Proc. 2001 IEEE-PES Summer Meeting, Vancouver, BC, July 2001.
Comparison of Voltage Security Constrained
Optimal Power Flow Techniques
Claudio Cañizares
William Rosehart
Alberto Berizzi
Cristian Bovo
IEEE Member
IEEE Student Member
IEEE Member
IEEE Student Member
University of Waterloo
Electrical & Computer Engineering
200 University Av. W.
Waterloo, ON N2L-3G1
CANADA
c.canizares@ece.uwaterloo.ca
Politecnico di Milano
Dipartimento di Elettrotecnica
Piazza Leonardo da Vinci, 32
20133 Milano
ITALY
alberto.berizzi@polimi.it
Abstract
This paper compares two different Optimal
Power Flow (OPF) formulations that consider voltage
security in power systems. The techniques are both based
on multi-objective optimization methodologies, so that
operating costs and losses can be minimized while
maximizing the “distance” to voltage collapse. The
techniques are described in detail and compared to study
their similarities, as well as advantages and
disadvantages. The comparisons are based on the results
obtained by applying these two methods to a modified
version of the 118-bus IEEE test system.
fact that optimization techniques can be readily used to study
the voltage stability problem [4], [5], has lead the authors to
propose OPF formulations that “optimize” a system
considering both costs and voltage stability criteria. Thus, in
[6] and [7], the authors propose two different OPF techniques
based on multi-objective optimization methodologies to
minimize operating costs and losses while maximizing the
“distance” to voltage collapse conditions. In the present
paper, these techniques are described in detail and compared
to study their advantages and disadvantages.
The
comparisons are based on the results obtained from applying
these two methods to a modified version of the 118-bus IEEE
test system [8].
This paper is organized as follows: Section II introduces
the basic theory associated with the two voltage security
constrained OPF techniques used in this paper, as well as the
actual mathematical formulation of these two optimization
problems. In Section III, a detailed comparison of the results
of applying these formulations to a test system are presented,
highlighting possible uses, advantages and disadvantages of
the two techniques. Finally, Section IV describes the main
contributions of this paper and discusses some possible future
research directions.
Keywords: Optimal power
optimization, voltage stability.
flows,
multi-objective
I. INTRODUCTION
Optimal Power Flows (OPF) have been widely used in
planning and real-time operation of power systems for active
and reactive power dispatch to minimize generation costs and
system losses and to improve voltage profiles [1]. Typically,
these two problems have been assumed decoupled and thus
treated independently. However, as the system operates
closer to its stability limits, such as its voltage collapse point,
due to market pressures, this assumption does not apply any
longer, and hence there is a need to consider these limits
within the OPF. By including these stability limits in the
OPF problem, optimization procedures can also be used to
improve the overall system security while accounting at the
same time for the costs associated with it, which is becoming
an important issue in open electricity markets.
The voltage stability problem in power systems has been
widely studied, and the basic mechanisms that lead to a
network voltage collapse have been identified and are now
clearly understood [2]. It has been demonstrated that the
overall stability of the system is closely associated with the
proximity of a system to a voltage collapse condition, i.e., as
the system approaches a voltage collapse point, its stability
region becomes smaller, resulting in a system that is less
likely to survive contingencies (e.g., [3]). Hence, as a first
approximation, one can use voltage stability criteria to
account for the overall system stability. This added to the
II. BASIC BACKGROUND
The basic voltage stability and OPF theory on which the
material on this paper is based is briefly discussed in this
section. The two diverse optimization formulations in which
the classical OPF is restated to account for system voltage
security are then explained in detail.
A. Voltage Stability
The voltage stability problem typically consists on
determining operating conditions where “equilibrium” points
of a nonlinear dynamic model of the power system merge [2].
