR daily-average rate and RTPs, as demonstrated in the proof
of Theorems 2, consumers in the same demand-profile cluster
also pay the same daily-average rate when the RTPs are used.
Corollary 3. Consumers in the same demand-profile cluster
pay the same daily-average rate when the RTPs or PBR are
implemented.
The conclusion in Corollary 3 can be generalized to any
economically optimal price scheme in the following theorem,
which is the core conclusion of this paper. In this theorem,
we conclude that an implemented retail price is economically optimal if, and only if, every consumer’s daily-average
rate calculated according to the implemented retail price is
their MCI. Consequently, if an economically optimal price
scheme is implemented, the partition of consumers according
to consumer-demand profiles is a refinement of the partition of
consumers according to daily rates. When consumers in two
clusters have different MCIs, the two partitions must be the
same.
Theorem 3. A price scheme is economically optimal if, and
only if, every consumer’s daily-average rate is their MCI.
Therefore, an economically optimal price will lead consumers
who share the same demand profile to pay the same rate for
their 1-KWh consumption even if they have different demand
levels.
Proof. Theorem 2 proves the sufficiency. Here, we prove the
necessity.
We assume that Γ is an economically optimal price scheme.
The daily-average rate of consumer i is γi , which is a function
of consumer i’s hourly demands and can differ between
consumers. Consequently, consumer i chooses their hourly
demands under the price scheme Γ by solving for the following
welfare optimization problem:
max Ui (li,1 , ..., li,24 ) − γi (li,1 , ..., li,24 )
(li,h ,∀i)
24
X
li,m .
(14)
m=1
Then, consumer i’s hourly demand bundle is solved from the
first-order conditions.
24
∂γi X
∂Ui (li,1 , ..., li,24 )
=
li,m + γi , ∀h.
∂li,h
∂li,h m=1
(15)
Because Γ is economically optimal, every consumer’s
∂Ui (li,1 ,...,li,24 )
marginal utility
in each hour must be equal
∂li,h
to the this hour’s RTP λh . Consequently, according to the
IV. C RITERIA TO ASSESS THE PERFORMANCE OF AN
IMPLEMENTED RETAIL PRICE SCHEME IN A DS
The economic linkages connecting the consumer-demand
profile, the MCI, and the economically optimal price scheme
in Theorem 3 are useful to assess the economic performance
of an implemented retail price scheme in a DS. We can
induce the most important corollary of this paper directly from
Theorem 3, which allows us to develop a sequence of criteria
to assess how different a current price scheme is from the
economically optimal price scheme.
Corollary 4. When the implemented price scheme in a DS is
economically optimal, the partition of consumers according to
their demand profiles must be a refinement of the partition of
consumers according to their daily-average rates. If consumers
with two demand profiles always have different MCIs, those
two partitions must be the same.
Proof. Assuming consumers i and k belong to the same
demand-profile cluster. We demonstrate that consumers i and
k also must belong to the same daily-average-rate cluster when
the implemented retail price is economically optimal.
Because consumers i and k share the same demand profile,
their marginal system-cost impacts are the same, so that
M CIi = M CIk . If consumers i and k pay different dailyaverage rates ri 6= rk , we conclude either ri 6= M CIi or
rk 6= M CIk , which is contradictory with the assumption
that the implemented price is economically optimal according
to Theorem 3. Thus, consumers i and k must pay the same
daily-average rate and belong to the same daily-average-rate
cluster.
To a large extent, Corollary 4 allows us to assess the
economic efficiency, even if the information about the RTPs
is completely unknown.
The economic efficiency of an implemented retail price
can be examined by comparing the partitions of consumers
clustered according to their demand profiles and the partitions
of consumers clustered according to their daily-average rates.
We use Ωp to represent the demand-profile-based partition
and Ωr to represent the average-rate-based partition. When
consumers with two demand profiles always have different
MCIs, the implemented retail price is economically efficient
only if Ωp and Ωr are identical. The difference between Ωp and
Ωr indicates the difference between the market equilibrium
induced by the implemented retail price and the market
equilibrium induced by the economically optimal price. The
larger the difference, the more the consumers having the same
demand profile are charged differently.
When consumer’s MCI is an injective function of their
demand profile, we define the degree of consistency between
the implemented price and an economically optimal price as
the difference between Ωp and Ωr . We assume that the size of
the profile partition Ωp is T , and the size of rate partition Ωr is
S. Thus, there are T types of consumer profiles and S types of
6
daily-average-based rates. The tth demand-profile-based type
is denoted by ωp,t , and ωp,t ∈ Ωp . The, respectively, the sth
daily-average-rate type is denoted by ωr,s , and ωr,s ∈ Ωr .
a measure assessing the difference between two rankings [18].
