IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 17, NO. 1, FEBRUARY 2002
41
Components of Nodal Prices for
Electric Power Systems
Luonan Chen, Senior Member, IEEE, Hideki Suzuki, Tsunehisa Wachi, and Yukihiro Shimura
Abstract—We present a method to provide a detailed description of each nodal price, by breaking down each nodal price into
a variety of parts corresponding to the concerned factors, such as
generations, transmission congestion, voltage limitations and other
constraints or elements. This full information for nodal prices can
be used not only to improve the efficient usage of power grid and
congestion management, but also to design a reasonable pricing
structure of power systems, or to provide economic signals for generation or transmission investment. Several numerical examples
demonstrate this approach.
Index Terms—Active set, congestion, decomposition, deregulation, marginal cost, nodal price, optimal power flow.
I. INTRODUCTION
E
LECTRIC utilities have experienced a period of rapid
changes especially in market structure and regulatory
policies in many parts of the world. Because of the emergence
of independent power producers (IPP) as well as the changing
structure of the electricity supply industry, the electric power
industry has entered an increasingly competitive environment
under which it becomes more realistic to improve economics
and reliability of power systems by enlisting market forces [1],
[2], [11], [15]. To induce efficient use of both the transmission
grid and generation resources by providing correct economic
signals, a nodal price or spot price theory for the deregulated
power systems was developed [1], [2]. Different from the conventional single-price structure, the nodal price varies in both
space and time and is composed of the variable operation costs
and any additional charges for maintaining quality and reliable
electricity services [5]. So far a considerable number of literature designing the nodal price have been published by taking
into the consideration of not only the short-term operation
costs but also values of the ancillary electrical services, such
as [1]–[5]. Most of the existing theories of nodal price use the
Lagrangian multipliers as shadow prices to evaluate the equivalent values of constraints or factors for security, reliability
and quality. Although these Lagrangian multipliers provide
valuable information to some extent based on the shadow price
of each active constraint (e.g., transmission congestion, voltage
limits), it cannot give a detailed description of each nodal
Manuscript received January 19, 2000; revised June 28, 2001.
L. Chen is with the Department of Electrical Engineering and Electronics, Osaka Sangyo University, Osaka, Japan (e-mail: chen@elec.osakasandai.ac.jp).
H. Suzuki and T. Wachi are with the Department of Power Systems, KCC
Ltd., Tokyo, Japan.
Y. Shimura is with the Electric Power Development Company, Ltd., Tokyo,
Japan.
Publisher Item Identifier S 0885-8950(02)00888-X.
price, which is demanded by the power industry. In other
words, it is still unclear which generator or constraint a nodal
price is influenced from and how much each factor of a power
system contributes to the nodal prices.
To break down or trace nodal prices into various components,
a number of useful computation methods, such as [12]–[15],
have been developed. However, all of these methods either involve some heuristic implementations that make the decomposition of nodal prices not unique due to the dependence on the
heuristic settings, or are only partial decomposition of independent factors. Unlike ac power flow which generally cannot be
traced to the sources or routes, we will show in this paper that
the nodal prices can actually be theoretically and uniquely decomposed into all independent components associated with not
only each generator but also each constraint or element of the
whole power system due to their linearity to all factors.
This paper aims theoretically to propose a method to provide a detailed description of each nodal price, i.e., we break
down each nodal price into a variety of parts corresponding to
the concerned factors, such as generations, transmission congestion, voltage limitations and other constraints. The decomposition is unique and components in each nodal price are identical
to their increment values from the economics viewpoint because
the derivations are based on the marginal conditions [11]. This
full information for nodal prices can be used not only to improve the efficient usage of power grid and congestion management, but also to design a reasonable pricing structure of power
systems, or to provide economic signals for generation or transmission investment. In the next section, we give a general explanation by deriving nodal prices from an Optimal Power Flow
(OPF) based model. Section III proposes a theoretical method
to determine each component of nodal prices with a number of
examples, and Section IV is numerical simulation. Finally, we
give several general remarks to conclude this paper in Section V.
II. DERIVATION OF NODAL PRICE
In this section, we first define the operation problem of a
power system as an OPF problem and then derive nodal prices
for both real and reactive power at an optimal solution.
A. Formulation
buses system, let
and
, where
and
represent real and
reactive power demands of bus- , respectively. Define the
,
variables in power system operation to be
such as real and imaginary parts of each bus voltage (or voltage
value and its angle).
