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IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 15, NO. 2, MAY 2000 879
Optimum Voltage Regulator Placement in a RadialPower Distribution NetworkAnastasia S. Safigianni, Member, IEEE, and George J. Salis
AbstractA computer algorithm for optimal voltage controlwith voltage regulators, suitable for large radial distribution net-works is given in this paper. An objective function concerning thetotal cost of the voltage regulators (investment and maintenancecost) as well as the cost of losses of the examined networks is de-veloped and constitutes the base of the algorithm. This algorithmmakes theinitial selection, installationand tapsetting of thevoltageregulators, which provide a smooth voltage profile along the net-work,utilizing former algorithmssuitably modified and optimized.Then it attempts to minimize the number of the initially selectedvoltage regulators as much as possible, by moving them in sucha way as to control the network voltage at the minimum possiblecost (maximization of the objective function). Thealgorithm is fast,efficient and reliable as its application to practical distribution net-works shows.
Index TermsRadial primary distribution networks, Voltagecontrol, Voltage regulators, Cost minimization.
I. INTRODUCTION
I F THE reinforcement of a network were required becauseof excessive voltage drop, any such reinforcement could bedeferred if the voltage drop could be sufficiently reduced by
some means, subject to economic as well as technical consid-
erations. Various devices such as capacitors and voltage regula-
tors (VRs) can be installed to reduce the voltage drop, experi-
enced at critical points of medium voltage networks. Conductor
replacements at network segments can also be used to maintain
the voltage along the entire network.
The papers [1], [2] propose reconductoring of currently
operating primary distribution networks in order to optimize
them technically as much as possible defraying the minimum
cost. Representative papers dealing with the optimization of
medium voltage networks operation by selecting (kind and
size), installing and controlling the appropriate number of ca-
pacitors, are [3][10]. Fewer papers deal with the determination
of the optimal locations and real-time control (tap positions)
of a minimum VR number, in order to minimize the peak
power and energy losses and provide a smooth voltage profile
along a distribution network with lateral branches, under timevarying conditions. Recent papers of this kind are [3], [4], [5],
[6]. Specifically an integrated method for the optimal reactive
power and voltage control of radial distribution networks by
using capacitors and VRs is given in [3], [4], [5], which are
parts of a complete study, as well as in [6]. All these papers
decouple the capacitor problem from the VR problem and
Manuscript received September 28, 1998.The authors are with the Departmentof Electrical and Computer Engineering,
Democritus University of Thrace, Xanthi (67 100), Greece.Publisher Item Identifier S 0885-8950(00)03833-5.
propose VRs for a network completely compensated with
capacitors.
The network voltage is the criterion for the selection of the
optimal VR number, locations and tap positions in [3], [4], [5].
An economic function, which estimates the power and energy
losses, is given in retrospect but this function is not utilized
during the main problem-solving process concerning the op-
timum VR placement and installation. The power flow at the
network segments is calculated by an approximate method (V-P
model, [11]).
In [6], the voltage regulation is initially attempted by
changing the tap positions at the substation and solving againthe capacitor problem. If the desirable voltage regulation is
not achieved in this way, a VR is placed at the main feeder,
next to the node where the subfeeder with the heaviest load is
connected and then the proper tap position of this VR is de-
termined. Sometimes VRs are placed at very long subfeeders
to improve their voltage profile. Consequently in this paper
the VR positions are in a way predetermined and they are not
indicated by the method.
The present paper utilizes the basic philosophy of the algo-
rithm of [3], [4], [5] only with regard to the initial VR selection,
placement and tap setting. But even in this stage it differenti-
ates the method of the above papers by calculating the currents
at the network segments by load flow analysis instead of an ap-proximate method as these papers do. After the determination of
the initial VR number and locations the optimization procedure
does not finish in opposition to [3], [4], [5], because it is true
that the resultant solution solves the problem of the excessive
voltage drop technically but it is not surely the most econom-
ical one. For this reason an objective function is developed in
the context of this paper, which includes the VR total cost and
the cost of losses of the examined network (evaluating the power
losses for thepeak load and the energy lossesaccording to an ap-
proximate annual load curve consisting of twelve distinct mean
monthly load values). Based on this function a procedure in-
vestigating the possible reduction of the number of the initially
selected VRs is evolved so as the finally proposed solution willbe the most economical one.
