IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 25, NO. 3, AUGUST 2010
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Min-Max Fair Power Flow Tracing for
Transmission System Usage Cost Allocation:
A Large System Perspective
M. S. S. Rao, S. A. Soman, Member, IEEE, Puneet Chitkara, Rajeev Kumar Gajbhiye, Student Member, IEEE,
N. Hemachandra, and B. L. Menezes
Abstract—Power flow tracing has been suggested as an approach
for evaluating 1) transmission system usage (TSU) cost and 2)
loss (MW) cost for generator and load entities in the system. Recently, optimal power flow tracing methods have been proposed to
“explicitly” model fairness constraints in the tracing framework.
This paper, further, strengthens the tracing-compliant min-max
fair cost allocation approach. The min-max model proposed in
this paper is robust. It addresses concerns like scalability, numerical stability and termination in a finite number of steps while
searching the optimal solution. We also propose a methodology
to model DISCOMs and GENCOs as coalition within min-max
framework. Case studies on an all India network of 1699 nodes
and comparison with average participation and marginal participation methods bring out the better conflict resolution feature of
the proposed approach. A method to model HVDC lines within the
marginal participation scheme is also proposed. Quantitative and
qualitative comparison of various TSU cost allocation methods
on such a large system is another noteworthy contribution of the
paper.
Line cost and cost per MW flow for a line
from bus to bus .
Index Terms—Cooperative game theory, equity, linear programs, marginal participation method, min-max fairness, power
flow tracing, transmission systems, usage cost allocation.
Power flow on a line
Penalty factor
Number of HVDC lines.
Set of entities whose price cannot be reduced
further in the th LP problem.
.
Number of buses, loads and generators.
Total number of entities, i.e.,
Contribution of load
.
in generator
Contribution of generator
in load
.
.
Price charged to an entity .
.
Set of entities whose prices have to be
th and subsequent
determined in the
LPs.
NOMENCLATURE
and
.
Set of tracing vectors for the
th LP.
Fraction of the TSU cost borne by the
generators.
Transmission system usage cost.
Cost of the th HVDC line.
Payoff to the th entity in a cost allocation
.
game
Tracing and price vectors.
Auxiliary variable in the th LP and its optimal
value.
Change in the usage cost of the th entity.
Tolerance parameter
.
Characteristic function of a game.
I. INTRODUCTION
Manuscript received July 17, 2009; revised December 21, 2009. First published February 18, 2010; current version published July 21, 2010. This work
was supported by PowerAnser Labs, IIT Bombay. Paper no. TPWRS-005582009.
M. S. S. Rao, S. A Soman, and R. K. Gajbhiye are with the Department of Electrical Engineering, IIT Bombay, Mumbai, India (e-mail:
[email protected]; [email protected]; [email protected]).
P. Chitkara is with Mercados Energy Markets International, New Delhi, India
(e-mail: [email protected]).
N. Hemachandra is with the Department of Industrial Engineering and Operations Research, IIT Bombay, Mumbai, India (e-mail: [email protected]).
B. L. Menezes is with the Department of Computer Science and Engineering,
IIT Bombay, Mumbai, India (e-mail: [email protected]).
Color versions of one or more of the figures in this paper are available online
at http://ieeexplore.ieee.org.
Digital Object Identifier 10.1109/TPWRS.2010.2040638
T
RANSMISSION system usage (TSU) cost allocation
problem is a complex problem. It involves recovery
of the embedded or fixed cost of transmission system from
multiple users of the network which are loads and generators.
Traditionally, this cost was recovered at a flat or postage stamp
rate where in TSU cost is shared by an user in proportion to
MW connected to the grid. Critique of this simple scheme has
been that it does not account for actual usage of the grid by a
load or a generator. This led to the development of a MW-mile
framework [1] which evaluates usage as a product of MW flow
on a line and length of the line. Consider that for a specific
power flow scenario, cost of a transmission line from bus “ ”
to bus “ ” to be recovered is $
and corresponding power
flow in MW on the line is
. Then, we can define the per
0885-8950/$26.00 © 2010 IEEE
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IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 25, NO. 3, AUGUST 2010
unit line usage cost by
$/MW.1 The network
usage cost to be recovered now is given by
II. TRANSMISSION SYSTEM PRICING: A REVIEW
A. Marginal Participation Method
(1)
Throughout the paper, we assume that “from” and “to” nodes,
.
i.e., and , respectively, of a line are so chosen that
Quantification of the network usage by an entity can be done
by one of the following approaches:
1) marginal participation (MP) method [2]–[4];
2) proportionate power flow tracing or average participation
(AP) method [5]–[8];
3) with and without transit flow methods [9];
based transmission network cost allocation [10];
4)
5) transmission charges based on bilevel programming [11];
6) cooperative game theory based methods [12]–[14];
7) min-max fair tracing method [15].
All the above methods require load flow solution, usually a dc
load flow, to compute the TSU cost. It has to be mentioned that
nodal prices or locational marginal prices (LMPs) obtained from
the solution of optimal power flow (OPF) with social welfare
maximization objective cannot recover this cost [2] because in
an ideal network which is uncongested and lossless, LMPs will
be identical at all nodes.
A comprehensive review of the existing mechanisms of transmission pricing, which is presented in Section II reveals that it
would, probably, be impossible to suggest a single allocation
method which outscores the rest. However, we feel that with
present state of the art, min-max fair tracing approach strikes
the right balance between fairness and tractability of computation. Hence, the aim of this paper is to strengthen the modeling framework of min-max fair tracing first proposed in [15].
The min-max model proposed in this paper is robust. It addresses concerns like cycling, convergence to suboptimal and
requirement of several iterations to reach the optimal, associated with model in [15]. We also compare and contrast min-max
fairness approach with MP and AP schemes. As MP method
cannot directly compute HVDC-cost allocation, we also propose a methodology for HVDC cost allocation in MP framework.
