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IEEE TRANSACTIONS ON POWER DELIVERY, VOL. 19, NO. 3, JULY 2004
Numerical Analyses of a Directed Graph Formulation
of the Multistage Distribution Expansion Problem
Mohammad Vaziri, Member, IEEE, Kevin Tomsovic, Senior Member, IEEE, and Anjan Bose, Fellow, IEEE
Abstract—In this paper, the numerical analysis of a multistage
formulation of the distribution expansion problem developed
is presented. The approximate objective function is employed
without sacrificing optimality. A complete nonlinear mathematical
programming optimization is employed to identify the global
optimum. The algorithm is applied here to two practical test
cases. Details of the input data and configuration of test systems
as well as the solutions are listed and discussed. Computational
considerations are addressed.
Index Terms—Distribution expansion planning, mixed integer
mathematical programming, multistage planning.
I. INTRODUCTION
D
ISTRIBUTION expansion planning is a difficult practical
problem with a more than 40-year history of continued
efforts and contributions for improved solutions. A practical solution should consider expansion of the substations and feeders
simultaneously in a multistage fashion to represent the natural
course of development while at the same time following all established industry goals and standards. The state-of-the-art in
distribution expansion planning has been summarized in [1], [2].
Based on the analyses of the previous contributions [3]–[15], a
new formulation as a directed graph network flow problem was
given in [16]. This new formulation has been summarized in
Appendix A.
In this paper, an approximate linear objective is introduced
that allows one to obtain the identical solution to the nonlinear
objective. That is, the quadratic term associated with network
losses can be approximated as linear without impacting the final
solution. A branch-and-bound algorithm is applied to find the
global minimum of the resulting mixed integer linear programming problem. The developed model is applied to two practical
test cases of moderately sized distribution planning areas having
multiple routing and conductor options. Numerical results indicate that a multistage expansion formulation works efficiently
for practical cases with desired results. This contradicts several
earlier research efforts that have focused on employing heuristic
methods that find suboptimal solutions in order to avoid computational complexity. In the following, various implementation concerns are discussed and analyzed. Several practical case
studies are presented that illustrate the viability of the approach.
Manuscript received April 15, 2003.
K. Tomsovic and A. Bose are with Washington State University, Pullman,
WA 99164-1067 USA.
M. Vaziri was with Washington State University, Pullman, WA. He is now
with Pacific Gas and Electric Company, Vallejo, CA 94591-5076 USA.
Digital Object Identifier 10.1109/TPWRD.2004.829948
II. IMPLEMENTATION CONCERNS
A. Linearized Objective
In previous research, the quadratic term arising from losses
was either, completely neglected (e.g. [8], [17]), linearized by
the conventional piecewise linearization techniques (e.g. [4],
[6], [9]), or handled by other quadratic approximations as in [3].
Conventional linearization techniques proliferate the number of
variables in proportion to the number of steps. Ignoring the
losses or other approximations introduce inaccuracies. Here, a
different approach is introduced. In [18], it is shown that estimated bounds for the ratio of the quadratic term to the linear
term, defined as the quadratic dominance ratio can easily be
determined. The quadratic dominance ratio, Q/L ratio, may be
used as a measure of linear or nonlinear dominance in a QP.
Q/L ratios larger than 1 indicate quadratic dominance, while ratios less than 1 indicate linear dominance. Specifically, a single
linear program (LP) having positive coefficients in the objective function can accurately approximate a linearly dominant
quadratic, which is a strictly increasing function of the continuous variable. The bound on error can be determined with only
the information available from the solution of this approximate
LP [19]. Further, it was noted that deviations between LP and
QP solutions occur at points where the QP solution starts to
contain cycles (loop flows) on the graph. For cases where the
solution is radial as in the distribution expansion problem, the
minimizer is the same for a substantially wider range of the Q/L
variations. Thus, for the purpose of finding the minimizer, the
quadratic objective function, may be replaced by a single linear
function without the need for conventional piece-wise linearization. Once the minimizer has been determined, the exact value
of the original QP objective may be found.