This can be stated mathematically using a system model
represented by the following set of differential-algebraic
equations (DAE):
•
x = f ( x, y , ρ , λ )
0 = g ( x, y , ρ , λ )
1
(1)
where x∈ℜn stands for the system state variables such as
generator angles and angular speeds, associated with the
nonlinear field f:ℜn×ℜm×ℜk×ℜl→ℜn; y∈ℜm corresponds to
the system algebraic variables, i.e., variables such as bus
angles and voltages that are not associated with any system
dynamics and that are directly defined by a set of nonlinear
algebraic constraints characterized by the nonlinear function
g:ℜn×ℜm×ℜk×ℜl→ℜm; ρ∈ℜk represents system controlled
parameters such as AVR set points; and λ∈ℜl stands for
“uncontrolled” system parameters such as loading levels.
Equilibrium points of (1) are usually determined by
solving the set of nonlinear equilibrium equations
F ( z , p, λ ) = 0
equations; and G(z,p,λ) stands for a general cost function,
which typically corresponds to generation costs or system
losses.
The OPF problem (3) can be used to represent a voltage
collapse condition by simply defining the objective function
to be
G ( z , p, λ ) = − λ
or
−1
{J }
G ( z , p, λ ) = σ min
B. Optimal Power Flow
The classical OPF problem can be stated as a general,
nonlinear, constrained optimization problem as follows:
G ( z , p, λ )
F ( z , p, λ ) = 0
z min ≤ z ≤ z max
(5)
where σmin stands for the minimum singular value of the
Jacobian J = DzF(z,p,λ).
The objective function (4) basically defines a maximum
loading value, which can be formally and readily shown to
correspond to either a SNB or a LIB [10].
The objective function (5) also forces the system to either
a SNB or a LIB, depending on the system limits, as the
system is singular or “almost” singular at these points. The
OPF formulation (5) is mathematically and numerically more
complicated to implement and analyze than (4), as it requires
the computation of the minimum singular value and
associated sensitivities. The use of the singular value makes
it possible to perform an “absolute” analysis of the proximity
of the system to a collapse point, without assumptions
regarding the load and generator change patterns, which may
be needed in some particular cases when using (4). It is worth
noticing that the issue of how to use the singular value as an
absolute proximity indicator to collapse is critical, because it
depends on the system and loading conditions. However, the
operators’ knowledge of the power system should allow for
the detection of critical situations from the direct analysis of
this value [11].
The actual implementation of both of these procedures in
a multi-objective optimization formulation that considers
operating costs and voltage security is discussed in more
detail in the next sections.
Observe that when using the OPF to study the voltage
stability problem, more complex system models than the
usual power flow model could be used, i.e., the equality
constraint F(z,p,λ) = 0 can represent actual steady-state
equations of the system.
(2)
where ideally F = [f g]T, z = [x y]T, p = ρ. However, in
practice, (2) stands for the power flow equations, i.e., F, z and
p are actually only a subset of the original state and algebraic
equations and variables and control parameters, respectively.
Thus, F typically represents the active and reactive power
mismatches at all buses; z stands for the phasor voltages and
angles at all buses; and p represents active power levels and
voltage set points of generators. Nevertheless, better steady
state models can be used if accuracy is an issue, particularly
when controls and their associated limits need to be
represented in more detail, as in the case of power
electronics-based controllers [9].
Multiple solutions for (2) can be obtained at given values
of the control parameters p and loading level λ. Voltage
collapse points can then be associated with solution points for
the corresponding p and λ values, say (zc,pc,λc), at which two
solution points merge, and then disappear when λ is
“slightly” changed (usually increased). This point may be
associated theoretically with a saddle-node bifurcation
(SNB), which is a point where the Jacobian Jc = DzF(zc,pc,λc)
is singular, or with a limit induced bifurcation (LIB), which
corresponds to a point where certain system limits are
reached (e.g., a generator reactive power) [2]. In practice,
whether the system collapses by a SNB or a LIB, the
Jacobian Jc tends to be either singular (SNB) or rather close
to singularity (LIB) at the collapse point.
min
s.t.