The application of the proposed method to a great number of
practical radial primary distribution networks has proved that
there are no restrictions according to the size of the examined
networks as the method is easy to use, very fast and efficient.
II. INITIALVOLTAGEREGULATORPLACEMENT ANDCONTROL
A power flow analysis method precedes the procedure of the
VR selection and installation in a radial distribution network, in
08858950/00$10.00 2000 IEEE
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880 IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 15, NO. 2, MAY 2000
order to evaluate the exact node voltages and segment currents.
Thus it is possible to avoid the fallacious assumption that a ra-
dial distribution network is optimally compensated using par-
ticular capacitors, because it is impossible to have a capacitive
current absolutely equal to the inductive current at each lateral
or sublateral branch due to the tree structure of the said network.
The power flow method used in this paper, which is based on
the direct application of Kirchhoffs voltage and current laws,is analytically described in [12] and has excellent convergence
characteristics when applied to large scale weakly meshed or
radial networks. According to this method, the voltage at the
source as well as the active power and the reactive power
at all the other nodes are given. The network is swept back-
ward and forward and the segment currents as well as the
node voltages are calculated. This power flow technique pre-
supposes that the examined networks are recorded as tree data
structures, so that they can be processed without special node
numbering. This simplifies the node sweep and increases the
flexibility of the method according to the addition or subtrac-
tion of branches. The voltage at each network node i must be
limited between specific limits and affected by thesize and the character of the load, in order that the total voltage
drop along the network and for each one of the network ends is
less than a definite percentage % of the rated voltage.
If this requirement is not satisfied a voltage regulation method
must be applied.
If the maximum possible voltage level were to be achieved,
as the power losses are proportional to the current squared and
the current is inversely proportional to the voltage, it would be
theoretically desirable to put one VR at each network node. This
would lead to the minimum cost of losses . In fact VRs
are installed only at some network nodes and they maintain the
voltage every moment according to the load. This means re-
spective time variation of the network current and relative losses
saving. The costof losses after the VR installationis given
by the relation:
(1)
where :
: the annual demand cost [dr/(kW a)]
: the energy cost [dr/kWh]
: the number of the network segments
: the network losses at peak load with VRs on the
network [kW]: the network losses during the time period with
VRs on the network [kW]
: time period during which the network losses are
constant
and
h (2)
The cost lies between two limits
(3)
where is the cost of losses before the VR installation
(4)
where:
: the network losses at peak load without VRs [kW]: the network losses during the time period T without
VRs [kW]
The VR problem consists of two subproblems, that of the
optimum selection and that of the optimum control. The first
subproblem concerns the determination of the VR necessary
number and locations on the network, to achieve the desirable
voltage profile, while the second subproblem concerns the se-
lection of their tap positions. The procedure for the first VR se-
lection and installation is relative to that described in [3], [4],
[5]. The following steps are executed by sweeping the network
forward (from the source to the ends):
Localization of the nearest node to the source where the
voltage drop is out of the predetermined limits.A VR installation at this node and tap setting of this VR so
that the voltage of the node is as close as possible to the max-
imum permissible voltage, without exceeding it.
Calculation of the node voltages by load flow analysis, which
takes into account the already existing VRs. The initial node
voltages, at the beginning of the repetitive load flow proce-
dure, do not necessarily need to be identical to the source
voltage . They may be equal to the node voltages before
the VR installation at the step II.
Examination of the node voltages to see if the desirable limits
are violated. If not the VR installation procedure comes to an
end or else all the steps are repeated.
A VR tap position at the node i can be generally determined by
the relation (5).