In [19], the coefficients used for the linear function that
approximates the quadratic objective function were arbitrarily
chosen to be the slopes of the secant line through the function at
lower and upper limit points of the continuous variables. Here,
which are the slopes of the
coefficients values of
tangents to the quadratic objective evaluated at the values of the
network flows have been chosen for the continuous variables
in the approximating LP. The cost of these losses is relatively
small compared to the fixed costs. As such, the accuracy for
the choice of the Loss Load factor (LLf) in the formula below
becomes less significant. A valid question then is if the cost of
the energy losses, which gives rise to the quadratic term may
often be ignored without a significant change to the solution,
then perhaps it is better to simply neglect the energy losses as
is done in [8]. Still, inclusion of a linear term penalizes for
the energy losses to some degree and accounts for the unusual
0885-8977/04$20.00 © 2004 IEEE
VAZIRI et al.: NUMERICAL ANALYSES OF A DIRECTED GRAPH FORMULATION OF THE MULTISTAGE DISTRIBUTION EXPANSION PROBLEM
1349
planning cases where consideration of losses would change the
routing option. Further, neglecting the losses will not improve
computational efficiency.
B. Computational Considerations
The distribution expansion modeled here as a network flow
problem with binary variables is NP-Complete. Branch and
bound techniques are usually used to solve mixed integer linear
programs (MILP) or quadratic programs (MIQP). In MIQP
formulations such as in [20], whether single stage or multistage,
every bounding QP sub-problem is proven NP-Complete [21].
In MILP formulations, as in this formulation, every bounding
LP sub-problem has been proven to have polynomial boundedness [21] if using an interior point algorithm. It should be noted
that, although polynomial bounded interior point algorithms,
as opposed to simplex, are less complex computationally, they
might perform less efficiently in practice [22]. For example,
the ellipsoid algorithm has not performed as well as the
simplex algorithm in practice for similar problems [22]. So
typically one employs the simplex algorithm for either MILP
or MIQP. Note that, as will be seen in the practical numerical
examples here, knowledge of, and proper implementation of,
the operational constraints often render NP-complete problems
computationally tractable.
III. NUMERICAL RESULTS
Two test systems are introduced. The first is a simple five
node example consisting of one substation and three load centers. The second is a moderately sized 14 node system consisting of three substations. In the second example, the number
of source possibilities for the load centers as well as the number
of routings have been exaggerated to increase problem size and
complexity in order to explore the limits of the proposed approach. A duration of two years between the first and the second
stages, and duration of eight years between the second and third
stages have been assumed for all cases.
A. Test Case 1
Consider the following simple case illustrated by Fig. 1 consisting of one existing substation designated by node 2 and one
existing load center, node 3. Future load center 4 will have six
possibilities for being served from node 3, and two additional
possibilities to be served from node 2. Load center 5 has three
source possibilities from any of the nodes 2, 3, or 4 but with only
one routing and one size per source possibility from nodes 2 and
4, and three additional size possibilities of service from node 3.
Note that unidirectional links indicate that flow can be in one
direction. Table I shows the diversified peak load data for the
various load centers at various stages. Table II is the feeder link
and transformer physical, characteristic, and unit cost data. The
unit costs are average value of the unit costs used by major investor owned and municipal utilities operating in northern California, USA. For all the simulated cases, a duration of two years
between the first and the second stages, and a duration of eight
years between the second and the third stages has been assumed.
Cumulative energy losses are based on uniform growth rates
using the initial and final projected loads for each load center.
Fig. 1.
System configuration—test case 1.
TABLE I
SYSTEM PEAK LOAD DATA—TEST CASE 1
Accumulations were over the periods between the subsequent
stages and a standard 30-year period beyond the final stage. For
the expense and the capital expenditures, an inflation rate of 5%
along with a fixed charged rate of 14% has been used.