The results indicate that min-max fair tracing outscores other
methods in providing equity. Corresponding computational
time is approximately 5 min for 1699-bus all-India network
which is reasonable. It can also model DISCOMs and GENCOs
explicitly which other methods do not.
This paper is organized as follows. Section II reviews existing methods. Sections III and IV develop min-max fair tracing
model and algorithm. An illustrative example is presented in
Section V. Modeling to improve robustness of min-max solver
by explicitly handling numerical precision issues is discussed
in Section VI. Section VII deals with modeling DISCOMs and
GENCOs in min-max tracing framework. Section VIII develops
a methodology to model HVDC lines in MP scheme. Case
studies on 1699-bus all-India network is detailed in Section IX.
Section X concludes the paper.
1An alternative is to define constant c
as a ratio of line cost to MW capacity
of line. This definition leads to better reflection of usage costs. However, it can
lead to under recovery as meshed networks have high redundancy for guaranteeing reliability under outages.
We begin with review of MP method. In this method, we evaluate incremental line (or network) usage for a generator or a
load when it injects or draws an extra MW into or from the grid.
Then, cost of the line (or network) is apportioned among load
and generator entities in proportion to their weighted marginal
participation, weights being MW injected or drawn by the generator or load. Note that loads or generators which reduce a line
flow (or network usage) by their marginal usage are exempted
from bearing cost of the line as, no entity should be paid for
usage of grid. Limitations of this scheme are as follows.
1) It is sensitive to choice of slack bus. This brings in arbitrariness in evaluation of quantity of usage of the network
by a load or generation entity. Usage of dispersed slack
bus to reduce price for loads and generators distant from a
conventional slack bus is suggested in [16]. An approach
independent of slack bus selection is proposed in [4] and
discussed in [17].
2) It is difficult to allocate cost of HVDC lines as marginal
cost of an HVDC line from for any load or generator entity
will always be zero. This is because flow on an HVDC line
is specified by its current or power order. HVDC controls
in a closed loop controller action adjust firing angle of converter and transformer tap to maintain this flow.
B. Power Flow Tracing
Power flow tracing methods trace the flow of power from a
source to a sink and vice-versa. It is known to have multiple solutions. In fact, in [18], it is shown that the complete space of
tracing can be modeled by linear equality and inequality constraints. Power flow tracing solvers can be further subclassified
as proportionate and optimal tracing solvers.
Proportionate tracing treats a node as a perfect mixer of commodity flows and in generation tracing allocates total incoming
generation commodity flow to outflows in proportion to their respective branch flow values. Load tracing is a dual of generation
tracing. The proportionate tracing principle has been justified
using game theoretical rationale in [19]. However, it is a greedy
algorithm. Formally, an algorithm is said to be greedy [20], [21]
if it follows the problem solving metaheuristic of making the locally optimal choice at each stage with the hope of finding the
global optimum. For many problems greedy algorithms fail to
produce the optimal solution. A generic definition of proportionality as used in bandwidth allocation problem in communication
network [22] requires maximization of a concave function subject to linear constraints (also see [15]).
An alternative to proportionate tracing rule is optimal power
flow tracing introduced in [18]. In this approach, fairness criterion is modeled explicitly by an objective function and an optimization problem with linear constraints is solved to determine the most fair solution. In [18], the aggregate deviation from
postage stamp rate is used as an objective function. However, its
critique, is that certain higher priced load or generator entities
subsidize network usage rate for the rest, which is a consequence
of aggregate rationality. This concern is properly addressed by
min-max fairness model.
RAO et al.: MIN-MAX FAIR POWER FLOW TRACING FOR TRANSMISSION SYSTEM USAGE COST ALLOCATION
1) Min-Max Fairness: Min-max fair optimal tracing has been
proposed in [15]. It has a strong fairness foundation as it guarantees that within tracing framework no high price taker subsidizes
a low price taker. A more detailed discussion follows in subsequent sections. Authors have extensively tested model proposed
in [15]. Critique of the model proposed in [15] is its lack of scalability. The algorithm proposed in [15] is an iterative approach
which can lead to 1) convergence to a sub-optimal solution or 2)
cycling. The algorithm in [15] can also fail due to numerical precision problems. Appendix B brings out these difficulties. The
model proposed in this paper overcomes these limitations.
C. With and Without Transit Flow Method
This method has been discussed in [9] to compute compensation due to a country, in Europe, for cross border flows. The
basic idea is to compute the MW-miles consumed by the network of a country through which power is transiting under two
and 2) without cross
scenarios: 1) with cross border flow
border flow
. If cross border flows in the network are such
, then compensation to be provided to the netthat
, where
is
work is proportional to
the country’s cost of the transmission network to be recovered.
The advantage of the method is that it is intuitive and straightforward. It only requires two dc load flow runs, one when the network is being utilized for transiting and another when it is not.
The second scenario is simulated by appropriately adjusting imports and exports to curtail cross border flows. In this work, we
also propose to use this principle to compute HVDC line usage
cost allocation in, otherwise, MP framework.
D. TSU Cost Allocation Using Cooperative Game Theory
be the characteristic function used to measure the
Let
network usage cost for a coalition , which is a set of load and/or
. Usually, evalgenerator entities. Further, let
uation of
requires at least a dc load flow solution. If is
the cost to be recovered from entity , then individual rationality
and coalition rationality demands that
demands that
. A allocation vector
is said
to be in core if it meets individual rationality for all entities and
coalition rationality for all possible coalitions. An allocation in
core can be computed by solving an LP problem with exponential number of constraints to model all possible coalitions. Depending upon choice of
, core may be 1) empty, 2) unique,
or 3) have multiple solutions. Thus, while being in core is desirable, it does not per se solve fairness problem.