For a
0885–8950/02$17.00 © 2002 IEEE
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IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 17, NO. 1, FEBRUARY 2002
Therefore, the operation problem of a power system for
the given loads ( , ) can be formulated as an OPF problem
[8]–[10]
(6)
Min
s.t.
for
(1)
(2)
(3)
and
have
and
equations, respectively, and are column vectors while
stands for the transpose of a vector .
: scalar, short-term operating cost, such as fuel
cost;
: vector, equality constraints, such as bus
power flow balances (Kirchoff’s laws);
: vector, inequality constraints including
limits of all variables
includes all variable limits and function
where
limits, such as upper and lower bounds of transmission lines,
generation outputs, stability or security limits [10], etc. When
is a fuel cost function of the system, can be expressed as
where is the fuel cost of
the th generator. Notice that can also be a benefit function
although this paper takes as a cost function.
Obviously, (1)–(3) are a typical OPF problem as far as the
demands ( , ) are given. There are many efficient approaches
which can be used to obtain an optimal solution, such as
successive linear programming, successive quadratic programming, the Newton method, the – decomposition approach
[6], [7], the surrogate constraint and functional transformation
approaches [8]–[10].
where
B. Nodal Price
Define the Lagrangian function (or system cost) of (1)–(3) as
, then
(4)
and
where
are the Lagrangian multipliers (or dual variables) associated to (2) and (3), respectively, and are usually
explained as shadow prices from the viewpoint of ecoand
nomics. Hence,
. Actually, the Lacan also be viewed as an equivalent
grangian function
system cost.
) and for a set of given
Then at an optimal solution (
( , ), the nodal prices of real and reactive power for each bus
are expressed below for
(5)
and
are nodal prices of real and reactive power
where
and
are usually
at bus- , respectively.
considered as the real and reactive power transaction charges
from bus- to bus- .
Notice that the nodal prices derived by differentiating the
maximized social welfare function with respect to the real and
reactive demands, yield the same results as (5) and (6). That is,
first replace (1) with “
for
” subject to the same constraints as (2) and (3),
is short-term value-added function of the
where
customers connected to bus- . Then we can prove that nodal
prices are the same as (5) and (6), by letting the marginal
equal to the nodal prices of real
benefit
and reactive power at the optimal solution, respectively [5].
the nodal price of real
Therefore according to (5),
power at bus- can be viewed as the system marginal cost
plus a set
(the sum of a marginal generation cost
of premiums corresponding to their respective constraints)
created by an increment of real power load at bus- . The
same explanation is also applicable to the nodal price of
.
reactive power
A significant property for nodal prices is that each nodal price
is actually defined simply as a linear summation of all factors
can
according to (5) and (6) because each nodal price, e.g.,
be rewritten as
This property is completely different from that of AC power
flow which is generally nonlinear to each source or route, and
is also fundamental to the decomposition or coloring of nodal
prices in this paper. Therefore, theoretically it is possible to trace
the contributions of all factors involving in the operations of
power systems to each nodal price.
C. Problem and Example
Equation (5) or (6) seems to have given a full description
of each component for a nodal price even without any further
analysis. However, in contrary to intuitive observation, we will
show late that it is incorrect. Actually there are generally only
one or two terms remaining nonzero at an optimal solution
for (5) and (6) depending on the formulation, even though
many constraints become active (or binding) and many generators contribute to this nodal price. The main reason is that
) in the Lagrangian function (4) [or
the variables (
OPF equations (1)–(3)] are all independent variables and only a
few equations among (2) and (3) or have direct relation with
certain bus demand or . Therefore, most of the terms in (5)
or (6) are eliminated after differentiating with certain demand
or , irrespective of any solution. We take a four-bus system
shown in Fig. 1 and Table I, as an example to show this point.
CHEN et al.: COMPONENTS OF NODAL PRICES FOR ELECTRIC POWER SYSTEMS
43
By solving OPF (1)–(3), we have an optimal solution where
,
and line flow from bus 3 to bus 2 reaches its limit –
.
In addition, the voltage values of bus 2 and bus 4 also reach
and
their lower and upper bounds, respectively, i.e.,
. Therefore, there are only a few Lagrangian multipliers nonzero (related to four equalities and three active inequalities) which are
(8)
(9)
(10)
Fig. 1. Four-bus test system.
