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IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 15, NO. 3, AUGUST 2000
Improved Sensitivities in MW Dispatch
for Control of Voltage
Keith R. W. Bell and Daniel S. Kirschen
Abstract—This paper describes the derivation of linear sensitivities used in the dispatch of MW controls for alleviation of voltage
violations. Such a set of sensitivities is important for the selection
and co-ordination of necessary changes to MW generation, phase
shifter settings and, as a last resort, load shedding where reactive
power controls are insufficient. The sensitivity analysis has been
embedded in a qualitative reasoning based decision support tool
in which operators’ considerations such as cost, control effectiveness, preservation of control margin and ease of implementation
are modeled in such a way that priorities can be adjusted and results achieved quickly. Results are presented demonstrating the accuracy and effectiveness of the proposed approach both for a standard test system and a system representing a real network.
Index Terms—Fuzzy reasoning, knowledge-based systems,
power system operation, power system security, sensitivity analysis, voltage control.
I. INTRODUCTION
I
N ORDER to ensure the security of the transmission network, power system operators have at their disposal several
types of control actions [1]. In general, they will use control actions of the active power type (such as generation redispatch and
load shedding) to deal with thermal overloads and transient stability limit violations. When faced with voltage problems, they
will rely primarily on control actions of the reactive power type
(e.g. reactive power injections and transformer tap changes).
However, this clear separation cannot always be maintained and
active controls must occasionally be called upon to deal with
severe voltage problems. Since active controls are considerably
more costly than reactive controls, they are normally used as a
last resort.
One of the reasons why operators have become accustomed
to thinking of control actions in terms of decoupled active and
reactive subsystems has been the need to quickly find simple responses to emergency situations which can be easily dispatched
when automatic generation control (AGC) is unavailable, as
is often the case in unbundled competitive electricity markets.
Further, in some countries, notably the UK [2], moves have
been undertaken to distinguish the “active” energy market from
the provision of additional “reactive” services for support of
system voltage for which a separate market has been developed,
meaning that the effects of “active” and “reactive” controls must
be assessed independently.
Conceptually, in a competitive market, the operator’s first
job is to enable the secure transfer of active power. The operator
Manuscript received February 14, 2000.
The authors are with UMIST, Manchester, UK (e-mail: keith.bell@ngc.co.uk;
daniel.kirschen@umist.ac.uk).
Publisher Item Identifier S 0885-8950(00)07740-3.
must ensure that loading of network branches is not excessive,
both for thermal reasons and where stability limits have been
set, and this would mainly imply control of expensive active
power. Once dispatch of active power controls—the active
power output of generators and the angles of phase-shifting
transformers—has been decided upon to remove any loading
limit violations, an operator would expect to find any low
voltage problems have been ameliorated since the reactive
loss associated with heavily loaded branches will have been
reduced. Direct attention will then be given to ensuring that
voltage magnitude limits are respected, and this will be done,
as far as possible, without making the active dispatch any more
expensive or complex, i.e. by utilizing “reactive controls” such
as voltage set points and transformer tap ratios.
Such a separate dispatch of “active” and “reactive” subsystems is usually an efficient way of breaking the dispatch
problem down and solving it quickly and manageably. In
some circumstances, however, the use of “reactive” controls
is insufficient to relieve voltage problems and the operator
is forced to modify active power flows, in the last resort by
shedding load. Moreover, if the operator does this, it must be
possible to demonstrate that this action was the most effective
way of dealing with the actual problem.
While it is well-known that the relation between active
power injections and voltage magnitudes is nonlinear, sensitivities based on a linear approximation of this relation can be
very useful. This paper describes a new formulation of these
sensitivities which makes no approximation other than those
inherent in a first order analysis. This new method of computation has been incorporated into an operator decision support
tool based on a qualitative reasoning approach [3]. This tool
is therefore capable not only of adjusting the active controls
to solve active power problems and the reactive controls to
correct voltage difficulties, but also to recommend changes in
the active controls if no other ways exist to obtain a satisfactory
voltage profile. It appears that most other knowledge-based
approaches to the dispatch of corrective actions described in the
literature currently lack this ability [4]–[6]. These sensitivities
can also be embedded in a decoupled linear programming (LP)
based optimal power flow program [7].