An alternative to core is nucleolus [23]. It involves defining
s.t.
an excess cost allocation vector
(2)
The vector is of dimension
. The coordinate
reflects the “attitude” of a coalition to an allocation vector .
The coalition which objects most to the allocation is the one
with largest . If
, then we say that allocation is more acceptable than . It
raises less of an objection. Therefore, in the first step of computing nucleolus, we restrict the search to a subset of those
imputations for which
is minimum, where refers to the set of imputations. If this set is not a
singleton, then we proceed to compute a subset of wherein
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we minimize next highest component of . Recursive application of this process leads to a unique allocation vector called nucleolus. An advantage of nucleolus is that it exists even if core is
empty. Also, nucleolus and min-max fairness have strong philosophical similarities as both of them work on the principle of
devising a cost allocation scheme with minimum regret.
An alternate game theoretic cost allocation method is Shapley
value [24]. For a player, it assigns usage cost as average of
marginal entry cost in possible permutations leading to grand
coalition. However, a drawback of methods which compute
core, nucleolus or Shapley value are that resulting problems are
intractable, i.e., NP-hard. In core and nucleolus, all possible
to be precise. Further,
coalitions have to be modeled,
each step in determination of nucleolus involves solving an
LP problem with exponential number of constraints [12]. In
the worst case, the number of LPs to be solved also grow up
exponentially. In case of Shapley value allocation, all possible
permutations have to be modeled. Hence, in general, it is not
possible to compute core, nucleolus or Shapley value based
allocation for a large system.
Recently, Aumann-Shapley value approach has been proposed for TSU cost allocation problem [14]. It addresses two
well-known concerns of Shapley value allocation scheme viz.
1) the problem of isonomy and 2) computational intractability.
In a sense, the model proposed in [14] also addresses the
problem of fair selection of slack bus of the MP method. The
slack bus selection is driven by the objective to minimize
aggregate TSU cost. In a dc framework, this translates into
solving a series of LP problems parameterized by the size of
generators or loads. If at all, critique of the objective function
chosen in [14] is that rather than minimizing only TSU cost
in the parameterized LP problem, the sum of total generation
cost and TSU cost should have been minimized. However, such
details are not available while solving cost allocation problem
and hence, such an approximation becomes necessary.
E. Other Methods
Approach: The technique proposed in [10] allocates
1)
. The method uses the contribution
TSU cost based on the
of the nodal currents to line power flows to apportion the use of
the lines. The method is both slack bus independent and does not
require an exogenous a priori proportion to split transmission
costs between loads and generators. However, the formulation
charges for counter-flows also.
2) Bilevel Programming Approach: In this approach, cost of
the carrier is integrated with the cost of content, such that the
optimal nodal energy prices are distorted minimally [11]. Authors suggest a bilevel programming approach for transmission
revenue constrained economic dispatch problem. The goal is to
minimally modify the nodal prices so as to recover the revenue
requirements of the transmission system without disturbing the
optimal schedule obtained from traditional OPF. This method
is different from the transmission cost allocation mechanisms
(discussed in this paper) based upon determining the physical
utilization of each transmission asset.
We conclude the following.
• Multiplicity of the solution space in the TSU cost allocation
makes the problem challenging.
• MP method is attractive because it is based upon principle
of marginal participation; also, it is easy to implement.
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However, there is a difficulty in justifying any “fixed”
choice of slack bus, individual or dispersed.
• AP method is attractive because it is based upon principle
of proportionality. Also, it is easy to implement. However,
the proportionate rule may not lead to a fair solution.
• Min-max fair tracing method is attractive because it improves fairness model in tracing. However, present model
in literature [15] may not converge to a fair solution and it
also lacks scalability.
• In cooperative game theoretic framework, there is no
unique way of characterization the cost of a coalition, i.e.,
. With the exception of Aumann-Shapley allocation
method, these methods cannot be easily scaled to large
systems.
III. MIN-MAX MODEL DEVELOPMENT
Following [15], we define a “tracing solution to be min-max
fair if any reduction in, say, per unit cost for an entity computed
within tracing framework will lead to increase in per unit cost of
another entity, which has to pay either equal or higher per unit
cost”.
A. Desirable Attributes of Min-Max Fairness
We now explain our preference for min-max fair solution
which, no doubt, is computationally much more demanding than
average participation tracing method.
Let be the price paid by the th entity and is a price vector
given by
. If it has to pay highest price for
,2 then, clearly, the th entity is most
TSU cost, i.e.,
unhappy.
The price can be computed as follows. Let the ratio of TSU
. If the th entity is a
cost of generators and loads be
generator injecting
MW or a load drawing
MW, then
corresponding prices are
further illustrates it. Then, in the first step, we seek a subset
of set , such that
(4)
where
“ ”. Let
is a price vector parameterized by tracing solution
(5)
provided that
, a requirement
Note that
which is always fulfilled.3 If is a singleton, i.e., it corresponds
, then we
to a unique tracing solution or if
have achieved the most fair solution as the price vector cannot
be improved any further. However, in general, neither the set
will be singleton nor all tracing solutions of
will give same
price vector , i.e.,
.
Now let
denote a reduced vector extracted from vector
by removing the determining coordinate corresponding
to
of for those entities whose price cannot be re. Such entity or entities are identified by
duced below
solving (10), which is elaborated later on. Then, price of
second-most-unhappy entity, i.e., second highest price taker, is
. Thus, in the second step, we seek a subset
given by
of tracing solutions
of , i.e.,
such that
(6)
provide simultaneous justice
Tracing solutions of the set
to the worst two price takers in the cost allocation scheme. Let
. Proof of existence of this minimum has already been outlined. Following similar logic, we
. If
is a singleton or
,
get
then min-max fair solution is achieved. Otherwise, following
the above process, at a step , we seek a subset
(7)
(3)
such that
Henceforth,
by
grouping
commodity
flows
and
in a tracing vector “ ”, we denote (3)
as
. Thus, “ ” is an
dimension column vector of commodity flows consistent with
linear constraints described in [18], where
is the
number of branches in the network.