(4)
C. OPF with Voltage Security Constrains
Based on (4) and (5), the OPF can be used to both
minimize system costs and increase system security by
simply restating the optimization problem as follows:
(3)
p min ≤ p ≤ p max
1. Maximizing the Distance to Collapse: Based on the
loading level represented by the parameters λ, a voltage
security constrained OPF can be formulated using a multiobjective approach as follows [6]:
where z∈ℜ stands for the system power flow variables or
dependent variables, which are usually bus voltages and
angles; p∈ℜK represents the system parameters; λ∈ℜl stands
for “uncontrolled” system parameters; F:ℜN×ℜK×ℜl→ℜN is
a nonlinear function that typically stands for the power flow
N
2
min
(1 − ω ) C ( z o , p ) − ω λ o − λ c
s.t.
F ( z o , p, λ o ) = 0
F ( z c , p, λ c ) = 0
zo
≤ zo ≤ zo
zc
≤ zc ≤ zc
min
min
most appropriate given the system conditions. The Pareto set
can be computed based on the Weights method or the εConstraint method [12]. In the Weights method, which is the
method used in (6), the Pareto set can be found by varying
the value of the weight ω from 0 to 1 in
(6)
max
max
p min ≤ p ≤ p max
min
−1
(1 − ω ) C ( z o , p ) + ω σ min
{J o }
s.t.
F ( z o , p, λ o ) = 0
zo
where o stands for the “current” operating point and c for the
collapse point; C(zo,p) represents the operating cost function
that usually depends on some of the system control
parameters (e.g., active powers of generators) and system
variables at the current operating point (e.g., transmission
losses). The second term in the cost function guarantees that
the “distance” between the current operating point and the
collapse point is maximized, and its effect on the
optimization is controlled depending on the weighting factor
ω, which is a scalar chosen to control the relative importance
and scaling of the different terms in the objective function.
Thus, as the current operating point gets closer to the collapse
point, the value of ω should increase, so that stability takes
precedence over cost as the system loading increases. This
formulation is known as the “Weights” method and it is used
in multi-objective optimization to find the Pareto set [12].
Usually, the choice of ω is important, as it defines the relative
importance of the different components of the objective
function, especially when these components have rather
different values due to their base units.
Observe that in (6), given the nature of the optimization
problem at hand, both the current and a corresponding
collapse point must be represented, making the constraints
highly nonlinear, as the active and reactive powers are tightly
coupled at the collapse point; this can create numerical
problems during the optimization procedure. Furthermore, if
a direction of load increase is chosen, λ becomes a scalar,
reducing the search space for the optimization procedure by (l
− 1); in this case, assumptions regarding load and generation
change patterns must be made, which is usually not a
problem given that these can be chosen based on appropriate,
common, and well-known load forecasting techniques.
Finally, the parameters p are basically the same at the current
and collapse points, as a pattern of generation increase is
chosen to respond to the desired loading pattern increase, i.e.,
a distributed slack bus approach is used. The latter is a
reasonable assumption, as one is not interested in optimizing
operating costs at the collapse point but rather at the current
operating point.
min
≤ zo ≤ zo
(7)
max
p min ≤ p ≤ p max
In the ε-Constraint method, one must choose a main
objective function, while the other components of the original
multi-objective function are treated as constraints. For
example, using this technique with the cost as the main
objective function, formulation (7) can be transformed into
min
s.t.
C ( z o , p)
−1
{J o } ≤ ε
σ min
F ( z o , p, λ o ) = 0
zo
≤ zo ≤ zo
min
p min
(8)
max
≤ p ≤ p max
where ε is a threshold value chosen by the user for the
“secondary” objective function. Choosing a value for ε is
usually easier than selecting appropriate values for ω in the
Weights method, as ε defines a minimum stability margin,
which has a more “physical” meaning. A similar approach is
proposed in [6] for the OPF formulation (6), where voltage
security if accounted for in the OPF formulation by adding a
minimum distance to collapse constraint.
These OPF problems are solved using the Han-Powell
procedure, which requires a second order approximation of
both the objective functions and the constraints. In this case,
a critical aspect in the solution of (7) and (8) is the
computation of the singular value and its derivatives at every
iteration, which can be done using the Hessian of the power
flow equations as described in detail in [13]. The problem
with this approach is that only approximations of the actual
derivatives needed in the solution process are used, which
could lead to convergence problems, especially if one
considers that σmin may be highly nonlinear, approaching zero
rapidly when close to the collapse point. This is a
disadvantage of this method when compared to (6).