The accuracy of the new method for computing these sensitivities and the robustness of the knowledge-based tool are demonstrated using the IEEE Reliability Test System [8] and a model
of the South West portion of the UK national grid [9].
II. SENSITIVITY ANALYSIS
A power system with buses can be described by a vector
of complex power balance equations. These equations involve
0885–8950/00$10.00 © 2000 IEEE
BELL AND KIRSCHEN: IMPROVED SENSITIVITIES IN MW DISPATCH FOR CONTROL OF VOLTAGE
complex state variables
balanced operation [10],
and
control variables . For
(1)
is applied, a small
If a small change to the control vector
will result. For
change in the vector of dependent variables
balanced operation to continue following the perturbation,
(2)
Using a Taylor series expansion and neglecting higher order
terms, equation (2) can be re-expressed as
(3)
and
are the and vectors before the change and
where
and
are the Jacobians of with respect to and respectively. Since the system was balanced before the change,
is zero allowing equation (3) to be re-written as
(4)
so that the sensitivity matrix
relating
to
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B. Relation of Active Power Controls to Active Powerflows
Many references describing the formulation of a matrix of
sensitivities of active power flows to active controls have required some pseudo-inverse matrix to be found, for example
[12]. The approach adopted here, however, avoids that and allows fast sparse matrix techniques to be used.
A vector
is defined as that of controllable active power
and, since
injections (including generation and load)
they primarily affect active power flows, phase shifter angles
. Bus 1 is defined as the reference bus. A sensican then be found relating small changes in
tivity matrix
to small changes in the active state vector
comprising
the active power generation at the reference bus and the nodal
is such that
voltage angles at all the others.
(11)
and
(12)
where
is
(13)
(5)
(14)
resulting from the change
Once is known, the change
can be estimated simply as
.
in control
A. Relation of Reactive Power Controls to Voltage Magnitudes
If the well-known principle of decoupling the active and reactive power equations of a power system is utilized and the
system is balanced, equation (3) can be re-expressed for the reactive subsystem as
and
comprises only the active power equations.
The changes in slack bus power and nodal voltage angles can
be related to changes in active power transmitted along each
from general
transmission line or through each transformer
node to node by
(6)
(15)
(7)
directly relating changes in active power
Hence, a matrix
injections and phase shifter angles and changes in active power
flow may be found such that
(8)
(16)
where the Jacobians
and
are
and
comprises only the reactive power balance equations.
comprises the dependent voltage magnitudes at PQ buses
contains controllable
and the reactive powers at PV buses.
voltage magnitudes at generators, synchronous compensators or
and SVC’s, and adjustable shunt susceptances. It also includes
the tap ratios of tap transformers since these primarily affect the
reactive characteristics of the system.
reactive sensitivity matrix
is found by
The
and
(9)
In practice,
is found by re-expressing equation (9) as
(10)
and solving using sparse matrix methods [11]. sense
(17)
If equation (12) is re-expressed as
(18)
can be found by sparse matrix methods and premultithen
to obtain
.
plied by
,
can be used to determine the
Since
effects of changes inactive power controls on active power flows
measured at each end of each branch.
C. Relation of Active Power Controls to Voltage Magnitudes
Whenever it is found that movement of reactive controls such
as generator terminal voltages, transformer tap ratios, SVC
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IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 15, NO. 3, AUGUST 2000
set points and switchable shunt susceptances is insufficient
to remove violations of load bus voltage magnitude limits, it
becomes apparent that the contribution of the transmission
system itself to reactive demand must be altered. This can only
be achieved by moving MW controls to alter flows.
In order to make use of these controls effectively, some sensitivity of the quantities to be controlled to movements in MW
generation, phase shifter angles and shedding of load should be
found. Any load shedding required will be achieved by shedding
proportionate amounts of both active and reactive load since,
in general, a transmission operator interacting with large customers such as distribution companies will not have any control over which individual loads are disconnected and the subsequent effect on load power factors, so that an assumption that
power factors remain constant is the most reasonable one.