Thus, we use the Kendall tau distance to measure the extent of
the distortion caused by an implemented retail market price.
Definition The degree of consistency (DOC) between the
implemented price and an economically optimal price in the
DS is
PT PS
2 × t=1 s=1 ρts log2 ρρtts
ρs
,
DOC = PT
PS
− t=1 ρt log2 ρt − s=1 ρs log2 ρs
if ρt ρs < 1
Definition We define the distortion (Dt) caused by the implemented retail price as
=1, otherwise
(16)
Here,
ρt =
|ωp,t |
|ωr,s |
|ωp,t ∩ ωr,s |
, ρs =
, and ρts =
.
N
N
N
|ωp,t | is the number of consumers having the tth demand
profile. |ωr,s | is the number of consumers paying the sth daily
average rate. |ωp,t ∩ωr,s | is the number of consumers that have
the tth demand profile and pay the sth daily average rate.
DOC is the normalized mutual information between Ωp and
Ωr . Normalized mutual information is a non-negative index
used in information theory to compare two partitions on the
same set [17]. DOC measures how much the diversity of a
consumer’s daily-average rate is reduced by when we know
this consumer’s daily profile. When the implemented price
scheme is economically optimal, DOC is equal to 1, which
means that Ωp and Ωr are identical. DOC is equal to 0 if Ωp
and Ωr are independent of each other, which is the worst case.
The higher the value of the DOC, the less the difference there
is between the market equilibrium induced by the implemented
price and the economically optimal market equilibrium.
It is rare that consumers with different profiles have the
same MCI. Therefore, the DOC can be used broadly for
various DSs and markets.
The DOC can test the efficiency of the implemented retail
price, even if we have no information about the system
→
−
→
−
daily cost C( L ). When more information about C( L ) is
revealed, further measurements can be established to assess
the economic efficiency and impacts in more detail.
→
−
If we know that C( L ) is monotonically increasing with a
→
−
→
−
linear combination of L ’s M features {φj ( L ), j = 1...M },
we can measure the economic distortion caused by the implemented retail price, even if we do not know the exact form
→
−
→
−
of C( L ). Assume that system cost C( L ) monotonically inPM
→
−
creases with j=1 µj φj ( L ), where µj > 0. Then, consumer
PM
i’s MCI monotonically increases with Φi = j=1 µj M F Ii,j .
When an economically optimal price scheme is implemented,
consumer i pays a higher daily rate than consumer k if, and
only if, Φi > Φk . Therefore, we call Φi the MCI index.
We can sort consumers in a list according to the ascending
order of their MCI index. If this list is different from the
list of consumers that is sorted in the ascending order of
daily-average rates ri , then the implemented retail price is
not economically optimal. The greater the difference between
the two lists, the more distortion caused by the implemented
retail price. In information theory, the Kendall tau distance is
Dt =
|{(Φi , Φk )|Φi > Φk ∧ ri < rl }|
× 100%.
N (N − 1)/2
(17)
If we have complete information about the function of
→
−
C( L ), such as the local marginal prices, then we can accurately measure the inefficiency of the implemented retail price.
Consumer i is subsidized by the implemented price scheme if
M CIi > λi and taxed if M CIi < λi . The amount
P24 of money
subsidized to consumer i is (M CIi − λi ) × h=1 li,h .
V. E MPIRICAL ANALYSIS ON A C HINESE DISTRIBUTION
SYSTEM
A. Data background and processing
We use real-time meter data from an industrial county of
one of the largest cities in China to analyze the consumer
marginal system impacts and to assess the effectiveness of the
current retail price scheme in this distribution system. In this
county, there are 2, 110 consumers, all of whom are equipped
with smart meters. The data ranges from January 2014 to
December 2014, and consists of the electricity consumption
of commercial, industrial, public-service, and residential customers measured at one-hour intervals. The total number of
daily demand profiles is 516, 494 [19].
This county is representative of a Chinese distribution
system. In contrast with the United States, China’s electricity is mostly consumed by large commercial and industrial
consumers. In fact, 70% of China’s electricity is consumed by
the industrial and commercial sectors [20]. In addition, most
distribution systems in China usually serve industrial, commercial, and residential consumers simultaneously. In contrast
with residential consumers, large industrial and commercial
consumers are easier to manage individually and to price
differently. In the distribution system studied in this paper,
the majority of consumers are industrial and commercial
companies. However, there is still a significant number of
residential and public-service users.