The optimal solution and the budget requirements are shown
in Table III and Table IV, respectively. The solution has link 2-4
to supply load center 4 and link 3-5 to supply load center 5 as
the optimal choices. In Table IV, the variable costs as calculated
by the approximate MILP have been shown under the heading
of Expense LP. Knowing the minimizer, the more accurate variable costs may be determined by the MIQP objective function,
which has been shown under Expense QP in the table. Note that
the costs determined by the MILP objective, are overestimated
at early stages, and underestimated at the horizon stage. This is
reasonable as the quadratic objective function begins to dominate the linear approximation at higher loading conditions. Note
also that the cost of total energy losses cumulated over 40 years,
plus an average $3.00 per foot added charges considered in all
cases, is below 10% of the total costs. This further justifies the
use of the linear approximation in the cost of losses.
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IEEE TRANSACTIONS ON POWER DELIVERY, VOL. 19, NO. 3, JULY 2004
TABLE II
LINK/TRANSFORMER DATA—TEST CASE 1
TABLE IV
BUDGETARY REQUIREMENTS—TEST CASE 1
TABLE V
SOLUTION W/O MULTIPLE SIZE OPTIONS—TEST CASE 1
TABLE III
OPTIMAL SOLUTION—TEST CASE 1
a much higher cost. This is because the voltage constraints can
not be satisfied with the single size option of 4/O AL that was
given as input. For the case without voltage constraints however,
the solution was the same as the one given by the case with multiple size options. That is, node 3 remained the preferred source
as given by Table V, with end of line voltage of 117.4 Volts,
which is just below the allowable maximum voltage drop. Additionally, when the single size option was changed to 715.5 AL,
which is the size solution indicated by the first case having multiple size options, the solution was identical to the first case. If
no size gradation capability had been considered, then imposition of explicit voltage constraints should have been avoided.
That is, if one does not model all the planning options fully, one
must be careful about imposing the constraints. It should also be
noted for this case that the existing transformer (designated as
1-1 for the link 1-2) could adequately serve the load until stage
3. This shows the importance of developing an upgrade capability in the program, as temporary, economical solutions that
are upgraded in future are commonly practiced by the utilities.
B. Test Case 2
To investigate the effects of multiple size gradation in
conjunction with the existence of explicit voltage constraints,
two additional studies were conducted on this test case. In both
studies, the multiple size option for the link 3-5 was reduced to
only one size (as has been customarily done in the past studies).
The explicit voltage constraints were included in only one of
the studies. Further, to ensure that the voltage constraints were
impacting the solution space, the allowable voltage drop limit
was tightened from 12 volts to 8 volts (on a 120 V base) for
both cases. Results are shown in Table V.
Note that with voltage constraints present, load center 5 is
served from node 2 instead of the much closer node 3, and at
In this case, a 14 node system having one existing substation and two candidate sites for future development was studied.
Fig. 2 shows the network configuration possibilities for this
system. Note that to avoid confusion, only the routing options
designated by type of construction have been shown for each
link. The adjacent number refers to the number of sizes considered for the particular routing option. For example, a routing
option designated by OH-3, means that for this particular overhead routing, three conductor sizes have been considered.
Also note that the number of source possibilities for several
of the load centers has been exaggerated from the more typical
one or two. This adds to the problem complexity as the link
possibilities increase rapidly due to the additional multiplicity
in routing and size options considered [2]. System load data
has been provided in Table VI and the detailed input data can
VAZIRI et al.: NUMERICAL ANALYSES OF A DIRECTED GRAPH FORMULATION OF THE MULTISTAGE DISTRIBUTION EXPANSION PROBLEM
1351
Fig. 2. System configuration—test case 2.
TABLE VI
SYSTEM PEAK LOAD DATA—TEST CASE 2
TABLE VII
BUDGETARY REQUIREMENTS—TEST CASE 2
has been to develop programs that create the input file for the optimization program. Several interactive MATLAB routines have
been developed with adequate error checking to assist the user
build the system topology and input the system data. The computational performance for the cases studied has been summarized in Table IX. The results indicate that even with the exaggerated routing options computational speed is more than adequate.
be found in [23]. Table VII gives the budgetary requirements.