In (7) and (8), only the “current” loading conditions, as
defined by λo, are considered in the optimization procedure,
i.e., the optimization is carried out without regard for the
possible direction of load and generation changes, which is
not the case in (6). This is could be an advantage of this
method.
2. Maximizing the Singular Value: Of the different objective
functions proposed in [7], the OPF formulations described in
this section focuse on maximizing voltage security through
the use of the minimum singular value σmin{Jo}. The
procedures presented in [7] concentrate on the determination
of the entire Pareto set, called also non-inferior set, so that
various possible alternatives can be looked at to choose the
3
III. RESULTS
λc
The OPF formulations (6), (7), and (8) were implemented
in MATLAB and applied to a modified version of the IEEE
118-bus system [8], which has 55 generators and 9 LTCs.
Although a simple power flow model with limits in bus
voltages and active and reactive powers of generators was
used, the OPF procedures can be applied to more complex
steady state models.
The OPF formulation (6) was implemented based on the
algorithm presented in [14], which is a nonlinear primal-dual
predictor-corrector interior point method.
The OPF
formulations (7) and (8) were implemented using a HanPowell second order method [15].
The results obtained by applying (6) to the test system are
depicted in Figs. 1 and 2 for different values of λ. In this
case, the active and reactive powers in all load buses are
assumed to increase in the same proportion as the base
loading value, and generators are assumed to all pick up
power in addition to their base loading using a distributed
slack bus approach, i.e., for all loads L and generators G,
Max.Dist. (w=1)
w=0.9997
Min.Cost (w=0)
0.8
0.9
1
1.1
1.2
1.3
w=0.9999
w=0.999
1.4
1.5
1.6
1.7
1.8
λo
Fig. 1. Loading factor λc at collapse versus base loading level λo in p.u. for
the Maximum Distance to Collapse OPF formulation with different
weighting factors ω when applied to the 118-bus test system.
35000
30000
Costs
25000
PL = PLo λ
Q L = Q Lo λ
2
1.8
1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
0
Max. Distance
(ω =1)
20000
15000
10000
(9)
PG = PGo + K G ∆P
Min. Costs
(ω =0)
5000
0
where PLo and QLo correspond to the base load powers; PGo ∈
p stands for the generator powers at the current operating
point, which are being optimized at the loading level defined
by λo; KG is a value that defines the generation increase
pattern (in the simulations presented here KG was assumed to
be the same for all generators); and ∆P is a scalar variable
representing the distributed slack bus. Based on (9), the
power flow constraint F(zo,p,λo)=0 in (6), is associated with a
load level (PLo + j QLo) λo and generation settings PGo,
whereas F(zc,p,λc)=0 is associated with (PLo + j QLo) λc and
PGo + KG ∆P.
Observe in Fig. 1 that as the value of the weight ω in the
multi-objective optimization formulation increases, the
solution procedure puts more emphasis on maximizing the
distance to collapse, i.e., the values of λc increase. Two
extremes in costs are depicted in Fig. 2; here one can see that
maximizing only distance to collapse leads to significantly
larger operating costs than when minimizing costs, as
expected.
Figure 3 depicts the different costs obtained from the OPF
formulation (6) for ω=0, ω=0.999 and w=0.9999; observe
that the optimization procedure is very sensitive to the weight
factor ω (see Fig. 2), which is a problem of the Weights
method. Furthermore, as the base loading level λo increases,
the procedure puts more emphasis in minimizing costs, as the
distance | λo− λc| gets smaller, and hence has less weight in
0.8
0.9
1
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
λo
Fig. 2. Operating costs versus base loading level λo in p.u. for the Maximum
Distance to Collapse OPF formulation with different weighting factors ω
when applied to the 118-bus test system
3500
Min.Cost (w=0)
Dist.Collapse (w=0.9999)
Dist.Collapse (w=0.999)
Sing.Val. (w=0.999)
3000
Costs
2500
2000
1500
1000
500
0
0.8
0.9
1
1.1
1.2
1.3
1.4
1.5
λo
Fig. 3. Operating costs versus base loading level λo in p.u. for to the 118-bus
test system using the standard OPF (Min. Cost, ω=0), Maximum Distance to
Collapse OPF (Dist. Collapse with ω=0.999), and Minimum Singular Value
OPF (Sing. Val. with ω=0.999) formulations.