TABLE I
TRANSMISSION LINE DATA OF A FOUR-BUS TEST SYSTEM
Example 1: For the system shown in Fig. 1 and Table I, upper
and lower bounds for generators G1 and G4 are
,
;
,
.
The voltage values for all buses are bounded between 0.95 and
1.05. Besides, the real power flow in line 2–3 is also restricted
between 0.3 and 0.3. All of the values are indicated by p.u.
The fuel cost function for generators G1 and G4 is expressed as
.
For this four-bus system, there are four equalities for (2) corresponding to their respective real and reactive power balances
(or Kirchoff’s laws) of load-buses 2 and 3, and 18 inequalities
for (3) corresponding to four pairs of voltage, 2 2 pairs of
generation output, and one pair of line flow upper and lower
bounds, respectively. In this paper, we take real and imaginary
which has 2 4 elparts of bus voltages as state variable
and
in
,
ements. Therefore,
and
can be represented in terms of
and
by using real power balances of their respective buses.
Regardless of the solution, according to (5), the nodal prices of
real power have the forms as
(7)
and
are the Lagrangian multipliers related to
where
and
real power balances of the respective buses 2 and 3.
are the Lagrangian multipliers related to the upper and
lower bounds of real power generation for generator , respecand
are for generator
.
tively, while
Obviously, nodal price at demand bus- is expressed only
by the Lagrangian multiplier corresponding to the power
flow balance of bus- , and all other terms in (5) vanish due to
their relation independent of . It is also true for the nodal
prices of reactive power.
and
are the Lagrangian multipliers related to
where
the lower and upper bounds of buses 2 and 3, and – corresponds to the line flow constraint from bus 3 to bus 2. Since the
at bus 4 is cheaper than the generator
, electric
generator
through bus 3 toward bus
power is mainly transferred from
2. As a result, power flow in line 3-2 reaches its limit (0.3 p.u.)
which causes the congestion problem, although the cheaper genstill has generation capability.
erator
Substituting the values of the Lagrangian multipliers and
) into (7), we have the nodal prices of real power
(
(11)
.
where
Hence, even although we can obtain the nodal prices
according to (5) and (6), it is still unknown exactly what
components of each nodal price are and how the constraints
(congestion or other limits) and generators influence the value
of each nodal price.
III. COMPONENTS OF NODAL PRICE
There are many factors or constraints affecting the operation of power systems, e.g., generators, voltage limits, line flow
limits, power flow balance conditions (Kirchoff’s laws). Some
of them (e.g., voltage limits) have market values which may
be relaxed (e.g., from 1.00 1.05 to 0.95 1.05) and taken as
tradable goods depending on market needs. The relaxation for
these limits may be realized by technology innovations or facilities investments, etc. But some of them actually cannot be
traded, e.g., for real power flow balance condition at each bus,
the summation of all injected real power at each bus must be
zero which cannot be relaxed or violated because it is a physical
law. Therefore the evaluation or pricing for the factors with no
market value is meaningless, even although we can theoretically
trace the contributions of all factors involving in the operations
of power systems to each nodal price. Hence, before breaking
down the nodal prices, we have to classify all constraints in the
operations of power systems into two groups, i.e., tradable constraints which should be components of each nodal price, and
nontradable constraints which are mandatory constraints during
the operation and are not components of nodal prices.
In this section, we theoretically propose a method to break
down the nodal prices into a variety of components by using
the marginal conditions of operation, which are derived from
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IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 17, NO. 1, FEBRUARY 2002
nontradable constraints and Karush–Kuhn–Tucker (KKT) conditions [10]. In other words, we try to identify or trace all factors from a nodal price not qualitatively but quantitatively.
A. Conditions of Optimality
Let and be the active inequalities among , and their respective Lagrangian multipliers among at an optimal solution
for a set of the
of (1)–(3). Then the optimal solution
satisfies
given demands
(12)
(13)
(14)
where the other Lagrangian multipliers in but not in are
all zero because their respective inequalities are not active (or
nonbinding). is a column vector while is a row vector.
According to the definition of nodal prices, (12)–(14) are necessary optimality conditions, which must be satisfied when evaluating nodal prices.
algebraic equations which have the same number as variables
.
and
have the same elements as those of
and
, respectively, according to their definitions.