As in Sections II-A and II-B, it is convenient to describe the
power balance equation in terms of control vector and state
vector partitioned into active and reactive components. In this
case, however, since the aim is to relate “active” controls to
is defined to comprise active power involtage magnitudes,
will comprise only
jections and phase shifter angles while
consists
reactive loads to properly represent load shedding.
of busbar voltage angles and the reference bus active power, and
includes busbar voltage magnitudes for PQ buses and reactive generation for PV buses so that changes to both active and
reactive injections and flows can be fully represented.
The power balance equation is then
and
with
as given in equation (7),
in (13),
in (14).
and
Equation (23) can be re-arranged to give
in (8)
(25)
and then
(26)
Substituting (26) into (24),
(27)
Grouping terms involving
,
and
together,
(19)
is changed by
and
by
, for balance to be
If
and
maintained, there will be small changes in the states
such that
(20)
Taking the Taylor series expansion of this expression and neglecting higher order terms,
(28)
consists only of changes to reactive load and these
Since
can be
are in proportion to the changes in active load,
represented by
(29)
(21)
where is a column vector where the terms corresponding to
are
changes in active generation and phase shifter angles in
zero and the others, as load power factors are assumed to be constant, are tan where is the power factor angle of the load at
contains only changes to reactive load
the th bus. Further, as
represents the active power balance equations,
is
and
zero. Hence, equation (28) reduces to
where
and all these derivatives are evaluated at
,
,
and
. Since the system was initially in balance, the first term in
equation (21) is zero leaving
(22)
If is also partitioned into active and reactive equations
and
, it can be seen that this represents two sets of equations
where
(23)
(24)
(30)
A sensitivity matrix relating changes in MW controls and load
shedding to load bus voltage magnitudes and PV bus reactive
generation can be found using the following relation
(31)
BELL AND KIRSCHEN: IMPROVED SENSITIVITIES IN MW DISPATCH FOR CONTROL OF VOLTAGE
1037
where
(32)
In practice,
is found by solving
(33)
where
(34)
(35)
A previous formulation of sensitivities relating active
controls to voltage magnitudes has made an approximation
has two
by considering that the term
cross-coupling terms between the active and reactive subin
systems and is therefore likely to be insignificant [7].
. While this speeds
equation (33) will then be simply
the finding of the sensitivity matrix considerably since the
will have already been found in
triangular factors of
the use of reactive controls for voltage control which preceeds
the invocation of any MW controls for voltage, the use of
active controls for control of voltage is likely to be needed
only for stressed systems where the decoupling of active and
reactive subsystems is weak. The full formulation of shown
in equation (34) will therefore be needed. This is illustrated in
Section IV where, in some cases, the approximate version leads
to voltage changes being predicted in the wrong direction.
III. INTEGRATION WITH A DECISION SUPPORT TOOL
This new method for computing sensitivities has been embedded in a knowledge-based decision support tool founded on
qualitative reasoning. This tool is described in more detail elsewhere [3] but its main characteristics are that the operator’s decision making process is modeled directly; operators’ priorities
can be easily adjusted; and the number of re-dispatches of settings, i.e. the complexity of the re-dispatch, can be controlled.
This latter is regarded as being particularly important in control
rooms lacking AGC under emergency conditions.
The tool’s structure comprises three dispatch sub-systems
and is illustrated in Fig. 1. The dispatch subsystems are
• active dispatch—dispatches settings of active power generation, shedding of load (active and reactive components
in proportion) and changes to phase shifter settings in
order to relieve overloads of transmission lines and cables;
• reactive dispatch—dispatches settings of reactive control
devices to correct violations of voltage magnitude limits;
• dispatch of active controls for correction of any outstanding voltage problems.
The required control moves in each sub-system are derived
with reference to the appropriate sensitivity matrices which are
Fig. 1. Corrective actions scheme including changes to MW’s for voltage.
also used to ensure that there are no secondary effects such as the
creation of new violations or the worsening of any existing violations. Which moves are chosen is decided by balancing the criteria of low cost, preservation of control margin and simplicity
of implementation.