Int (5)
where:
Int : the function which returns the integer part of its argu-
ment
: the voltage at the node i before a VR installation at
this node under peak load conditions
: the VR voltage step (0.006 25 p.u.)
During the VR installation procedure the voltage refersto peak load conditions, which means that this voltage takes its
minimum value given by the relation:
(6)
where:
: the percentage voltage drop from the source to the
node i resulting from power flow analysis.
When themethod proposes forthe VR a tappositiongreater than
its possibilities, a location closer to the source must be searched
out for this VR, which will lead to a smaller tap position, even
if the voltage drop at this location is not unacceptable.
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SAFIGIANNI AND SALIS: OPTIMUM VOLTAGE REGULATOR PLACEMENT 881
The profit because of the losses saving, after the VR installa-
tion is:
(7)
This profit increases as the voltage at the network nodes ap-
proaches its maximum permissible value. That means the tap
positions must be adjusted during the year, so that the VR output
voltage approachesits maximum value without exceeding
it. The necessary variation of the tap positions , in order to
have the maximum output voltage, is given by the relation:
int
(8)
where:
: the previous VR tap position
(9)
The above described method for the VR placement is defective
because it does not take into account the annual investment and
maintenance cost for all the installed VRs which is rela-
tively high. An objective function taking into account this cost
can be written as:
(10)
with:
(11)
where:
: the number of the network nodes
: theVR annualinvestment andmaintenancecost
[dr/a]
when a VR exists at the node i
when there is no VR at the node i
: the VR investment cost [dr]: the VR annual maintenance cost [dr/a]
: the annual capital recovery factor given by the
relation:
(12)
where :
: the annual interest rate (no inflation)
: the expected VR life.
After theVR installationaccording to theabove mentioned steps
I-IV the initial value of the objective function (10) is calculated.
Fig. 1. Block diagram of the Pull_Back_Regulators procedure.
When the method proposes more than one VR it must be deter-
mined if the voltage can be limited between the predeterminedlimits using a smaller VR number in a way that maximizes the
objective function (10).
III. ALGORITHM FOR THE RESTRICTION OF THEVOLTAGE
REGULATORS
A recursive procedure named Reg_Recursive( ,Reg) is
used to examine whether fewer VRs contribute an acceptable
voltage profile to the network whilst simultaneously increasingthe value of the objective function (10). The arguments of this
procedure, which constitute the state variables of the problem,
are the initially proposed VR number and relative information
Reg [ ] for each one of these VRs such as the cor-responding network node and tap positions. The first time the
above recursive procedure is executed, the VR number and tap
positions are those resulting from the algorithm previously de-
scribed which are referred to as Scheme 1 in the block dia-
gram of Fig. 1.
The first stage of the Reg_Recursive( ,Reg) procedure
is the Find_path( ,Reg) procedure as in Fig. 1 is shown.
During this procedure each one of the VRs is successively
moved to nodes closer to the source, which without fail belong
to the route joining directly the source and the VR location. The
total movement route for one VR depends on the current stage
of the recursive procedure and whether or not VRs are present
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882 IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 15, NO. 2, MAY 2000
at the nodes which are further away from the source and stem
from the initial position of this VR. The contribution of these
VRs to the network voltage profile depends directly on the lo-
cation and the tap position of the examined VR. Every time the
VR is moved, its tap positions are adjusted since its rated input
voltage tends to increase as the VR approaches the source and
the voltage at the new VR location must approach the maximum
permissible value without exceeding it, according to the relation(8).
The next step is the voltage computation at each node
belonging to network branches, which stem from the new
VR location and an investigation of whether some voltage
drop problem exists. If no problem exists this new location
is acceptable and the VR is moved to the next node toward
the source within a procedure trying to find the route of the
permissible VR movements. If a voltage drop problem exists
the Find_path( ,Reg) procedure for the examined VR
terminates. In this case the end of the route of the permis-
sible movements is the closest node to the source, where
the VR installation keeps between the acceptable limits the
voltage of the nodes which stem from the VR position. Theexamined VR is moved back to the node it was at before the
beginning of the above described procedure, acquiring again
the tap position which corresponds to this location and the
Find_path( ,Reg) procedure goes on examining the next
VR. When this procedure finishes, each VR is accompanied
by a path of its permissible movements. This path may be
regarded as fictitious due to the fact that both the origin and the
end of the path are situated at the initial VR location, making
actual movements redundant.
