Computer Science Technical Reports Computer Science
6-27-1997 Power System Security Margin Prediction Using Radial Basis Function Networks Guozhong Zhou Iowa State University
James D. McCalley Iowa State University, [email protected]
Vasant Honavar Iowa State University
Follow this and additional works at: http://lib.dr.iastate.edu/cs_techreports Part of the Artificial Intelligence and Robotics Commons, Information Security Commons, and the Systems Architecture Commons
Recommended Citation Zhou, Guozhong; McCalley, James D.; and Honavar, Vasant, "Power System Security Margin Prediction Using Radial Basis Function Networks" (1997). Computer Science Technical Reports. 154. http://lib.dr.iastate.edu/cs_techreports/154
This Article is brought to you for free and open access by the Computer Science at Iowa State University Digital Repository. It has been accepted for inclusion in Computer Science Technical Reports by an authorized administrator of Iowa State University Digital Repository. For more information, please contact [email protected]. Power System Security Margin Prediction Using Radial Basis Function Networks
Abstract This paper presents a method to predict the postcontingency security margin using radial basis function (RBF) networks. A genetic algorithm-based feature selection tool is developed to obtain the most predictive attributes for use in RBF networks. The proposed method is applied to a thermal overload problem for demonstration. Simulation results show that the proposed method gives satisfactory results and the running time decreases by a factor of 10 compared with using multilayer perceptrons.
Keywords Security margin, power systems, radial basis function networks, genetic algorithms, feature selection
Disciplines Artificial Intelligence and Robotics | Information Security | Systems Architecture
This article is available at Iowa State University Digital Repository: http://lib.dr.iastate.edu/cs_techreports/154
Power System Security Margin Prediction
Using Radial Basis Function Networks
TR 97-10
Guozhong Zhou, James D. McCalley and Vasant Honavar
June 27, 1997
ACM Computing Classi cation System Categories 1991:
I.2.6 [Arti cial Intel ligence] Learning | connectionism and neural nets
Keywords:
power systems, security margin
feature subset selection, genetic algorithms, radial basis function networks
Dr. McCalley's research is partially supp orted through grants from National
Science Foundation and Paci c Gas and Electric Company. Dr. Honavar's
research is partially supp orted through grants from National Science Foun-
dation and the John Deere Foundation.
This pap er will app ear in: Pro ceedings of the 29th Annual North American
Power Symp osium, Oct. 13-14. 1997, Laramie, Wyoming.
Arti cial Intelligence Research Group
Department of Computer Science
226 Atanaso Hall
Iowa Sate University
Ames, Iowa 50011-1040, USA
Power System Security Margin Prediction
Using Radial Basis Function Networks
Guozhong Zhou James D. McCalley Vasant Honavar
[email protected] [email protected] [email protected]
Department of Electrical and Computer Engineering Department of Computer Science
Iowa State University Iowa State University
Ames, Iowa 50011 Ames, Iowa 50011
Abstract
This pap er presents a metho d to predict the p ostcontingency se-
curity margin using radial basis function networks with a fast training
metho d. A genetic-based feature selection to ol is develop ed to obtain
the most predictive attributes for use in RBF networks. The prop osed
metho d is applied to a thermal overload problem for demonstration.
The simulation results show that the prop osed metho d gives satisfac-
tory results and the running time decreases by a factor of 10 compared
with using multilayer p erceptrons.
1 Intro duction
1.1 Power system security margin prediction
Security is a very imp ortant problem for power system op erations. Power
system states can be identi ed as secure or insecure. In addition, it is very
useful for op erators to know the margin, i. e., how far the state is from
the security b oundary. Knowledge of security margins is b ecoming more
and more essential as transmission lines are op erating closer and closer to
their capacities in to day's deregulated environment. The security margin
prediction problem can b e expressed as follows:
Given an operating condition characterized by a set of critical parameters
chosen with respect to a particular contingency and postcontingency perfor-
mance measure, nd the \distance" between the operating condition and the
boundary de ned by a threshold on the postcontingency performancemeasure.