Now, the highest price taker will seek redressing, i.e., a more
favorable tracing solution in which its TSU cost is reduced. In
other words, if we are able to move to a tracing solution in
which the price corresponding to highest price taker can be reduced, then it should be sought for, as it addresses the concern
of the worst-off or the most unhappy price taker in a cost sharing
problem.
Let set denote the set of all tracing solutions. Its modeling
is discussed in [18] and the numerical example in Section V
2Infinity
norm of “p” is given as max
j
p j.
(8)
Clearly, the process will terminate in a maximum of
steps
where
is the dimension of -vector. Also, minimum sequence
determines the optimal prices. A
very desirable property of min-max fair solution is that it is
unique [26].
A question of interest, regarding min-max fair tracing solution, is whether it can be explained from the framework of cooperative game theory. For example, is the min-max fair solution in
3Note that constraint space of tracing (T ) is non-empty, closed and bounded
(refer to [18]), and function p(t) defined by (3) is continuous. Hence, by Weierstrass theorem [25] minimization problem of (5) has a solution. Thus, T = ft :
t 2 T and p(t) z g 6= fg. Further, T is closed and bounded. It is closed
because it is described completely by linear equality and inequality constraints
only. It is bounded because T T .
RAO et al.: MIN-MAX FAIR POWER FLOW TRACING FOR TRANSMISSION SYSTEM USAGE COST ALLOCATION
the core of a cost-allocation game? This question is further dealt
within Appendix C, where, by defining an appropriate tracing
game, we show that the set of all pricing vectors derived from
tracing framework are in the core. However, due to multiplicity
in the solution space of tracing, core need not have a unique
solution. With the additional specification that pricing vector
should be min-max fair, we are able to resolve the dilemma and
this also leads us to a unique solution. Indirectly, the Appendix
also brings out the difficulty of defining reasonable characteristic functions which are meaningful, realistic enough, easy to
compute and at the same time lead to a unique cost-allocation
vector.
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Note that by complementary slackness conditions of KKT,
are at . Proof of the lemma is given in
all prices in
Appendix A.
Remark 1: The lemma has significant computational implineed not be solved at all. This saves
cation, i.e., subproblem
solution of many LPs and hence, the proposed method becomes
attractive for large systems:
(11)
Then, there is a scope to perform maximum price reduction in
given by
set
(12)
IV. OPTIMIZATION MODEL FOR COMPUTING SET
Let
, then a tracing solution corresponding to the set
as defined in (4) can be computed by solving the following
optimization problem:
(9)
The set
corresponds to set of entities whose price has not yet
have
been determined. Note that the prices of entities in set
been determined by the previous LP and equals
. The process
is similar to one outlined so far. Algorithm 1 summarizes the
proposed min-max algorithm for tracing.
Algorithm 1: Min-max algorithm for TSU cost allocation
In above LP, is an additional variable. Since can be modeled by linear constraints, the above formulation represents an
LP problem. Note that this model is different from one proposed
in [15]. Its initialization does not depend upon an instance of
is a sparse LP
tracing solution. Hence, it is generic. Since
program, it can be solved efficiently. While this step is quite
straightforward, identification of entity which “truly” should
be price taker of , i.e., one whose price cannot be lowered
below , requires additional work. Since such an entity is one
whose price cannot be lowered below in any tracing solution,
it can be computed by solving the following subproblem
:
Initialize:
-the fraction of cost that generators must bear.
while
do
1) Solve the LP problem
:
(13)
(14)
(10)
If
, then clearly entity is not the worst price taker as
its price can be improved (i.e., reduced) from . Conversely, if
is such that
, then we have identified the entity whose
price cannot be reduced below . Further, there can be multiple price takers of . There after, we can now proceed to next
step to compute . However, a closer inspection reveals that
it is not necessary to solve subproblems in (10) to identify the
min-max price taker entities of step-1. These can be identified
itself, which are available at the
from the dual variables of
solution of (9). Dual variable contains sensitivity information of
the objective function to constraint function [25]. The following
lemma formalizes the intuition.
Lemma 1: Let
. Let in the problem
the Lagrangian multipliers associated with min-max constraints
be given by set . Let at the optimal solution
the subset of of positive Lagrangian multipliers be . Let the
corresponding index set of entities, a subset of , be . Then,
set of prices
at optimal of
cannot be
reduced below .
where
is a linear function as defined in (3). Let
the solution be given by .
2) Compute
corresponding to dual
variable of constraint set
.
is the set of entities, whose price cannot be
improved beyond .
3) Update
end while
Notice that min-max computation involves recursive LP computation. At each step, price for at least one entity is fixed. Instead of termination condition
, we should set it
to
, where
indicates the cardinality of the set.
Then, the algorithm terminates in at most
steps. As an LP
can be solved by a polynomial time algorithm (e.g., an interior
point method), this is a polynomial time algorithm. The implication of the above statement is that even a very large min-max
fairness problem can be solved (in finite time) using a state
of the art sparse LP solver. However, many times with a very
large system (e.g., all-India system), it is not necessary to model
price at each bus. Rather, zonal price, or a DISCOM/GENCO
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Similar equations can be written for other load drawals and
generator injections.
4) Loss Inequality Constraints: These inequalities model
nonnegative relationship on loss. The following equation
and load
:
models this constraint between generator
(20)
Other generation and load combinations will have similar relations. The linear constraints of type (15)–(20) model the set of
all traceable solutions .
5) Price Vector Description: For the generator
Fig. 1. Four-bus example system.
price, is desired. This also significantly reduces the computational burden of the algorithm. This aspect is further discussed
in Section VII.