Equation (15) is necessary condition to keep the Lagrangian
function (or equivalent system cost) always at a minimum while
(16) can be viewed as compulsory constraints for which the operation of a power system must obey. On the other hand, may
be considered as the factors which can be traded (or tradable
goods) and have market values, in contrast to . Therefore,
from the viewpoint of economics, (15) and (16) are the marginal conditions for the decomposition of nodal prices, which
are identical to (12)–(14) at the optimal solution.
C. Description of Nodal Price
and
Next, we first establish the relations between
based on the marginal conditions (15) and (16), and then
use these relations to decompose the nodal prices.
of
Assume that (17) holds at an optimal solution
(1)–(3)
(17)
B. Classification of Constraints
From the viewpoint of economics, the Lagrangian function
in (4) is the sum of operating cost plus a number of the equivalent charges related to their constraints. Needless to say, each
in is an important factor which we
element of
intend to distinguish from nodal prices. Besides, to break down
the nodal prices in a more detail manner, as mentioned before,
we have to decide which components we (or participants of the
market) are interested among all of constraints in . For instance
in Example 1, there are two voltage and one line flow constraints
active in addition to four equalities. We are interested in, how
the generators G1, G4 and voltage limitations and line congestion influence each nodal price of (5) and (6), comparing to the
terms of (5) related to the power balance constraints (or Kirchoff’s laws) of load-buses 2 and 3. We will show these in Example 2.
be the constraints (nontradable constraints) among
Let
which we are not interested or we do not intend exbe the
plicitly to count their charges for nodal prices, and
remaining constraints (tradable constraints). Define to be the
Lagrangian multipliers corresponding to the constraints of ,
and to be the remaining Lagrangian multipliers corresponding
except ,
to . Then we drop other equations among
and rewrite (12)–(14) as follows:
Then according to the implicit function theorem, there exists
,
in open
a unique differentiable mapping
. In other words, the variables
neighborhoods of the given
are not independent of
but functions of
as far as (15) and (16) hold. Actually (17) is a partial Hessian
matrix of , which is usually used in sensitivity analysis of OPF
solution.
and
as functions of
By considering
, (15) and (16) can be rewritten as follows:
(18)
(19)
,
Let
,
,
,
. The above notion is appli-
or
.
cable to
Differentiating (18) and (19) with respect to
is an element of , we have
that
and noting
(20)
(15)
(16)
where
Therefore, by solving (20) and considering (17), we have
at the optimal solution as follows:
(21)
for the simplicity of expression, and
are all column
vectors while , are row vectors. Note that (15) and (16) are
where
and
.
CHEN et al.: COMPONENTS OF NODAL PRICES FOR ELECTRIC POWER SYSTEMS
As the same way,
D. Algorithm and Examples
can be obtained
(22)
and
.
where
Then we are at the position to decompose the nodal prices.
and
are functions of
, the
Noting that
Lagrangian function of (4) is reformulated as follows:
(23)
Therefore, differentiating of (23) with respect to
nodal prices of (5) and (6) become
45
and
,
(24)
(25)
where we take into account of the second part of (20), i.e.,
or
for , and also
and
at an optimal solution, for the derivation
use
of (24) and (25).
Next, we show that (24) and (25) are actually the deof
composed nodal prices. If the objective function
(1) is constructed by many factors (e.g., many gen, then
erators), i.e.,
for the first term of
(24), or
for the first term of (25). Furthermore, let
and
where
and
are the th equation
of and its respective Lagrangian multiplier. Then the second
term of (24) or (25) can be represented as
or
. Therefore,
and
are the components associated to the factor
(e.g., the th generator) for real and reactive power, respecand
tively, while
represent the terms of
for real power and reactive power,
the respective constraint
respectively. Generally, each term in (24) and (25) is nonzero
at an optimal solution, in contrast to the terms of (5) and (6).
includes all of the constraints among both
and ,
If
then all of the factors appearing in are nonzero and will explicitly expressed in the nodal prices. On the other hand, if
includes all of the constraints among both and , then all
of the terms except those among disappear in (24) and (25).
In the extreme, if none of (12)–(14) is used in deriving (21) and
and are all independent due to the elim(22) [i.e.,
ination of (15) and (16)], then (24) and (25) simply reduce to
(5) and (6) which have no information left except the values of
nodal prices, e.g., (24) becomes (7) for Example 1.