Fig. 1 shows the pro cedure for security margin prediction. 1 Intelligent Information Processing
Step 1 Step 2 Step 3 Step 4
Identify AUTOMATIC SECURITY FEATURE SELECTION Security Construct ASSESSMENT SOFTWARE (GANN) Basecase KEEP Problem - structured Monte Carlo FILE - genetic algorithm sampling (Large + accuracy - industrial grade analysis Database) + cardinality software +controllability - modularity of analysis - simple neural network
Step 6 Step 5
TRAIN NEURAL NETWORK VISUALIZATION & PRESENTATION Encoded Input/ x1 - margin prediction Out put Relation R x2
x3
Figure 1: Pro cedure for security margin prediction
There are various security problems in p ower systems. We consider only
thermal overload problem here. In this case, the p ostcontingency p erfor-
mance is the ow on the line at risk of overload and the threshold is the
emergency rating of this line. Let x denote the critical parameter candi-
s
dates, vector x denote the selected critical parameters, and R denote the
s
that p erformance measure, our goal is to obtain a relationship R = f x
predicts the contingency e ects using knowledge of only precontingency con-
ditions. Neural networks are used for this purp ose b ecause of their capability
to handle nonlinear functions of many variables.
A sample system is shown in Fig. 2. The load in the area, during high
loading conditions, is greater than the generation capacities in this area.
Therefore, a signi cant amount of power must be imp orted into the area to
meet the demand. There are several ties b etween this area and the remaining
part of the system. Thermal overload o ccurs on tie line 5 when tie line 3 is
outaged. So the p ostcontingency p erformance measure is the ow on tie line
5.
The emergency rating of tie line 5 is I = 600A. Therefore, the threshold
0
value for this constraintisI = 600A, and when the p erformance measure is
0
normalized according to
I I
0
R =
I
0
wehave R = 0 on the b oundary,i. e., the margin is zero. Thus, the value of
0
R represents the security margin. The more negative the margin, the more
secure the system. 2 Intertie 1
Area Tie 1 Gen B
Tie 2 Subarea
Tie 3 Gen A Gen C
Tie 4
Load
Tie 5
Intertie 2
Figure 2: A sample system
1.2 Feature Selection for Security Margin Prediction
Security assessment studies have traditionally dep ended on engineering judg-
ment to select the critical parameters, i.e., to select the parameters to be
used in predicting the margin for each security problem. Our goal here is to
develop an automatic approach to critical parameter selection, i.e., feature
selection. We do not intend that automatic feature selection replace engi-
neering judgment, but rather enrichitby con rming and extending physical
understanding.
Feature selection is a searchover the space of all combinations of features
where a function is used to evaluate what is b est. So two comp onents of
feature selection are evaluation function and search approach.
For the purp ose of feature selection in power systems, op erating param-
eters can b e classi ed as independent or dependent. A parameter is indep en-
dent if it is included in the input data to a power ow program; examples
include MW injection or voltage magnitude at a generator bus or load level
MW or MVAR injection at a load bus. A parameter is dep endent if it is
computed as a result of a power ow program solution; examples include
bus voltages at a load bus or line ows. Indep endent critical parameters can
be further divided according to op erator controllability. MW injections and 3
voltage magnitudes at generator buses are controllable indep endent param-
eters; load levels at load buses are noncontrollable indep endent parameters.
The problem for feature selection in power system security assessment
can b e describ ed as follows:
Given a database having columns consisting of a number of features and
a single performancemeasure, determine a subset of the total attribute space
that can be used to train a neural network that predicts the postcontingency
performance measure such that the fol lowing criteria are satis ed:
Set Suciency accuracy: The feature set must contain sucient in-
formation to allow prediction of the p ostcontingency p erformance mea-
sure within a desired accuracy for all op erating conditions within the
study scop e.
Set Cardinality: The feature set should b e chosen as the set of minimum
size that satis es the set suciency criterion.
Control lability Constraint: At least one feature within the set must
be controllable by the op erator so that the op erating p oint can be
adjusted, with resp ect to the b oundary, using preventive actions.