V. NUMERICAL EXAMPLE
Consider a lossless system shown in Fig. 1. The tracing constraints, as per model described in [18], are as follows.
1) Flow Specification Constraints: These constraints for
transmission line flows for generator and load entities are as
follows.
• Generator Tracing: The line decomposition equations can
be written as follows:
(15)
(21)
where
denotes price of generator
. In this example,
is set to 0.5. Similarly, price relations for the generator
and
other loads can be written.
6) Min-Max Constraints: The min-max fairness constraints
as proposed in this paper are as follows.
Step 1: In step 1 of min-max fair algorithm, we define varis.t.
able
(22)
The objective in
is to minimize the maximum price, i.e.,
Similar equations can be written for other branches.
• Load Tracing: Similarly, for load entities, we have
(23)
(16)
Similar equations can be written for other branches.
2) Source and Sink Specification Constraints: They describe
decomposition of injection and drawals in terms of drawals and
injections, respectively. The equations are as follows.
• Generator Tracing: For loads, we have the following decomposition:
subject to constraints (15)–(22). The objective function at
optimal, i.e., , is 6.1601. The price vector at optimal is
[6.1601, 5.0400, 4.9600, 6.1160, 5.3520] for the entities
, respectively. We also found that the
constraint in (22) corresponding to generator
is strongly
binding. Hence
(17)
Step 2: In this step, we minimize
additional constraints as follows:
subject to (15)–(21) and
(18)
Generator
will have similar decomposition.
3) Conservation of Commodity Flow Constraints: These
constraints are analogous to KCL. For load
, let
The above subproblem minimizes the maximum price
among remaining cost bearing entities while constraining
below its current level. Optimal value
the price of
is 5.6213 and the corresponding price vector is
. Both constraints
and
are strictly binding. Hence
Then conservation of commodity flow equations for load
at all nodes is given by
Step 3: Similarly, in this step we minimize
(15)–(21) and additional constraints as
Other loads will have similar decomposition.
• Load Tracing: Similarly, for generators, we have
Optimal value
is
(19)
subject to
is 5.0400 and the corresponding price vector
. Further
RAO et al.: MIN-MAX FAIR POWER FLOW TRACING FOR TRANSMISSION SYSTEM USAGE COST ALLOCATION
Since no more redistribution of cost is possible, the min-max
fair cost allocation has been found out.
The results obtained from proposed method and that of [15]
are identical. However, the proposed algorithm requires only
three LP solutions to reach the optimal solution while method
in [15] requires eight LP solutions. This is because the proposed
min-max approach being a direct approach does not require any
check for satisfying optimality condition. Also, the proposed
method does not begin from any instance of a tracing solution.
The price vector obtained from AP method is given
while the corby
responding price vector from MP method is given by
. This also shows that
worst price taker in min-max fairness approach gets a better
price (6.1601) than the worst price taker in the AP and MP
methods which are 6.552 and 6.639, respectively. The standard deviations for prices in min-max tracing, AP, and MP
methods are 0.492, 2.044, and 1.723, respectively. Thus, price
distribution with min-max is more equitable than the other two
methods.
, generator
at bus B
It may be argued that for load
itself supplies 10 MW and hence its utilization cost should be
set to zero. However, in tracing it is observed that this 10 MW
, as well, using
could be partially or completely routed from
line AB. We notice that min-max approach which emphasizes
equity, in fact, prefers such a solution. In essence, fairness has
both societal and individual implications and this trade-off can
be addressed in many ways; in this paper, notion of min-max
fairness has been advocated as a reasonable approach.
A case study on a large system will be taken up in Section IX.
For such systems, approach [15] is not viable.
VI. ROBUSTNESS OF PROPOSED MIN-MAX MODEL
An extensive testing of proposed direct min-max fair tracing
approach on many systems and scenarios has brought out robustness of the proposed scheme. However, in certain large systems and scenarios, it was found that intermediate LPs were
infeasible, implying infeasibility of min-max solver. Theoretically, it is known that min-max fair solution exists and it is
unique. Hence, the problem is possibly a numerical precision
problem. Further, it could arise out of the following reasons:
1) the nonzero residuals remaining in net power injection at
nodes after convergence of load flow. Ideally this should
, used as a conbe zero. It arises due to -tolerance
vergence check in ac load flow program. When using dc
load flow, this residual is a consequence of finite precision
arithmetic. Here, the residuals are really small in magnitude, e.g., less than
MW;
2) the numerical ill-conditioning associated with min-max
specific constraints, specifically inequality (14).
The first problem can be resolved as follows. Add a fictitious
bus to the system and connect it to all nodes in the system. The
flows on corresponding arcs are set to the corresponding residuals. If
, where residual is the difference of injected and calculated value at bus “ ”, then the fictitious bus is a
load bus; otherwise, it is a generator bus. Also, this bus should
not be used in pricing. Cost of corresponding fictitious arcs are
set to zero. The flows on these branches are practically zero and
so is the net injection on the bus. But, it accounts for numerical
precision problems which otherwise could make the problem
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Fig. 2. Modeling a DISCOM in min-max tracing.
infeasible. This modeling is particularly relevant for zero injection buses because on other buses, the loads or injections could
have been directly adjusted to account for the residual.
To handle infeasibility arising out of numerical precision
problems associated with min-max fairness constraints of the
, the following model is proposed. Let be a
type
variable used to relax constraint
, i.e., replace the
constraint
by
(24)
Under ideal condition, will be zero. Therefore, we penalize
any deviation of from zero value, in the objective function, i.e.,
we set objective function in
as
where
is
a large number. In our experiments, the value of
used is
.
It was seen in simulations that was usually set to zero, and
never exceeded 0.00021. With these modeling enhancements,
authors did not find any situation wherein LPs in min-max fair
tracing problem did not converge.