4
the objective function, which is a disadvantage of this
method; this problem can be addressed using fixed stability
margins or goal-programming techniques as proposed in [6].
This figure also shows the results of applying the OPF
formulation (7) with ω=0.999. Observe that these results are
in general agreement with those obtained using (6), even
though the “security” component of the objective function is
different; similar voltage security levels are attained in both
procedures.
Figure 4 depicts the results obtained when
applying the OPF formulation (7) with ω=1. Observe that
σmin-1increases as the loading increases, and hence the system
gets closer to collapse, as expected. At the maximum loading
condition, i.e., at the collapse point, this value may become
rather large, especially if the system collapses due to a SNB
(σmin=0); this can create numerical problems in the
optimization procedure at heavier loading conditions, which
is a disadvantage of this method. On the other hand, since
σmin-1 gets larger as the system is loaded, the multi-objective
optimization procedure “automatically” puts more emphasis
on minimizing this value as opposed to minimizing costs
when the system becomes “less” stable; this is an advantage
of this method.
Finally, Fig. 5 shows the complete results of solving the
OPF formulation (8), for different values of ε, i.e., the
complete Pareto set at different loading levels. Notice that
operating costs and security levels are conflicting goals, i.e.,
improving security, i.e., lower σmin-1 values or higher | λo− λc|
values, results in higher costs. Thus, one can observe that as
the system loading increases, similar security levels can only
be obtained at higher costs. The effect of system security on
operating costs can be obtained from the slopes of the Pareto
sets, or from the analysis of the weights in problem (6) and
(7), or the Lagrange multipliers in problem (8); all these
values indicate how the cost changes at different security
levels.
4.66
4.65
4.64
4.63
σmin-1
4.62
4.61
4.6
4.59
4.58
4.57
4.56
4.55
1
1.1
1.2
1.3
1.4
1.5
λo
Fig. 4. Inverse of the minimum singular value σmin-1 versus base loading
level λo in p.u. for the Minimum Singular Value OPF formulation with ω=1
when applied to the 118-bus test system.
5.5
5.4
5.3
σmin
-1
5.2
5.1
5
4.9
4.8
4.7
4.6
1000
λ=1.5
λ=1
2000
3000
4000
Costs
5000
6000
7000
Fig.5. Pareto sets as determined by the solution of the multi-objective OPF
problem for different values of ω and of λ when applied to the 118-bus test
system.
REFERENCES
IV. CONCLUSIONS
The paper presents a detailed description and comparison
of two different OPF techniques that consider voltage system
security represented by basic voltage collapse conditions.
Advantages and disadvantages of both methods are discussed
based on the results obtained from applying the presented
optimization procedures to a test system. From the results
and comparisons discussed in this paper, the authors are
developing improvements to the Voltage Security
Constrained OPFs presented here.
It is important to highlight the fact that the proposed OPF
techniques could be readily adapted to determine some of the
security costs associated with the operation of a power
system, which is of great interest in electricity open market
environments.
The authors are currently working on
methodologies to implement these ideas.
5
[1]
M. Huneault and F. D. Galiana, “A Survey of the Optimal Power Flow
Literature,” IEEE Trans. Power Systems, Vol. 6, No. 2, 1991, pp. 762-770.
[2]
C. A. Cañizares, editor, “Voltage Stability Assessment, Procedures and
Guides,” IEEE/PES Power System Stability Subcommittee, technical
report draft available at http://www.power.uwaterloo.ca, July 2000.
[3]
C. A. Cañizares, “Calculating Optimal System Parameters to
Maximize the Distance to Saddle-node Bifurcations,” IEEE Trans.
Circuits and Systems I: Fundamental Theory and Applications, Vol 45,
No, 3, March 1998, pp. 225-237.