Based on representation of adjectives describing the quality
of a variable—“linguistic values”—by “fuzzy sets” defined by
“membership functions,” fuzzy reasoning gives the facility to
encapsulate operators’ knowledge in fuzzy rules which can be
implemented in software. A set of rules can then be used to
assess appropriate responses to a control problem based on a
number of criteria which are combined by means of various
fuzzy operators [13]. Fuzzy descriptions of changes to control
settings satisfying a number of objectives are found which are
then “defuzzified” to yield crisp changes to the inputs to the
process being controlled.
Since being first proposed, “fuzzy control” by means of
qualitative reasoning expressed through fuzzy rules has found
widespread application [14]. In contrast to conventional production rule techniques, fuzzy sets capture expert knowledge
concisely and the rules provide smooth mappings between
input and output which can be adjusted on-line [15].
A further development has seen the use of the “cardinality” of
a fuzzy set to rank different fuzzy rule outputs [16], e.g. to determine which single control device would best suit the objective
of relieving a number of violations of power system operating
limits while costing little and preserving control margin [3].
Each dispatch sub-system is implemented in the manner
shown in Fig. 2 and is based on one of two sets of rules. The
rules for active dispatch for relief of overloads are as follows
where the terms in italics are fuzzy sets, the terms in sans
serif are crisp measurements or calculated crisp quantities
and
terms are margin-limited sugwhile the
gested actions for individual controls and specific violations
which are combined to form single fuzzy control moves for
each controller.
1) If loading is high and sensitivity is positive
and control margin is enough to lower and cost
is low then setting is
2) If loading is high and sensitivity is negative and
control margin is enough to raise and cost is low
then setting is
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IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 15, NO. 3, AUGUST 2000
Fig. 2. Operation of dispatch routine.
Fig. 3.
With appropriate inputs, the rules for correction of voltage
using reactive or active controls are:
1) If voltage is low and sensitivity is positive and
control margin is enough to raise and cost is low
then setting is
2) If voltage is high and sensitivity is positive and
control margin is enough to lower and cost is low
then setting is
3) If voltage is low and sensitivity is negative and
control margin is enough to lower and cost is low
then setting is
4) If voltage is high and sensitivity is negative and
control margin is enough to raise and cost is low
then setting is
The fuzzy sets are defined as in Fig. 3. In Fig. 3, is voltage,
is the sensitivity of the th dependent variable to a change
in setting on the th controller, min( ) is the most negative
sensitivity for any controller controlling the th dependent variand
are deadbands,
able and max( ) the most positive,
and
are maximum and minimum changes in
is the change necessary on
setting on the th controller,
is a margin
controller to correct a violation of variable ,
dead-band, and are some typical minimum and maximum
is the
costs, is the active power on a particular branch,
is some
maximum power permissible on that branch and
factor greater than 1 [3].
and
alVariation of the deadbands and the factors ,
lows the significance of different clauses in the rule antecedents
to be adjusted. The costs which are fuzzified are the costs of the
proposed actions—zero for reactive controls, or set according to
Fuzzy set definitions.
the price of the required change in MW’s for generation or the
value of lost load for load shedding. is the degree of membership of the fuzzy set in question.
By comparing the cardinality of each controller’s fuzzy proposed setting change, the controls may be ranked in order of
suitability for the power system scenario to be corrected considering the criteria specified in the rules [3]. The user may then
choose to use only the first controls so that implementation
of the suggested actions may be easily achieved (the program
will then use additional controls only where necessary to remove outstanding violations). The fuzzy control moves are then
“defuzzified” using the centroid method [13] to yield crisp control moves. Where MW generation is being re-dispatched, the
total increase in MW generation is matched by the sum of generation increases and load reductions to ensure that MW balance
is maintained.
Finally, before being tested in a full AC load-flow, the chosen
crisp control actions are checked with the appropriate sensitivity
matrix to ensure that in relieving one violation, another is not
created or made worse, and reactive power limits are not hit.
The actions are then reduced if necessary.