In this analysis, we normalize the daily demand profiles
and use the k-means algorithm to cluster all normalized daily
demand profiles into 36 profile types using the elbow method
[21]. In each cluster, the Euclidean distance from a demand
profile to the cluster kernel is less than 5% of the l2 norm of
the cluster kernel [22]. The k-means clustering is implemented
using R.
We calculate all consumers’ MFIs for the system-load
profile’s morning ramp range, evening ramp range, and peak
demand level, which are represented by MFIMR , MFIER , and
MFIPD , respectively. China does not have a wholesale electricity market while it does have a fully-regulated wholesale
price scheme. Consequently, the system cost is determined
mainly by the three features that we analyzed here. According
to Theorem 1, we only need to calculate each profile type’s
MFIs instead of calculating each consumer’s MFIs, rather than
7
needing to calculate each consumer’s MFIs. Therefore, the
computational load is significantly reduced.
We use consumers’ MFIs to explore high-marginal-impact
consumers and to examine whether these consumers are
large users. Before the consumers’ heterogeneous profiles are
realized, China designs their retail prices and DR projects
according to the demand-level-based principle to incentivize
consumers, especially industrial consumers, to reduce the
total consumption [23]. Since 2004, China has allowed local
governments to differentiate electricity price rates according
to consumers’ demand levels [24]. Various enforced and
voluntary energy conservation projects have been implemented
during the last decade [25]. We use our data to examine
whether it is economically efficient and fair to differentiate
consumers’ rates or to target consumers for a DR project
according to their demand levels.
(a) System-load profile
B. Consumers’ marginal system impacts
MFIs provide us the information about whose consumption
were more responsible for the aggregate load profile’s ramps
or peak level. For a given feature, the consumers’ MFIs are
significantly heterogeneous. In the Appendix A, we present
the histograms of MFIMR , MFIER , and MFIPD .
Some consumers have negative MFIMR and MFIER on particular days. These consumers’ consumptions are very valuable
in moderating the aggregate load’s morning or evening ramps
on those days. If those consumers demand more electricity and
keep their demand profiles, the aggregate demand will lead
to more moderate morning or evening ramps. For instance,
on April 18th, 2014, the demands of Type-6 consumers is
ramping down when the aggregate demand experiences the
morning ramp (Fig. 1). Consequently, Type 6 had a negative
MFIMR on that day.
Consumers’ demands affect the distribution system’s costs
through different dynamics. The costs serving some consumers
are mainly caused by dealing with their morning ramps while
the costs serving other consumers are mainly due to their
impacts on the system’s evening ramps or peak-demand levels.
In Fig. 2, we plot the MFIs of the two types of consumers
for March 24th, 2014. The system costs for dealing with the
morning ramp are caused more by the consumption of Type16 consumers rather than by the consumption of Type-20 consumers. We also see that the system costs for dealing with the
evening ramps and peak loads are due more to the consumption
of Type-20 consumers rather than the consumption of Type
16-s consumers.
Understanding the long-term distributions of a consumer’s
MFIs is necessary for examining how expensive it is to serve
this consumer and where the expense comes from. The longterm distributions of consumers are also useful for planning
distribution system’s facilities and assets, such as choosing
the sizes of the transformers. We summarize the medians and
variations of every type of consumer MFIs for three features
in Fig. 3. The median cost of serving some consumers is less
than that of serving others. For example, Type 1 consumers
have lower medians of MFIs for all three features than Type
3 consumers.
(b) Aggregate demand profile of Type-6 consumers
Fig. 1. The difference between the system-load profile and Type-6 consumer’s
demand profile On April 18th, 2014
Fig. 2. Type-16 and Type-20 consumer’s MFIs for three features on March
24th, 2014
Many consumer-profile types have MFIs with significant
fluctuations. For example, the median of Type 3’s MFIMR is
nearly 10 times larger than its 25% quantile value. Further,
MFIPD has smaller within-type variation than either MFIMR
and MFIER . The heterogeneities of MFIMR and MFIER are
mainly within types, rather than between types. In contrast,
MFIPD has more significant cross-type heterogeneity than
MFIMR and MFIER . The significant within-type variation of
MFIs is caused by the large difference between daily system
load levels and load profiles. If two days have different system
load levels or load profiles, the same profile type has different
MFIs of the same given feature for those two days.