[4]
C. A. Cañizares, “Applications of Optimization to Voltage Collapse
Analysis,” Panel Session, Optimization Techniques in Voltage
Collapse Analysis, IEEE/PES Summer Meeting, San Diego, CA, July
1998.
[5]
T. Van Cutsem, “A method to compute reactive power margins with
respect to voltage collapse,” IEEE Transactions on Power Systems,
Vol. 6, No. 1, February 1991, pp. 145-156.
[6]
W. Rosehart, C. Cañizares, and V. Quintana, “Costs of Voltage
Security in Electricity Markets,” Proc. IEEE/PES Summer Meeting,
Seattle, July 2000.
[7]
A. Berizzi, C. Bovo, P. Marannino, and M. Innorta, “Multi-objective
Optimization Techniques Applied to Modern Power Systems,” Proc.
IEEE/PES Winter Meeting, Columbus, January 2001.
[8]
The University of Washington Archive,
http://www.ee.washington.edu/research/pstca.
[9]
C. A. Cañizares and Z. T. Faur, “Analysis of SVC and TCSC
Controllers in Voltage Collapse,” IEEE Transactions on Power
Systems, Vol. 14, No. 1, February 1999, pp. 158-165.
Claudio A. Cañizares (M’86) received the Electrical Engineer diploma
(1984) from the Escuela Politécnica Nacional (EPN), Quito-Ecuador, where
he held different positions from 1983 to 1993. His M.Sc. (1988) and Ph.D.
(1991) degrees in Electrical Engineering are from the University of
Wisconsin-Madison. Dr. Cañizares is currently an Associate Professor at the
University of Waterloo, and his research activities concentrate on studying
computational, modeling, and stability issues in power systems with HVDC
links and FACTS controllers.
William D. Rosehart (SM’94) received his Bachelor's and Master's degrees
in Applied Science, Electrical Engineering in 1996 and 1997 respectively
from the University of Waterloo. From 1991 to 1995 through the
cooperative education program at the University of Waterloo, he worked in
the Power Industry in Canada, including GE Canada, Hammond
Manufacturing, and Waterloo North Hydro. He is currently studying for his
PhD degree at the University of Waterloo.
[10] W. Rosehart, C. Cañizares, and V.H. Quintana, “Optimal Power Flow
Incorporating Voltage Collapse Constraints,” Proc. IEEE/PES Summer
Meeting 1999, Edmonton, Alberta, July 1999.
Alberto Berizzi (M'93) received his M.Sc. degree (1990) and his Ph.D.
degree (1994) both in Electrical Engineering from the Politecnico di Milano.
Since 1992 he is being at the Electrical Engineering Department of the
Politecnico di Milano, where he is currently an Associate Professor. His
areas of research include power system and voltage stability analysis and
control.
[11] A.Berizzi, P.Bresesti, P.Marannino, G.P.Granelli, M.Montagna:
“System-area operating margin assessment and security enhancement
against voltage collapse,” IEEE Transactions on Power Systems, Vol.
11, No. 3, August 1996, pp. 1451-1462.
Cristian Bovo (SM’00) received his M.Sc. degree (1998) in Electrical
Engineering from the Politecnico di Milano. He is currently a PhD Student
at the Electrical Engineering Department of the Politecnico di Milano. His
areas of research include power system analysis and optimization, and
electricity markets.
[12] V. Chankong, Y. Y. Haimes, Multiobjective Decision Making: Theory
and Methodology, North Holland Series in Science and Engineering,
Vol. 8, New York, 1983.
[13] A. Berizzi, P. Bresesti, P. Marannino, M. Montagna, S. Corsi, and G.
Piccini, “Security Enhancement Aspects in the Reactive-voltage
Control,” Proc. IEEE Stockholm Power Tech, Vol. on Power Systems,
Stockholm, June 1995, pp. 674-679.
[14] G. L. Torres and V. H. Quintana, “Nonlinear Optimal Power Flow in
Rectangular Form via a Primal-Dual Logarithmic Barrier Interior Point
Method,” Technical Report 96-08, University of Waterloo, 1996.
[15] D. G. Luenberger, Linear and Nonlinear Programming, 2th edition,
Addison-Wesley, 1984.
6