VII. MODELING OF DISCOMS AND GENCOS
Cost or resource allocation methods like nucleolus and core
model all possible coalitions behavior to ascertain fair allocation. In contrast, min-max fairness for cost allocation models
only individual entities. However, in power systems, natural
coalitions do exist as DISCOMs and GENCOs. A DISCOM
draws power from a grid at multiple buses, which usually are
a contiguous set of buses, while a GENCO may inject power
at multiple buses without any adjacency relationship. Therefore, coalition modeling in terms of DISCOMs and GENCOs
is equally important and desirable. In this section, we extend
min-max framework to model such coalitions.
Consider a DISCOM serving buses
having
loads
MW. Then, subsequent to a load
flow solution, we introduce a fictitious bus A representing the
DISCOM (Fig. 2). The net load on the bus A is sum of loads
of buses
. Simultaneously, loads at the buses
are removed. Then, the bus A is joined
to buses
by fictitious arcs
which carry flow of
MW, respectively. These
arcs have zero cost.
Note that in this process, we have neither done any network
reduction nor altered the power flows on the existing network.
However, number of cost sharing entities and hence dimensionality of the tracing problem has reduced. Hence, degrees of
freedom in tracing has increased. The DISCOM’s price is now
the price of the load at bus A. Other DISCOMs and GENCOs
can be modeled in a similar way. In contrast, no such explicit
modeling of DISCOMs and GENCOs is done in AP and MP
methods.
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IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 25, NO. 3, AUGUST 2010
VIII. MODELING OF HVDC LINES IN MP METHOD
,
Let a system have HVDC lines with costs
respectively. Now, any power flow tracing method can model
HVDC system without any complication, as we only need to
know the flow value on the line. In addition, optimal tracing
methods can model circular flows without any difficulty. However, modeling an HVDC line in the MP method does pose
a challenge. This is because, flow on the HVDC line is regulated by power order and hence it remains constant for marginal
change in load or generation. Hence, marginal participation of
an HVDC line is zero. Thus, MP-method cannot directly recover
cost of an HVDC line. Therefore, to evaluate utility of HVDC
line for a load or a generator, we propose the following methodology.
Step 1: Evaluate the TSU cost allocation of ac subsystem for
loads and generators corresponding to base case which has all
.
HVDC lines in service. Set
Step 2: Disconnect the th HVDC line and again compute the
new flows on the ac system. Hence, evaluate the new TSU cost
allocation of the ac system for the loads and generators. Let the
. Then, the usage cost
usage cost of an entity change by
of the th HVDC line for the th entity is given by
(25)
In other words, negative contributions are ignored.
, stop; else connect back the th HVDC
Step 3: If
line, increment
, and go back to step 2.
Notice that the proposed approach is similar to with and without
transit flow method, except that herein without scenario corresponds to disconnecting a circuit element, rather than alteration
in import/export through the system.
Remark 2: HVDC line in a dc load flow is modeled as a load
with MW equal to P-order at the sending end and a generator
with corresponding MW at the receiving end. A “without” scenario for an HVDC line corresponds to disconnecting the corresponding load-generation pair. Sensitivities for these fictitious
loads and generators are not computed as they are not to be
priced.
IX. RESULTS
The proposed approach has been implemented in C++.
XpressMP package [27] has been used for solving sparse LP
problem. Simulations were carried out on an Intel Pentium-IV
PC with 1 GB RAM with Linux operating system. We consider
a 1699-bus all-India transmission system which corresponds
to year 2011–2012 scenario. The system has 805 loads and
499 generators. The transmission system has a total length of
188 834 kms with 3158 kms of 765 kV, 86 718 kms of 400 kV,
98 432 kms of 220 kV, and 527 kms of 132 kV lines. There are
ten HVDC lines with 8300 kms of length and having capacity
of 6600 MW. Simulation studies considered six scenarios with
load variation of 109 748 MW to 155 490 MW corresponding
to seasonal peak and off-peak conditions.
We compare and contrast results based upon AP, MP, and
min-max fair tracing method. Table I summarizes the key performance indicators for each method. The value of in tracing
methods for each scenario is so chosen that the total cost shared
Fig. 3. Summer peak prices sorted in non-ascending order.
between loads and generators is identical to that of marginal participation method. This facilitates comparison between different
methods; the values are close to 0.5. The performance indicators
used to evaluate various methods are:
1) equity: as a methodology to resolve conflict;
2) computation time: The algorithm should solve the TSU
cost allocation problem in a reasonable time;
3) effect on generation and load siting decisions.
In this study, for convenience, loads associated with a state
are grouped under one state DISCOM and similarly generation
within a state is grouped as a state GENCO. There are 32 states
and union territories in India. The salient observations are as
follows.
A. Equity
To achieve fair comparison on equity between the three
schemes, a dispersed slack bus was used with MP method, i.e.,
an unit increment in load is distributed among all generators in
the grid in proportion to their respective MW. Similarly, for a
generator, each unit increment in injection is distributed among
loads in proportion to their respective MW. However, no such
finer adjustments can be done in AP method.
Fig. 3 shows prices obtained from AP, MP, and min-max
tracing for the summer peak condition. The states are ordered
on X-axis so as to obtain non-ascending price distribution as we
move from left to right. Now the figure using min-max tracing
clearly brings out its additional conflict resolution property,
i.e., equity to the extent possible within usage based price
framework. We see clusters of state DISCOMs and GENCOs
with same price which brings out the cooperative nature of the
methodology within usage based pricing framework. Notice
that standard deviation (stdev) in price with min-max tracing
(Table I) is consistently lower than that corresponding values of
AP and MP method. Further, standard deviation of MP method
is consistently lower than the AP method. A likely reason for
this is the usage of dispersed slack bus in the MP method.