The next step of the Reg_Recursive( ,Reg) procedure
is the formulation of all the possible pairs (reg ,reg ) derived
from the set of all the installed VRs at the network.
Supposing thatset=[reg , reg , , reg ] theneach paircan
be described as (reg ,reg ), with
and
The total number of the pairs is:
(13)
For each one of these pairs the following are executed. It is
checked if there are common nodes between the routes path of
the VRs corresponding to each pair (Check_paths(reg ,reg )
procedure). If there are not, the Reg_Recursive( ,Reg)
procedure goes on dealing with the next pair (reg ,reg ). In
the opposite case, the common node furthest from the source,
named com is chosen as the node where one VR is installed
instead of the two VRs of the pair. All the VR tap positions are
adjusted according to (8) and a new power flow analysis takes
place. The network acquires a new form and the Reg_Re-
cursive procedure acquires new values of the state variables
( ,newReg), which are , and
Fig. 2. Radial feeder having four voltage regulators.
Fig. 3. The radial feeder of the Fig. 2 after the reduction of the number of the
voltage regulators.
newReg[ ], provided the VR number is decreased
by one. The value of the objective function (10) is calculated
afterwards (Calc_SF procedure) for the new situation and if
this situation is more beneficial economically than the previous
situations, it is kept as the best one resulting from the inves-
tigation procedure up to this moment. The Reg_Recursive
procedure is executed with the form Reg_Recursive( ,
newReg) despite what the result of the objective function is
(better or worse).
The Reg_Recursive( ,Reg) procedure comes to an
end when all the possible VR pairs, (reg ,reg ) have beenexamined. If this procedure has not been called inside a
previous Reg_Recursive( ,Reg) procedure, the total
Pull_Back_Regulators procedure, concerning the VR move-
ment toward the source in order to reduce their number,
finishes.
The Pull_Back_Regulators procedure can be made quite
clear with the following example. Lets suppose that four
VRs have been installed during some stage of the recursive
procedure at the nodes 7, 9, 10 and 16 of the feeder given
in Fig. 2, which is a quite hypothetical feeder (a practical
application of the optimum VR placement method to a realistic
feeder is given at the next section). The first VR set examined
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SAFIGIANNI AND SALIS: OPTIMUM VOLTAGE REGULATOR PLACEMENT 883
Fig. 4. Feeder examined as a practical application of the optimum VR placement method.
by the Reg_Recursive( ,Reg) procedure is set=[reg ,
reg , reg , reg ] with . Lets also suppose that the VR
paths for the first execution of the above procedure are:
path reg node 7-5-3
path reg node 9-6
path reg node 10-6
path reg node 16-14
and the six pairs to be relatively examined are:
reg reg reg reg reg reg
reg reg reg reg
reg reg
For each one of the above pairs it is checked if a common node
com exists. The node 6 is such a node for the pair (reg ,reg ).
So a VR is installed at the node 6 and the VRs of the nodes 9and 10 are subtracted. The feeder acquires a new form given in
Fig. 3, which is likely to constitute an improvement in the value
of the objective function (10). At this point the recursive proce-
dure Reg_Recursive ( ,newReg) with
is executed again for the network form given in Fig. 3. New VR
paths aredefinedand there are now three pairs to be examined.
Every time the Reg_Recursive( ,newReg) procedure
is completed the feeder acquires again its previous form.