Evidently the nodal prices of (24) and (25) hold as long as
the marginal conditions of (15) and (16) or (21) and (22) are
satisfied. Equations (24) and (25) are certainly identical with
(5) and (6) in terms of the values of nodal prices but have
more detail information for their components. We will use
several examples to show this fact next.
We straightforward have the following procedure computing
nodal prices as well as their components.
1) For a set of the given demands
, solve the optimization problem (1)–(3) to obtain an optimal solution and
.
their dual solution
whose costs will explicitly be
2) Choose the constraints
counted in the evaluation of the nodal prices, and then
or
according to (21) or (22).
calculate
and
and their compo3) Evaluate the nodal prices
nents according to (24) and (25).
Note that the Lagrangian multipliers are zero for nonbinding
equations, thereby having no influence on the nodal prices or
the Lagrangian function.
Example 2: Assume the same conditions as Example 1. And
then calculate the components of the nodal prices.
According to Example 1, there are seven binding constraints,
including three active inequalities (voltage lower bound at bus 2,
voltage upper bound at bus 4, line flow bound from bus 3 to bus
2) among (3), and four equalities (two pairs of real and reactive
power balances for demand buses 2 and 3) of (2). Besides, there
are two energy resources appearing in of (1), i.e., generators
G1 and G4.
[Case-1]: We intend to express the nodal prices in the following form:
charge from generator G1
charge from generator G4
compensation for voltage lower bound at bus-2
compensation for voltage upper bound at bus-4
congestion charge of line 3–2.
Nodal price at bus-
(nontradable constraints) includes all four
For this case,
power balance equalities of (2), and (tradable constraints)
is composed of the three active constraints among (3). Hence,
and
– . We
or
according to (21) or (22) at the opfirst calculate
timal solution of Example 1. Then we have the nodal prices
or
and their components according to (24) and (25)
summarized in Table II. The results in Table II are the same as
(11) for the values of nodal prices but have detailed descriptions for each term. For instance, the nodal price of real power
at bus 2 is composed of five terms corresponding to the equivalent charges or compensations for generator G1 (1.5599), generator G4 (19.6789), lower bound of bus 2 voltage ( 0.6433),
upper bound of bus 4 voltage (1.3856) and congestion of line
3–2 (5.8805), i.e.,
generator G1
generator G4
lower bound of bus-2 voltage
upper bound of bus-4 voltage
congestion of line 3–2
(26)
which shows equivalent charges for their respective factors
when there is an increment of real power load at bus 2, and
these charges are certainly identical with the cost increment
of the total system for one unit change of . From (26),
obviously power supply is mainly from generator G4 but is
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IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 17, NO. 1, FEBRUARY 2002
TABLE II
NODAL PRICES AND THEIR COMPONENTS FOR CASE 1
TABLE III
NODAL PRICES AND THEIR COMPONENTS FOR CASE 2
partially restricted due to the congestion of line 3–2 for an
increment of the real power at bus 2. In addition, from Table II
it is evident that the congestion of line 3–2 considerably incurs
the expense of customers at bus 2, in contrast to other buses.
[Case-2]: We intend to express the nodal prices in the following form:
Nodal price at bus-
TABLE IV
NODAL PRICES AND THEIR COMPONENTS WITHOUT LINE FLOW CONSTRAINTS
charge from generator G1
charge from generator G4
congestion charge of line 3–2
without explicit expression of compensations for voltage
bounds at bus 2 and bus 4 (i.e., further assume that bounds of
bus 2 and bus 4 are not tradable goods besides power balance
constraints of each bus).
includes six equations including all four
For this case,
equalities of (2) and two active inequalities related to voltage
lower bound at bus 2 and voltage upper bound at bus 4, respecis composed of only one active
tively. On the other hand,
,
constraint related to line 3–2 flow constraint. Hence
,
,
and
- . As the same way
as the computation of Case-1, we have the nodal prices
summarized in Table III, which shows the same values of nodal
prices as (11) or Table II of Case 1 but has different descriptions
due to the setting of this case. For instance, as shown in Table III,
the nodal price of real power at bus 2 is composed of three terms
corresponding to the equivalent charges or compensations for
generator G1 (2.5482), generator G4 (19.6326), and congestion
of line 3–2 (5.6809), i.e.,
generator G1
generator G4
congestion of line 3–2
(27)
which shows equivalent charges when there is an increment of
real power load at bus 2, as far as only these three factors are
concerned.