We also note that the variations of consumer MFIs are
8
(a) Variations of MFIMR within and between profile types
Fig. 4. Weak correlation between consumers’ MFIMR and demand levels in
2014
C. Consumer-demand levels and marginal system impacts
(b) Variations of MFIER within and between profile types
(c) Variations of MFIPD within and between profile types
Fig. 3. Variations of consumer’s MFIs within and between profile types
positively correlated with the absolute value of the mean of
their MFIs. Consumers that have significantly higher or lower
average marginal system impacts usually also usually have
significantly varying system impacts. In the Appendix B, we
provide the scatter plots of consumers’ yearly average MFIs
and the associated standard deviations.
We note that nearly all profile types have both negative and
extremely high positive MFIMR and MFIER . Therefore, every
type of consumer can be valuable in moderating system ramps
on some Key Days, while their demands significantly aggravate
system ramps on certain other days. For example, Type 33 has
the highest average MFIMR in 2014. However, this type still
has negative values of MFIMR for 5% of the days.
The demand-level-based principle must be replaced by the
marginal-system-impacts-based principle for designing retail
price. We demonstrate that a consumer’s MCI and MFIs are
determined by their demand profile rather than their demand
level. Empirical evidences from our distribution system further
confirm that a consumer’s demand levels are not positively
correlated with their MFIs. Thus, differentiating prices by
consumers’ demand levels will distort the price signal and
deepen the economic inefficiency.
In the distribution system under study, consumers’ MFIs are
not positively correlated with their demand levels. In Fig. 4, we
plot the correlations between consumers’ demand levels and
their MFIMR . Many large consumers have smaller MFIMR
than small consumers. We note that MFIMR ’s variations
significantly increase when consumers’ demand levels exceed
a certain level. The relationships between the consumers’
demand levels and their MFIER and M F IDR have similar
behaviors; this is shown in the Appendix D. Therefore, when
the system costs monotonically increases with the system ramp
ranges and the system-peak-demand levels, the marginal costs
of serving large consumers are not necessarily higher than the
marginal costs of serving small consumers. Thus, in this DS,
it would be highly inefficient if large consumers were simply
charged more than small consumers.
D. Assessing the efficiency of the implemented retail-price
scheme
Consumers in this DS are charged differently. Commercial
and industrial consumers are charged time-of-use rates while
residential and public-service consumers are charged a flat
rate. The detailed rates are listed in the Appendix C. Because
China does not have a local-marginal-price-based wholesale
market, the daily system costs of a DS is vague. Consequently,
the economic effectiveness of the implemented price schemes
cannot be confirmed by directly comparing consumers’ rates
with their marginal impacts on the system costs.
Thus, we use the DOC index to examine the effectiveness
of the retail prices, and use the Dt index to assess the extent of
the price distortion in this DS. In addition, these two indices
9
also reveal information about the cross subsidies caused by
the implemented price.
In order to calculate the DOC index of this DS in each
day, we calculate each consumer’s daily-average rates, which
are continuously distributed between 0.35 Yuan/kWh to 0.87
Yuan/kWh. Then, we equally split the interval [0.35, 0.87] to
36 segments, which is the number of consumers’ profile types,
and cluster the consumers into the same daily-average-rate
type if their daily-average rates are in the same segment. For
each day, we use the index given by (16) to calculate the
degree of consistency (DOC) between the profile clustering
and the daily-average-rate clustering.
The results of the DOC calculation demonstrate that the
implemented price is inefficient. In Fig. 5(a), we present the
monthly average DOC of both weekdays and weekends. For
all days, the profile clustering results are significantly different
from the daily-average-rate clustering results. The highest
daily DOC index in these 11 months is 0.3825, which is
significantly less than 1. Thus, in this DS, very few consumers
sharing the same demand profile pay the same daily-average
rate in the year of 2014. Therefore, the retail price scheme is
deemed inefficient, despite the lack of information regarding
the DS’s daily cost structure.
The DOC value vary by days. During the weekdays of June
to August, the DOC values are, on average, higher than on
the other days. Therefore, more consumers within the same
profile type paid similar rates in on summer weekdays than
on other days.
In this research, we assume that a consumer’s MCI on each
day is monotonically increasing with the MCI index Φi =
M F Ii,M R + M F Ii,ER + M F Ii,P D . With this assumption in
mind, we calculate the Dt index. We first sort consumers into a
list according to their values of Φi , and then we sort consumers
into another list according to their daily average rates. Nest,
we compare the two lists and calculate the Dt index of each
day according to (17). We plot the monthly average Dts of
weekdays and weekends in Fig. 5(b).