During the detection of the most economical way to have an
acceptable voltage profile along the examined network, using
the above described recursive procedure, it is possible for a spe-
cific network form to be examined more than once at different
execution levels of this procedure. One way to avoid this repeti-
tion is to store the network forms at each level of the procedure
and to compare the current examined form with the stored forms
to decide if it is possible to reduce the proceedings. However,this procedure is at least as time consuming as the repetitive
procedure described earlier, so there is no call for its implemen-
tation.
IV. CONCLUSIONS ANDRESULTS
A computer algorithm for the voltage control of large radial
distribution networks is presented in this paper. The algorithm
takes into account the actual network data such as conductor
sizes, lateral and sublateral branches as well as load distribution
and time variation and handles fast large distribution systems as
a total and notin parts making relative approximations. It aims at
an acceptable voltage profile along the network at the minimumpossible cost. That is obtained by moving toward the source and
readjusting the initially selected VRs, which probably leads to
a reduction of their number and a positive economical result.
A feeder of the primary power distribution network of the
area of Xanthi, Greece, which belongs to the Public Power
Corporation (P.P.C.) and has 229 nodes (load locations and
branch points), has been examined as a practical application
of the optimum VR placement method described above. The
furthest feeder end is 45 km away from the source and the
feeder segments have mainly conductors of the minimum size
( mm ACSR), as Fig. 4 shows. The segment lengths are
given in Table I. The rated voltage of the feeder is 20 kV and
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884 IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 15, NO. 2, MAY 2000
TABLE IDATA ANDRESULTS OF THEAPPLICATION OF THEOPTIMUMVR PLACEMENT METHOD ON THEMEDIUM-VOLTAGEFEEDER OFFIG. 4
% of the rated voltage is taken as the acceptable voltage
variation at each distinguished node as well as at all the terminal
nodes. A capacitor of 450 kVAr has already been installed at
the node 167 of the feeder in order to minimize the losses. The
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SAFIGIANNI AND SALIS: OPTIMUM VOLTAGE REGULATOR PLACEMENT 885
TABLE I (Continued)DATA ANDRESULTS OF THEAPPLICATION OF THEOPTIMUMVR PLACEMENT METHOD ON THEMEDIUM-VOLTAGEFEEDER OFFIG. 4
total peak load of the feeder at the moment the optimization
procedure begins is equal to 80 A. The installed loads at the
nodes of the feeder given in Table I are coincident with the
above peak load before the optimization procedure begins. The
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886 IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 15, NO. 2, MAY 2000
standardized VRs used by P.P.C. have nominal values 5 MVA
and 10 MVA, tap range of % and tap increments 0.625 %
( tap positions). Given in Table I is also the percentage
voltage drop before the VR installation. It is obvious that the
voltage drop exceeds the permitted limits in many feeder nodes.
The application of the VR optimization method given in [3],
[4], [5] after its differentiation, in order to include load flow
analysis, indicates three VRs with nominal values 5 MVA atthe nodes 100, 112 and 139 and tap positions for the peak load
16, 16 and 15 correspondingly. This solves the voltage drop
problem as the relative column of Table I shows. At this point
the Pull_Back_Regulators procedure, proposed in this paper,
is used aiming at the restriction of the necessary VR number
and consequently of the requisite cost. Specifically through the
Reg_Recursive procedure all the VR pairs are examined in
order to replace them by one VR closer to the source. The final
result is that the usage of only one instead of three VRs is pro-
posed which is obviously a more economical solution. This VR
has nominal value 5 MVA and tap position for the peak load 13,
it is placed at the node 37 and it solves the voltage drop problem.
It is also offering to the feeder more room to increase its induc-tive load without voltage problem, compared to the case of the
three VRs, as the relative columns of Table I show.
The algorithmdescribed in this paper was coded in the Turbo-
Pascal language. The time needed for the solution of the prac-
tical problem given above is 33s using a Pentium-233 MHz.
The optimization problem examined in this paper is part of
a more general problem, which has already been examined by
the authors of the paper and aims at giving the more econom-
ical combined optimization solution for currently operating pri-
mary power distribution networks, in an annual as well as in a
long-term basis, by using all the available means (capacitors,
VRs, network reconductoring).