Fig. 4 shows the overall system price of DISCOMs of
each state computed as a weighted aggregated prices of each
scenario. Weights are given in proportion to the duration of
RAO et al.: MIN-MAX FAIR POWER FLOW TRACING FOR TRANSMISSION SYSTEM USAGE COST ALLOCATION
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TABLE I
SUMMARY OF RESULTS FOR DIFFERENT SCENARIOS OF 1699-BUS SYSTEM
Fig. 4. Seasonal weighted average transmission prices of state DISCOMs using AP, MP, and min-max tracing methods.
scenario. It can be seen that maximum price in min-max tracing
approach (15.74 paise/kWh) is lower than that of average
participation and marginal participation methods which are,
respectively, 25.33 paise/kWh and 23.31 paise/kWh. This indicates that worst price takers are given due attention in min-max
tracing approach consistent with the minimum regret principle.
Also, maximum price from average participation method is
larger than maximum price from marginal participation method.
This is probably due to use dispersed slack bus in MP method.
Similar results are observed with GENCO prices.
We conclude that min-max fairness strives for equity emphasizing the cooperative nature of this usage cost allocation
scheme.
B. Computational Time
Table I shows that, computationally, AP method is the most
efficient method followed by min-max tracing and then MP
method. The computing times are in the range of 20–30 s for
AP, 5–6 min for min-max tracing, and 7–8 min for MP method.
While higher computational times for min-max is predictable,
the corresponding high range for MP method requires justification. A plain MP run ignoring HVDC lines requires only around
90–100 s of computational time. However, when the effect of
HVDC lines is modeled, the corresponding “without” scenarios
requires additional MP runs equal to the number of HVDC lines.
This increases overall computation time of MP method. It is
quite clear that 5–6 min computation time required by min-max
is very reasonable, considering that it provides an excellent conflict resolution feature.
C. Generator and Load Siting Signals
Under standard market design, siting signals for generation
can be based upon locational marginal prices (LMPs). However,
in India, no such mechanism has been implemented. Under
such circumstances, point of connection transmission system
price can be used to obtain some location signals. All the
three methods viz. AP, MP, and min-max tracing can provide
locational signals for siting generator and loads. Since TSU
cost of an entity (e.g., generator) cannot be quantified uniquely,
it implies that problems of fairness and economic efficiency
are coupled.4 By seeking equity in the point of connection
tariff, i.e., by clustering nodes into equal price clusters within
usage based pricing framework, the min-max fairness approach
can provide more freedom to generators to site themselves according to other important siting factors. This way of handling
trade-off between fairness and efficiency is unique to min-max
fairness approach.
X. CONCLUSION
This paper proposed a min-max fair tracing algorithm for
transmission system usage cost allocation problem, specifically
suitable for large system applications. The proposed improvements in the model lead to a direct, i.e., non-iterative approach
to min-max fair price determination. The model is equipped
to handle numerical precision problems expected in large systems. It can directly model coalition behavior of DISCOMs and
GENCOs. The proposed method has been tested on many systems including a large 1699-bus transmission system depicting
4Economic efficiency implies that investors build generation facilities at sites
which lead to the best overall use of the generation-transmission system [14].
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IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 25, NO. 3, AUGUST 2010
year 2011–2012 load flow scenario for India. It has been compared and contrasted with marginal participation and average
participation schemes, which can equally well be applied for
large systems. The results indicate the following.
1) Worst price takers in min-max method get a lower price
than the worst price takers in average participation and
marginal participation schemes. This behavior can be
theoretically justified when comparing with average participation scheme as AP method is also a tracing based
methodology. The favorable comparison with marginal
participation scheme brings out advantage of explicitly
modeling of min-max fairness constraints in TSU-cost
allocation problem.
2) The min-max fair algorithm seeks equal price clusters,
which implies that equity is factored within the usage based
pricing framework. This leads to better conflict resolution
feature in min-max approach.
3) On a large real-life test example, computational time of
min-max fair algorithm has been found to be reasonable.
It is in order of few minutes.
In totality, the work brings out min-max fair tracing as a good
alternative for TSU-cost allocation for large systems. In future,
authors plan to extend min-max fairness philosophy to the marginal participation scheme.
be given by
Proof: Let optimal solution point of
. Let active set of constraints at optimal
, where
The first example brings out the iterative nature while the
second example illustrates the problems of cycling and convergence to a suboptimal solution for the algorithm proposed in
[15].
1) Iterative Nature: Consider a problem to obtain min-max
which is governed by
fair allocation on vector
the following constraints set:
(29)
It is easy to verify that min-max fair allocation for this problem
is unique.
given by
Now for using model of [15] we need an initial feasible solu. Then, the first LP problem
tion which is, say,
is given by
(30)
Clearly, the current solution itself is optimal. In the second step
we solve the following problem:
(31)
APPENDIX A
PROOF OF LEMMA 1
be given by, i.e.,
APPENDIX B
SHORTCOMINGS OF ALGORITHM IN [15]
) and the optimal
The optimal cost is given by 10, (i.e.,
. No improvement is made
solution is given by
when we solve the next LP problem viz.
and
contains binding constraints of min-max type
. All other active constraints due to tracing set
and price vector definition (3) are represented by submatrix
and right hand side subvector .
Now assume to the contrary an entity
s.t. an alternate
solution
s.t.
. Then it implies that a
feasible direction
(32)
This concludes the first major iteration of the algorithm. Obis not the
serve that the current best estimate
. In the next major iteramin-max fair solution
tion, we first solve the problem (30) and as the current solution
itself is optimal, we can move to the next problem which is (32).
As the current solution is optimal with respect to (32) as well,
the next LP is given by
(26)
s.t. corresponding component
mality of
at
implies
. Now, opti-
(27)
where
. From (26)
and from (27)
(28)
Since for strict equality constraints of type
, we must
have
, (28) implies
. Since
(
would lead to infeasibility,
i.e.,
),
(by assumption) and
, we must have
, a contradiction
. Hence,
cannot be reduced below any further.