In this example, the constraints of voltage bounds at buses 2
and 3 as well as the four real and reactive power balance equalities may be viewed as compulsory conditions and cannot be
traded in the market.
Example 3: Assume the same conditions as Example 1 but
without the real power flow constraint in line 3–2. Then recalculate the components of the nodal prices.
There are four equalities of corresponding to their respective real and reactive power balances of demand buses 2 and 3,
and 16 inequalities of corresponding to four pairs of voltage
and four pairs of generation output, respectively.
By solving OPF (1)–(3), we have an optimal solution where
,
.
In this case, real power of generator G4 reaches its upper bound
0.7, and line flow from bus 3 to bus 2 increases to 0.31 more
than the previous limit of Example 1. In addition, the voltage
values of bus 2 and bus 4 also reach their lower and upper bounds
and
. Therefore, there are
respectively, i.e.,
seven Lagrangian multipliers nonzero, related to four equalities
of , the lower bound of bus 2 voltage, the upper bound of bus
4 voltage and the upper bound of real power for generator G4,
respectively, (instead of line 3–2 limit in Example 1).
In this example, we intend to express the nodal prices in the
following form:
charge from generator G1
charge from generator G4
compensation for voltage lower bound at bus-2
compensation for voltage upper bound at bus-4.
Nodal price at bus-
The nodal prices as well as their components are depicted in
Table IV, which shows that the price at bus 2 is significantly
reduced from 27.8616 to 24.6495 due to the elimination of the
congestion in line 3–2. Actually the generation of G4 has no effect on nodal prices due to its binding to the upper bound, and
can be dropped from the components. Different from Example
2, according to Table IV, the power supply as well as the generation charges comes completely from generator G1 for an additional increase of real power at any bus because generator G4
has reached its upper bound.
IV. SIMULATION AND POWER LOSSES
A. Numerical Simulation
A IEEE 30-bus test system shown in Fig. 2 and Table V is
used in this section for numerical simulation. The voltage values
for all buses are set between 0.95 and 1.05. Besides, the real
power flow in line 2–5 is also restricted between 0.6 and 0.6.
The main loads are at buses 5,7 and 8 while the four generators
CHEN et al.: COMPONENTS OF NODAL PRICES FOR ELECTRIC POWER SYSTEMS
47
Fig. 2. IEEE 30-bus standard test system.
are located at buses 1, 2, 22 and 27. All of the values are indicated by p.u. The objective function of (1) is the total fuel costs
.
of generators, i.e.,
For this system, there are 2
26 equalities of
corresponding to their respective real and reactive power balances
of the buses without a generator, and 78 inequalities of
corresponding to 30 pairs of voltage, 2 4 pairs of generation
output, and one pair of line flow upper and lower bounds,
respectively.
Table VI shows the optimal outputs of generators, demand of
each bus and nodal prices of real power. At the optimal solution, five inequalities become active, which are related to upper
, lower
bound of buses 11 and 27 voltages
and upper bounds of real power for generators at bus 1 and bus
, and upper bound of
27, respectively,
. The components of
line flow from bus 2 to bus 5
–
nodal prices in Table VI are obtained by setting five terms of the
nodal prices as follows:
Nodal price of real power at bus-
generators G2, G22
voltages
congestion of –
where we drop the components related to fuel costs of generators G1 and G27, which are actually all zero and have no influence on the nodal prices because they have reached their limits
of outputs and do not provide power for an increment of demand
at any bus as long as the necessary conditions of optimality of
(15) and (16) hold.
TABLE V
CONSTRAINTS AND FUEL COSTS OF GENERATORS
For this case,
includes all 2 26 equalities of (2) and
two active inequalities corresponding to lower and upper bounds
and
, respectively. Note
of real power
that the charges from generators include the power loss. From
Table VI, it is evident that the nodal prices at buses 5 and 7
are mostly influenced by the congestion of line 2–5, while the
voltage bound of bus 11 mainly affect the nodal price at bus 5.
On the other hand, the fuel (or generation) charges of demands
at bus 5 and bus 21 are mainly from generator G2 and generator
G22, respectively, while the fuel charge for the demand at bus
8 is almost equally shared by both generators G2 and G22 according to Table VI from the viewpoint of marginal cost. As the
, we can calculate the nodal
same way as the computation of
in terms of the concerned compoprice of reactive power
nents.