If our assumption is true, the implemented price creates
significant distortions in this DS. If we randomly select two
consumers on the same day, there is a 40% to 60% probability
that the consumer with a low M CI was charged more
than the consumer with a high M CI for 1 KWh electricity
consumption. Thus, a large number of low-M CI consumers
was forced to subsidize the electricity usage of the high-M CI
consumers.
Therefore, if the three features of the daily system load
impact the daily system cost equally, the implemented price
is unfair and leads to an economic inefficiency, as it distorts
the price signal by charging the consumers that caused high
marginal system costs lower rates. We notice that the Dt values for the summer season weekdays are relatively lower than
for other days. Thus, the distortion caused by the implemented
price is relatively smaller on those days than on other days.
(a) Monthly average DOCs of weekdays and weekends under the
implemented retail price
(b) Monthly average Dts of weekdays and weekends under the
implemented retail price
Fig. 5. The inefficiency and distortion caused by the implemented retail price
in 2014
VI. C ONCLUSION AND DISCUSSION : MANAGING
CONSUMERS ACCORDING TO THEIR MARGINAL SYSTEM
IMPACTS
In this paper, we explain how a distribution system’s daily
costs of purchasing electricity are affected by consumers’ daily
demand profiles. We clarify that a consumer’s marginal system
impacts are determined by their demand profiles rather than by
their demand levels. Thus, profile-based consumer clustering
clusters consumers according to their marginal system impacts.
We also demonstrate that a consumer’s daily marginal systemcost impact is the economically optimal daily-average rate.
If we design a profile-based price scheme whose rate is
equal to a consumer’s daily marginal system cost impact, that
profile-based price is equivalent to real-time pricing and is an
economically optimal price scheme.
We argue that clustering demand profiles is identical to
clustering consumers according to their marginal system impacts. When consumers with various demand profiles have
different marginal impacts on the system’s daily cost, demandprofile clustering is identical to daily-average-rate clustering
only if the implemented price is economically optimal. Based
on these theoretical analyses, we develop a criteria system to
10
evaluate the economic efficiency of an implemented retail price
scheme in a distribution system. Our criteria system can test
the economic efficiency of a retail price scheme, even if we do
not have information about the distribution system’s daily cost
structure. These criteria can also be used to target consumers
who either overpay or underpay for their electricity usage.
We analyze data from a real distribution system in China
and examine consumers’ marginal system impacts and the
efficiency of the retail price scheme there. The empirical results deepen our insights and understanding of the consumers’
various marginal system impacts on the features of the systemload profiles. The results strongly suggest that the retail price
schemes implemented in China are inefficient and should be
redesigned.
In general, the designs for retail-price schemes or demandresponse projects need to be smarter by considering the
following three issues.
First, it is very valuable for a distribution-system operator
or utility to accurately target and manage Key Days when a
profile type’s MFIs are either extremely low or high. Fig. 3
demonstrates that there are a small number of Key Days for
each profile type. Thus, every consumer will be affected on
very few days if the system operators or utilities target and
manage the Key Days. However, on these small number of Key
Days, it is either very valuable or extremely expensive to serve
the corresponding type of consumers. Therefore, distribution
systems can significantly improve economic performance by
developing demand-management projects for Key Days for
each profile type.
Second, the large within-type variations of MFIs remind
us that demand management, including retail-pricing design,
must make a trade-off between long-term median and variation
of a consumer’s system impacts. In the studied distribution
system in our research, consumers who have small MFIs
usually have stable MFIs over days. For these consumers, their
retail rates can be stable throughout the whole year. In contrast,
complicated designs for retail price rates or demand-response
projects are necessary in order to manage those consumers
whose MFIs have large values and significantly vary.
Finally, and most importantly, the marginal-system-impactbased principle should replace the demand-level-based principle in order to optimize the demand management. The consumers’ marginal system impacts reflect which consumption
behaviors are expensive to serve and need to be the foundation
for retail-price designs.
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A PPENDIX
A. MFI distribution
Consumers’ MFIs vary significantly over the entire 11
months. Thus, various consumers have significantly different
marginal impacts on the system load’s morning ramps, evening
ramps, and peak-demand levels. In Fig. 6, we summarize the
histograms of the consumers’ MFIs for three features in 2014
in the Chinese DS.