(33)
which is min-max
The optimal now is given by
fair solution. However, as the approach of [15] is iterative in
nature, one more pass (major iteration) is required to confirm
that no further improvement in min-max fair estimate can be
made. Thus, the algorithm requires three major iterations (nine
LP problems) to solve the problem. Note that the algorithm proposed in this work will solve the problem in just one pass, i.e.,
only three LPs are required.
2) Cycling Problem: We now provide an example to bring
out the cycling issue with the approach of [15]. Consider the
problem of min-max fair allocation in entities
subject to following constraints:
(34)
RAO et al.: MIN-MAX FAIR POWER FLOW TRACING FOR TRANSMISSION SYSTEM USAGE COST ALLOCATION
The min-max fair allocation is given by
the initial feasible solution be given by
the first LP is as follows:
. Let
. Then,
(35)
The optimal solution is given by
minimization problem is given by
. The next
(36)
The current solution, itself, is an optimal solution to problem
(36). Hence, the next optimization problem is given by
(37)
While minimizing , the optimal solution can remain at current solution, which concludes the major iteration 1. In major
iteration 2, the first formulation corresponds to problem (36),
and the optimal solution is same as the current solution. The
next optimization is given by
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or
. In case the vector arrived by the LP solver
, then the algorithm will terminate premawere
turely as the solution at the end of second iteration is identical
to that at the end of first iteration. However, note that min-max
.
fair solution is
Notice that in these illustrations, we have exploited multiplicity in optimal solution space of LP problem. In general, these
solutions represent different vertices or hyperplanes of the polytope.
We can now conclude that the approach of [15] which ed
min-max fairness constraints by balance type inequalities at
each iteration does not capture the generic min-max behavior.
APPENDIX C
TRACING GAME
We introduce a game based on tracing framework with the
intent to capture game theoretic rationale of min-max fairness.
. Define
Let be the set of players, , a coalition, i.e.,
characteristic function,
, of the TSU cost allocation game
as the maximum cost, a coalition
may have to pay under the
tracing framework. This is the upper limit on the usage based
allocation under tracing framework, i.e.,
(41)
(38)
The optimal solution of above problem is
next optimization problem is as follows:
. The
should be appropriately interpreted as load or generation at
a given bus. Let be the payoff (cost) for the player .
Lemma 2: Characteristic function defined by (41) meets the
requirement of it being sub-additive for cost allocation game.
Proof:
(39)
An optimal solution to (39) is
; this concludes the major iteration 2. To make sure that the there is no
further improvement in min-max fair solution estimate, we proceed to the next major iteration. The next optimization problem
is given by
(40)
Both
and
are optimal
solutions to this problem. A black box solver may reach
any one of them. To bring out the issue at stake, we choose
. The next optimization problem is same as
. Subsequent
(38), and an optimal solution is
optimization problem is given by (39), and an optimal solution
is
. This concludes major iteration 3. The
solutions at the end of major iterations 2 and 3 are not same, so,
the algorithm proceeds to the next major iteration. Observe that
the initial solution for the next major iteration is same as the
initial solution for major iteration 2; the steps are as in major
iteration 2 and then 3 and so on, which leads to cycling.
3) Convergence to Suboptimal Solution: The same example
can also be used to demonstrate the possibility of convergence to
suboptimal solution. In the last step of second major iteration in
(39), optimal solution could have been either
Further, even by a simple illustration, it can be shown that this
game is essential.
and
, then
Lemma 3: If
.
Proof:
Above derivation holds even for a singleton set
.
Since, the cost coefficients used in defining tracing based price
vector is so as to recover the total cost of the network, i.e.,
, we see that the cost allocation vector from
tracing space, in fact, belongs to set of imputations. The previous lemma further establishes that these imputations also meet
the coalition rationality requirement. Hence, price vectors derived from tracing solutions, in fact, belong to the core of the
tracing game. We capture this important result as a theorem.
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Theorem 1: All price vectors derived from the tracing solutions lie in the core of the tracing game.
Further, as a tracing solution (e.g., proportionate tracing) can be
computed in polynomial time, for this game, a pricing vector
belonging to the core can also be computed in polynomial time.
However, this is not true in general.
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M. S. S. Rao is currently pursuing the Ph.D. degree in the Department of Electrical Engineering, Indian Institute of Technology, Bombay, India.
His research interests includes transmission pricing, load forecasting, largescale power system analysis, and deregulation of power system.
S. A. Soman (M’07) is a Professor in the Department of Electrical Engineering,
IIT Bombay, Mumbai, India. He has authored a book on Computational
Methods for Large Sparse Power System Analysis: An Object Oriented Approach (Kluwer, 2001). His research interests and activities include power
system analysis, deregulation, and power system protection.
Puneet Chitkara is currently working with Mercados Energy Markets International, New Delhi, India. He was with Indian Institute of Technology, Kanpur,
India, as an Assistant Professor from 2000–2002. Among his research interests
are simulation of deregulated power markets, regulatory economics, and performance measurement.
Rajeev Kumar Gajbhiye (S’07) is currently pursuing the Ph.D. degree in the
Department of Electrical Engineering, Indian Institute of Technology, Bombay,
India. His research interests include large-scale power system analysis, power
system protection, and deregulation.
N. Hemachandra is an Associate Professor with Industrial Engineering and
Operations Research, IIT Bombay, Mumbai, India. His research interests are
broadly in operations research and its applications to supply chains, financial
engineering, logistics, and communication networks.
B. L. Menezes is a Professor in the Computer Science and Engineering Department at the Indian Institute of Technology, Bombay, India. His research areas
include forecasting, data mining, and network security.