It may be noted that, for each dispatch sub-system, the user
may choose the maximum number of iterations around the main
loop in Fig. 2 (that where violated states have been improved and
further actions are sought where necessary) before returning to
the main program control shown in Fig. 1. This may be used,
for example, to force the full dispatch system to seek further
“reactive” control moves after every iteration of active controls
in the “active dispatch for voltage” sub-system. (In the example
BELL AND KIRSCHEN: IMPROVED SENSITIVITIES IN MW DISPATCH FOR CONTROL OF VOLTAGE
TABLE I
ESTIMATES OF CHANGES IN VOLTAGE ON IEEE RTS
shown in Section IV-B, however, this maximum number of iterations is set to an arbitrarily high figure7 of 20 so as to reduce
the complexity of the dispatch.)
Where required, the knowledge-based system described
here can be adapted to simultaneously find corrective and
preventative actions [17]. For a given intact scenario, each
of a pre-defined set of contingencies may be applied in turn.
The set of actions which would be necessary to remove the
post-contingency violations is derived each time. The subset of
those actions which are carried out for violations which would
have to be prevented (e.g. transfer limits set for stability) are
then composed together to form one fuzzy set representing
control moves necessary for preventative action for each
controller.
When all the contingencies have been tested, each fuzzy set
for preventative actions is defuzzified. The crisp actions for each
controller are then all tested using the base-case (intact system)
sensitivity matrix to check that no new violations are created
on the base-case. They are then implemented to form a new
base-case. The process is then repeated until all violations have
either been prevented or are correctable. Where they are correctable, the necessary actions will already have been found.
IV. TESTING
A. Testing of Linear Sensitivities
Table I shows the changes to bus voltages for PQ buses and
Table II those for reactive generation at PV buses predicted by
the sensitivity analysis described in Section II-C for the IEEE
reliability test system (RTS) at peak demand [8]. The control
change was an 80 MW generation decrease at bus 12 and a simultaneous 80 MW, 16.25 MVAr reduction of load at bus 20.
and
, final values
and
found
The initial values
and
from a full AC load-flow and the actual changes
are also shown.
and
for method 1
In the tables, the estimates of
were given by the sensitivity analysis with in equation (33)
[7]. The estimates of
and
approximated as simply
by method 2 were given for as in equation (34). The
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TABLE II
ESTIMATES OF CHANGES IN REACTIVE GENERATION ON IEEE RTS
Fig. 4.
Section of England and Wales south-west test system.
errors for each method are the errors of the different estimates
with respect to the actual change found from the AC load-flow
expressed as a percentage of the actual change. Estimates which
are of the wrong sign (i.e. where the actual change is in the
opposite direction to that predicted) are shown in parentheses.
in method 1 can
The effect of the approximation of as
be clearly seen.
B. Application to Corrective Re-Dispatch
A model of the south-west portion of the England and Wales
system [9] was used to test the knowledge-based system. Fig. 4
shows a diagram of a section of this network. Table III shows the
violated states resulting from an outage of the double circuit out
of ABHA4 to LAND4 and INDQ4 concurrent with a planned
outage for maintenance of the branch connecting ALVE4R and
TAUN4. The successive states result from application of the
control actions listed in Table IV. (Note that FAWLEY is east
of the part shown in Fig. 4.)
These results demonstrate that the knowledge-based system
can correct a severe voltage problem in three steps. The third of
the steps involves a significant change in the active controls.
V. CONCLUSIONS
This paper has presented a development of standard decoupled approaches to the selection of corrective actions which permits changes to MW generation and load-shedding to be scheduled for relief of voltage violations. This is enabled by a new
formulation of first-order sensitivity analysis.
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IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 15, NO. 3, AUGUST 2000
TABLE III
VIOLATIONS ON ENGLAND AND WALES SW SYSTEM
TABLE IV
ACTIONS TO ALLEVIATE VIOLATIONS
The use of the new sensitivity analysis has been demonstrated
as part of a knowledge-based system founded on qualitative
reasoning to represent operators’ judgments. As well as
dispatching appropriate changes to settings of MW controls
for alleviation of voltage violations only when necessary, the
knowledge-based system improves on many existing knowledge-based approaches by enabling cost to be considered and
in providing a facility for the dispatch of preventative actions.