REFERENCES
[1] G. Salis and A. Safigianni, Economically Justified Modification of aRadial Network to an Optimal Form, Advances in Engineering Soft-ware, vol. 21, no. 1, pp. 4965, 1994.
[2] G. Salis andA. Safigianni, Economical Optimizationof a Radial PowerNetwork Based on Voltage Drop Criterion, Advances in EngineeringSoftware, vol. 22, no. 1, pp. 120, 1995.
[3] J. J. Grainger and S. Civanlar, Volt/Var Control on Distribution Systemwith Lateral Branches Using Shunt Capacitors and Voltage Regulators,Part 1: The Overall Problem, IEEE Trans. on PAS, vol. 104, no. 11, pp.32783283, November 1985.
[4] J. J. Grainger and S. Civanlar, Volt/Var Control on Distribution Systemwith Lateral Branches Using Shunt Capacitors and Voltage Regulators,Part 11: The Solution Method, IEEE Trans. on PAS, vol. 104, no. 11,pp. 32843290, November 1985.
[5] S. Civanlar and J. J. Grainger, Volt/Var Control on Distribution Systemwith Lateral Branches Using Shunt Capacitors and Voltage Regulators,Part III: The Numerical Results,IEEE Trans. on PAS, vol. 104, no. 11,pp. 32913297, November 1985.
[6] M. M. A Salama, N. Manojlovic, V. H. Quintana, and A. Y. Chikhani,
Real-Time Optimal Reactive Power Control for Distribution Net-works,International Journal of Electrical Power & Energy Systems ,vol. 18, no. 3, pp. 185193, 1996.
[7] M. E. Baran and F. F. Wu, Optimal Sizing of Capacitors Placed on aRadial Distribution System, IEEE Trans. on Power Delivery, vol. 4,no. 1, pp. 735743, January 1989.
[8] M. E. Baran and F. F. Wu, Optimal Capacitor Placement on RadialDistribution System,IEEE Trans. on Power Delivery, vol. 4, no. 1, pp.725734, January 1989.
[9] H. D. Chiang, I. C. Wang, and G. Darling, Optimal Capacitor Place-ment, Replacement and Control in Large-Scale Unbalanced Distribu-tion Systems: System Solution Algorithms A Numerical Studies,IEEETrans. on PWRS, vol. 10, no. 1, pp. 363369, February 1995.
[10] H. D. Chiang, I. C. Wang, and G. Darling, Optimal Capacitor Place-ment,Replacementand Control in Large-Scale UnbalancedDistributionSystems: System Modeling and A New Formulation, IEEE Trans. onPWRS, vol. 10, no. 1, pp. 356362, February 1995.
[11] J. J. Grainger, S. Civanlar, K. N. Clinard, and L. I. Gale, OptimalVoltage Dependent Continuous Time Control of Reactive Power onPrimary Feeders,IEEE Trans. on PAS, vol. 103, no. 9, pp. 27142722,September 1984.
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Anastasia S. Safigianni(M97) was born in Greece in 1957. She received herDipl.-Eng. degree and Ph.D. degree fromthe Electrical Engineering Departmentof the Democritus University of Thrace, Greece, in 1981 and 1988 respectively.She is currently Assistant Professor in the Department of Electrical and Com-puterEngineering of the DemocritusUniversity of Thrace, Greece. Her teaching
interests include power systems and electrical installations. Her research inter-ests include power system planning and optimization and short-circuit lossesand forces in metal enclosed arrangements.
George J. Salis was born in Zurich, Switzerland, in 1970. He receivedhis Dipl.-Eng. degree from the Electrical Engineering Department of theDemocritus University of Thrace, Greece, in 1992 and his Ph.D. degree fromthe Electrical and Computer Engineering Department of the DemocritusUniversity of Thrace, Greece, in 1998. His research interests include computerapplication to distribution system planning and optimization.
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