The optimal solution for case 2 is shown in Table VIII. Note
that the optimization program has selected one of the candidate
substation sites (node 10) for expansion.
For all MILP solutions, the commercial optimization package
CPLEX [24] has been used. Due to the enormity of data input
requirement for the distribution expansion problem, a challenge
C. Comments on Computational Performance
The distribution expansion modeled here as a network flow
problem with binary variables is NP-Complete. Branch and
bound techniques are usually used to solve mixed integer linear
programs (MILP) or quadratic programs (MIQP). In MIQP
formulations such as in [20], whether single stage or multistage,
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IEEE TRANSACTIONS ON POWER DELIVERY, VOL. 19, NO. 3, JULY 2004
TABLE VIII
OPTIMAL SOLUTION—TEST CASE 2
to the fact that in the distribution problem the reduced matrix A
is very sparse (only 8994 nonzero elements for the case mentioned here). Yet more extreme situations are unrealistic and
impractical mostly due to the enormity of acquisition, entry, and
management of the input data from users point of view. It should
be kept in mind that there are many NP-Complete problems of
practical significance that are too important to abandon merely
due to complexity. In many cases, especially with the advances
in computing technologies, NP-Complete problems of small to
moderate sizes are being solved efficiently [25]. For practical
problems of larger sizes, a perfectly acceptable near exact solution can often be obtained very efficiently. Further, proper implementation of operational constraints often makes NP-complete problems computationally tractable in practical cases.
IV. CONCLUDING REMARKS
TABLE IX
INPUT DATA & COMPUTATIONAL PERFORMANCE
every bounding QP sub-problem is proven NP-Complete [21].
In MILP formulations, as in this formulation, every bounding
LP sub-problem has proven for polynomial boundedness [21],
which is far less complex computationally. Still, typically one
employs the simplex algorithm for both the MILP and the
MIQP, which has a potential for exponential complexity. In
some cases, as the number of variables, , increases relative to
, the algorithm
the number of constraints, , such that
[26], i.e.,
can have exponential complexity of at least
NP-complete. In general the simplex algorithm has been shown
to work quite efficiently for solution of LPs. On the other hand,
some of the polynomially bounded interior point algorithms, as
opposed to simplex, are less complex computationally, but may
perform less efficiently in practice [22].
Note in Case 2 where the routing and source possibilities have
been exaggerated, simplex algorithm performed very efficiently.
This does not imply that for larger systems the algorithm will be
as efficient. It is conceivable especially when unusually large
number of routing and size options are considered, that be. Still in a yet larger problem
comes substantially larger than
,
, siminvolving upgrades (see [23]), with
plex algorithm performed quite efficiently. This is generally due
A complete multistage solution of the distribution expansion
problem designed for practical applications has been presented.
Modeling and formulation are based on the natural course of
progression in distribution expansion and aligned with the procedures and the standard practices of the industry. The multistage formulation has been designed and implemented within a
single mathematical programming algorithm. Global optimality
is guaranteed within the numerical tolerances of the branch and
bound algorithm without the need for heuristic or decomposition techniques. Capability for inclusion of multiple source options, multiple routing options pre source, and multiple size options per routing has been designed for each load center. These
options are necessary for practicality and correct application
of explicit voltage constraints. It was shown that the nonlinear
MIQP for this problem could be accurately modeled as an MILP
using a single linear function having the same minimizer. The
MILP formulation was applied to two different test cases, and
the results were analyzed. Results indicated that the formulation
reflects practical planning attributes and the optimal solution
can be found efficiently. For future enhancements, the formulation could be extended to include upgrades and optimal load
assignment possibilities, to aid the decision-maker in local loop
design and optimal switching arrangements. Further, the extension of the approach to accommodate the inherent uncertainties in load growth could greatly enhance the usefulness of the
methodology.