REFERENCES
[1] B. Stott, O. Alsac, and A. J. Monticelli, “Security analysis and optimization,” Proc. IEEE, vol. 75, no. 12, pp. 1623–1644, 1987.
[2] A. P. Malins, D. J. Coates, A. M. Chebbo, and K. M. Radi, “Reactive
power market and reactive plant control in England and Wales,” in Proc.
CIGRE Symposium on Working Plant and Systems Harder, London,
June 7–9, 1999.
[3] K. R. W. Bell, A. R. Daniels, and R. W. Dunn, “Modeling of operator
heauristics in dispatch for security enhancement,” IEEE Trans. on Power
Systems, 1999, to be published.
[4] L. M. F. Barruncho, J. P. S. Paiva, C. C. Liu, R. Pestana, and A.
Vidigal, “Reactive management and voltage monitoring and control,”
Int. Journal of Electrical Power and Energy Systems, vol. 14, no. 2/3,
pp. 144–157, 1992.
[5] R. Yokoyama, T. Niimura, and Y. Nakanishi, “A coordinated control
of voltage and reactive power by heuristic modeling and approximate
reasoning,” IEEE Trans. on Power Systems, vol. 8, no. 2, pp. 636–645,
1993.
[6] R. Cicoria, P. Migliardi, and P. Pogliano, “SELF: An expert system for
the verification of network operating constraints in power transmission
planning,” in Proc. ISAP’94, Montpellier, France, 1994, pp. 405–411.
[7] D. S. Kirschen and H. P. Van Meeteren, “MW/voltage control in a linear
programming based optimal power flow,” IEEE Trans. on Power Systems, vol. 3, no. 2, pp. 481–489, 1988.
[8] IEEE RTS Task Force, “IEEE reliability test system,” IEEE Trans on
Power Apparatus and Systems, vol. PAS-98, no. 6, pp. 2047–2054, 1979.
[9] N. T. Hawkins, “On-line reactive power management in electric power
systems,” Ph.D. dissertation, University of London, 1996.
[10] I. Hano, Y. Tamura, S. Narita, and K. Matsumoto, “Real time control of
system voltage and reactive power,” IEEE Trans. on Power Apparatus
and Systems, vol. PAS-88, no. 10, pp. 1544–1559, 1969.
[11] I. S. Duff, A.M. Erisman, and J. K. Reid, Direct Methods for Sparse
Matrices: Oxford Science Publications, 1986.
[12] A. O. Ekwue and M. J. Short, “Interactive security evaluation in power
system operation,” Proc. IEE Part C, vol. 130, no. 2, pp. 61–70, 1983.
[13] C. C. Lee, “Fuzzy logic in control systems,” IEEE Trans. on Systems,
Man and Cybernetics, vol. SMC-20, no. 2, pp. 404–435, 1990.
[14] M. Sugeno, Industrial Applications of Fuzzy Control. Amsterdam:
North Holland, 1985.
[15] D. Driankov, H. Hellendoorn, and M. Reinfrank, An Introduction to
Fuzzy Control: Springer-Verlag, 1993.
[16] H.-J. Zimmermann, Fuzzy Set Theory and its Applications: Kluwer Academic Publishers, 1991.
[17] K. R. W. Bell, A. R. Daniels, and R. W. Dunn, “A fuzzy expert system
for overload alleviation,” in Proc, PSCC 12, Dresden, Germany, August
1996, pp. 1177–1183.
Keith R. W. Bell received his B.Eng. in electrical engineering and electronics
and Ph.D. on “Artificial Intelligence and Uncertainty in Power System Operation” from the University of Bath in the U.K. in 1990 and 1995 respectively.
After a spell with Ansaldo Trasporti in Naples, he worked as a Research Associate at UMIST on computation of the value of security in large interconnected
power systems before taking up a post with the National Grid Company.
Daniel S. Kirschen received his electrical and mechanical engineer degree in
1979 from the Free University of Brussels in Belgium and his M.S. and Ph.D.
degrees from the University of Wisconsin-Madison in 1980 and 1985 respectively. After working for Control Data Corporation, Empros and Siemens between 1985 and 1994, he joined UMIST where he is a Senior Lecturer. His main
research interests are in the area of power system operation and economics.