11
(a) Mean and standard deviation of consumers’ MFIMR
(a) Histogram of consumers’ MFIMR during the 11 months
(b) Mean and standard deviation of consumers’ MFIER
(b) Histogram of consumers’ MFIER during the 11 months
(c) Mean and standard deviation of consumers’ MFIPD
Fig. 7. Mean and standard deviation of consumers’ MFIs
(c) Histogram of consumers’ MFIPD during the 11 months
Fig. 6. Histogram of consumers’ MFIs during the 11 months
The histograms demonstrate that a small proportion of
consumers have high marginal impacts on the system’s morning and evening ramps. We should target and manage these
consumers. Also, the variations of MFIPD , which is the
consumer’s marginal impact on the daily-peak-demand levels,
suggest that the demand charge could differentiate the rates
for efficiently managing the consumers’ peak-demand levels.
B. Mean and standard deviation of MFIs
For all three features, the absolute value of the mean and
standard deviation of a MFI are positively correlated. In Fig. 7,
we summarize the correlation between each consumer’s mean
and standard deviation of every MFI in the Chinese DS in
2014.
From the three figures in Fig. 7, we find that a consumer’s
MFI is either has a large average value but varying, or alternatively, has a small average value but stable. Thus, changing the
system load’s features requires us to target the high-marginalimpact consumers on Key Days. Consumers who have large
impacts on some days can have small impacts on other days.
Therefore, the targeting of consumers can be replaced by the
targeting of the pairs of (consumer, day).
C. Retail price in the studied China’s DS
China has various retail price schemes per cities. In the
city under study, consumers are classified into four types:
industrial, commercial, residential, and public-service consumers. Industrial and commercial consumers are charged by
time-of-use rates. Industrial consumers have higher rates than
commercial consumers in all time periods. Residential and
public-service consumers are charged using a flat-rate price
scheme.
Here, the off-peak period is 11pm − 7am. The high-plate
period is 10am − 3pm and 6pm − 9pm. During the high-plate
period, there are two peak-time periods defined for summer,
which is 11am − 1pm and 8pm − 9pm, during the months of
12
TABLE I
I MPLEMENTED RETAIL PRICES IN THE C HINESE DS
Peak
Industrial
Commercial
Residential
Public service
High-plate
0.97
0.93
0.48
0.48
Mid-plate
0.89
0.85
0.48
0.48
Off-peak
0.63
0.59
0.48
0.48
0.37
0.33
0.48
0.48
(a) Correlation between consumers’ MFIER and demand levels
(b) Correlation between consumers’ MFIPD and demand levels
Fig. 8. Correlation between consumers’ MFIs and demand levels
July, August, and September. The rest of the time is mid-plate
period.
D. Consumers’ demand levels and MFIs
Consumers’ MFIs for evening ramp and peak-demand levels
are also not positively correlated with demand levels. Many
consumers with high demand levels have low values for
MFIER and MFIPD .
We notice that when the demand level increases above
a certain threshold, the MFIs are clearly clustered into two
types. One type of consumer has high demand levels and
high marginal impacts. The other type of consumers has
high demand levels but small marginal impacts. This result
suggests that there are some other consumer characteristics
that determine their MFIs, which needs to be investigated
further.
E. Nomenclature
N : number of consumers;
li,h : consumer i’s demand in hour h.
−
→
Li : consumer i’s daily-demand vector.
→
−
L : system-daily-load vector.
P fi : consumer i’s demand profile.
→
−
φj ( L ): the extent of the feature j of system-daily-load
vector that influences the system-daily cost.
→
−
C( L ): the system-daily cost.
M F Ii,j : consumer i’s marginal system-feature impact on
→
−
φj ( L ).
M CIi : consumer i’s marginal impact on the system-daily
cost.
Ui (li,1 , ..., li,24 ): consumer i’s utility function.
Bi,cost : the total system benefit from consumer i’s demand
change.
Γ: an economically optimal price scheme.
γi : the daily-average rate for consumer i under Γ.
Ωp : the demand-profile-based partition.
Ωr : the average-rate-based partition.
ωp,t : the tth demand-profile-based type, which is an element
of Ωp .
ωr,s : the sth daily-average-rate type is denoted by ωr,s ,
which is an element of Ωr .
ρt : the proportion of total population in a DS that have the
tth demand profile .
ρt : the proportion of total population in a DS that have the
tth demand profile.
Φi : the MCI index of consumer i.