APPENDIX
OPTIMIZATION MODEL
The expansion problem is modeled as a minimum edge cost
network flow problem having multiple arc alternatives between
any two vertices (nodes) defined on a directed graph
where and are the vertices and arc sets, respectively. Here,
to allow multiple possibilities in routing and size for the arcs
(links), denotes a set of pairs of vertices for which multiple
arc routing options each having multiple size options may exist.
, at most one arc routing
Specifically, for any pair
option from a set
, having at most one arc routing size
is allowed. This allows variable muloption from a set
tiple routing options, each with multiple size options. Note that
is given by the
the total link possibilities between nodes
VAZIRI et al.: NUMERICAL ANALYSES OF A DIRECTED GRAPH FORMULATION OF THE MULTISTAGE DISTRIBUTION EXPANSION PROBLEM
product of the two sets
and
. Associated with some
,
vertices there exist external flows, so for each
denotes the external supplies or demands. If
, then there
is no external supply or demand at vertex . The multistage expansion model is summarized below [16].
(1)
such that
(2)
(3)
(4)
(5)
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where
is the arc set containing all possible links for the
system.
is the fixed cost of the
size of the
routing
of the link at stage
is the variable cost of the
size of the
routing of the link at stage
is the decision variable for the
size of the
routing of the link at stage
is the power flow from node to node in the
size of the
routing of the link at stage
.
is the length of the link
is the resistance of the link in Ohms per unit
length
is the nominal Line-Line operating voltage of
the link in KV
is the voltage drop coefficient for the flow variable.
is the total of the capital and the expense budgets
for all expansions at stage
is an arbitrary number in the range of
.
is the maximum flow allowable on
is the node voltage at stage .
is the set of all possible source nodes to .
is a posiitive number equal to the Maximum allowable Voltage drop.
,
if
Note that at the onset stage of planning
otherwise.
the link exists, and
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(6)
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(8)
(9)
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IEEE TRANSACTIONS ON POWER DELIVERY, VOL. 19, NO. 3, JULY 2004
Mohammad Vaziri (M’01) received the B.S. degree
in electrical engineering from the University of
California, Berkeley, in 1980, the M.S. degree in
electrical engineering from California State University (CSU), Sacramento, in 1990, and the Ph.D.
degree in electrical engineering from Washington
State University (WSU), Pullman, in 2000.
Currently, he is a Supervising Protection Engineer
at PG&E, San Francisco, CA, and Part-Time Faculty
at CSU Sacramento. He has 17 years of professional
experience at PG&E, and CA ISO, and more than ten
years of academic experience teaching at CSU Sacramento, and WSU Pullman.
He has authored and presented technical papers and courses in the U.S., Mexico,
and Europe. His research interests are in the areas of power system planning and
protection.
Dr. Vaziri is an active member and serves on various IEEE and other technical
committees.
Kevin Tomsovic (SM’00) received the B.S. degree in electrical engineering
from Michigan Technical University, Houghton, in 1982, and the M.S. and Ph.D.
degrees in electrical engineering from the University of Washington, Seattle, in
1984 and 1987, respectively.
Currently, he is a Professor at Washington State University, Pullman. From
1999 to 2000, he held the Kyushe Electric Chair for Advanced Technology at
Kumamoto University, Kumamoto, Japan. His research interests include intelligent systems and optimization methodologies applied to various power systems
problems.
Anjan Bose (F’89) received the B.tech. degree (Hons.) from the Indian Institute
of Technology (IIT), Kharagpur, in 1967, the M.S. degree from the University of
California, Berkeley, in 1968, and the Ph.D. degree from Iowa State University,
Ames, in 1974.
Currently, he is the Distinguished Professor in Power Engineering and Dean
of the College of Engineering and Architecture at Washington State University,
Pullman. He was with the Consolidated Edison Co., New York, from 1968 to
1970; the IBM Scientific Center, Palo Alto, CA, from 1974 to 1975; Clarkson
University, Potsdam, NY, from 1975 to 1976; Control Data Corporation, Minneapolis, MN, from 1976 to 1981; and Arizona State University, Tempe, from
1981 to 1993.