B. Consideration of Power Losses
Although the components of nodal prices include all independent factors in OPF formulation, there are still important depen-
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IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 17, NO. 1, FEBRUARY 2002
TABLE VI
DEMAND, GENERATION AND NODAL PRICES
STANDARD TEST SYSTEM
FOR
IEEE 30-BUS
a total reactive power balance equality
in (2), or evaluate
the real and reactive power losses in reactive nodal price
of (25).
V. CONCLUSION
dent elements, e.g., power losses, which are not explicitly handled in (24) and (25). Since power loss (real or reactive power
loss) is not an independent factor in the objective function or
constraints, it cannot be directly expressed as a component of
nodal prices.
If it is necessary to evaluate effect of real power loss, (2) can
be reorganized in the following way. Deleting one real power
balance equation at any bus, we instead add the total real power
balance equality
This paper proposes a methodology directly to link each
concerned factors to the nodal prices, i.e., we break down
each nodal price into a variety of parts corresponding to
different factors, such as generations, transmission congestion,
voltage limitations and other constraints. Different from the
ac power flow which cannot generally be identified to its
sources or routes, this paper shows that the nodal prices can
theoretically be traced to each factor based on the marginal
conditions from the economics viewpoint. The decomposition
is unique, and components of each nodal price are identical
to their increment costs or benefits for total system. These
detail information for nodal prices can be used not only to
improve the efficient usage of power grid, energy resources and
congestion management, but also to design a reasonable pricing
structure of power systems, or to provide economic signals for
generation–transmission investment [11]. Several numerical
simulations have been used to demonstrate our approach.
If the decomposed components are all required to be positive, the components can be recalculated by first letting the negative components be zero, and then proportionally allocating the
nodal prices to the positive components depending on their contributions (values).
Although this paper uses OPF-based model with static constraints as an example to demonstrate the decomposition of the
nodal prices which are actually based on the short-term marginal
cost principle, the nodal prices incorporating the long-term investment as well as dynamical constraints can be decomposed
in the same manner provided that the long-term expansion planning model or dynamical constraints are adopted instead of the
static OPF model. For example, the values or prices of control
devices such as PSS and AVR can also be evaluated as the same
way as (24) and (25) if the transient stability constraints are considered in OPF [10].
ploss
REFERENCES
into (2), where
and
are the summations of real
power generations and loads, respectively, and ploss is total real
. Then the power
power loss which is a function of
loss is explicitly expressed in the reformulated by (1)–(3), which
are theoretically equivalent to the original (1)–(3). From the dual
theory of mathematical programming, there generally exists a
for the new equality at any
nonzero Lagrangian multiplier
optimal solution of the reformulated (1)–(3).
Take real power loss ploss as an independent factor, i.e.,
assume ploss tradable. In other words, the total real power
balance equality should be added into . Then there is a term
in the components of nodal price
of (24), which corresponds to real power loss and can
be viewed as the value of real power loss. As the same way,
we can also evaluate the value of the reactive power loss as
in real nodal price
of (24) when
one reactive power balance equality of any bus is replaced by
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Luonan Chen (M’92–SM’98) received the B.S.E.E. degree from Huazhong
University of Science Technology, Wuhan, China, and the M.E. and Ph.D. degrees in electrical engineering from Tohoku University, Sendai, Japan, in 1984,
1988, and 1991, respectively.
He joined KCC, Ltd., Tokyo, Japan, in 1991. Since 1997, he has been a Faculty Member at Osaka Sangyo University, Osaka, Japan, where he is currently
an Associate Professor in the Department of Electrical Engineering and Electronics. His research interests include nonlinear dynamics and optimization for
power systems.
49
Hideki Suzuki received the B.S. degree from the Science University of Tokyo,
Tokyo, Japan, in 1987.
Since 1993, he has been working at KCC Ltd., Tokyo, as a System Engineer.
His interests include operation and planning of power systems.
Tsunehisa Wachi received the B.S. and M.S. degrees from Gakushuin University, Tokyo, Japan, in 1995 and 1997, respectively.
Since 1997, he has been with KCC, Ltd., Tokyo. His research interests include
operation and analysis of power systems.
Yukihiro Shimura received the B.S. degree from Tohoku University, Sendai,
Japan, in 1988.
Since 1988, he has been working at the Electric Power Development Company, Ltd., Tokyo, Japan. His interests are operation and analysis of power systems.