Researchers in the statistics and pattern recognition communities have
investigated feature subset selection problems for decades [10]-[12]. Many of
these metho ds are based only on the ability to predict the output. There
are no ecient way to account for cardinality constraints on the feature set
although there are several techniques that have attempted to nd minimum
feature subsets see [16] for a discussion. Furthermore, there is no capability
to preselect some parameters into the solution. Some of the mo dels select
the feature sets that satisfy the suciency condition, but no mechanism has
b een prop osed to ensure that the chosen subset will satisfy the cardinality
and the controllability conditions. To satisfy all three parts of the stated
criteria, we have develop ed a genetic algorithm approach that searches only
p ortions of the solution space of a sp eci ed cardinality level containing certain
attributes. This lo calized search technique ensures that the cardinality and
controllability criteria are satis ed. We used MLPs in our early work, but
it to ok to o long to obtain the results. Therefore, we turn to radial basis
function networks to sp eed up the solution. 4
2 Radial basis function networks
Neural networks can be classi ed into sup ervised and unsup ervised net-
works according to their learning strategies. The most widely used sup er-
vised neural network is the multilayer p erceptron MLP trained with back-
propagation algorithm. The design of this kind of neural network may be
thought of as an application of an optimization metho d known in statistics
as sto chastic approximation. However, the design of a sup ervised neural net-
work may be pursued in di erent ways. One imp ortant approach is to view
this design problem as a curve tting problem in a high-dimensional space.
From this p ersp ective, learning is equivalent to nding a surface in a multi-
dimensional space that b est ts the training data in the sense of statistics.
Generalization is then equivalent to using this multidimensional surface to
interp olate the test data. Such a viewp ointis indeed the motivation b ehind
the metho d of radial basis functions RBFs. Bro omhead and Lowe [2] rst
explored the use of RBFs in neural networks. Moody and Darken [3], Re-
nals and Rohwer [4], and Poggio and Girosi [5] among others made ma jor
contributions to the theory, design, and application of RBF networks.
ϕ(.)
x1
x2 Σ y
xn
Figure 3: A radial basis function network
The basic architecture of a RBF network is shown in Fig. 3. It includes
three entirely di erentlayers. The rst layer is an input layer of which each
no de corresp onds to an attribute of an input pattern. The second layer is
a hidden layer that serves a di erent purp ose from that of the output layer.
The third layer is an output layer resp onding to the input patterns. The 5
transformation from the input layer to the hidden layer is nonlinear, whereas
the transformation from the hidden layer to the output layer is linear. An
activation function for a hidden layer no de is a lo cally radially symmetric
function typically Gaussian function, whose output decays to zero as the
distance usually Euclidean distance b etween the input vector and its center
increases, see Fig. 4. There are two imp ortant parameters, center and width,
asso ciated with an activation function. A center is a vector with the same
dimension as the input pattern, which represents a cluster center of the input
space. A width is used to control the spread of the RBF so that its output
decays more slowly or more rapidly as the distance b etween the input vector
and the center increases. Each RBF ' resp onds to a small convex region of
the feature space. A large numb er of these functions cover the entire feature
space so that the output layer neurons can join them with di erent weights
to accomplish the function approximation or classi cation. Fig. 5 shows a
p ortion of two-dimensional feature space covered by RBFs.
Output
0
Distance to center
Figure 4: A radial basis function in one-dimensional space
Without loss of generality, we consider a RBF network with only one
output. The output of such a network can be expressed as
M
X
w ' kx c k 1 f x=
i i i
i=1
where x is an input pattern, c is the center for hidden no de i, w is the weight
i i
between hidden no de i and the output no de, and w is a bias weight. M is the
0
numb er of hidden no des, ' is the activation function for the hidden layer.
The norm kk can b e Euclidian or Mahalanobis distance when the densities 6 x ϕ 2 ϕ 2 1
ϕ Μ
x1
Figure 5: A p ortion of two-dimensional feature space covered by RBFs
have a common covariance matrix. We consider RBF approximation using
Gaussian functions.
When ' is a Gaussian function, 1 can be seen as approximating
a probability density by a mixture of Gaussian functions. The Gaussian
function can b e expressed as
u