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Abstract— In smart gr ids, the power supply and demand are
balanced through the electricity market to promote the
maximization of social welfare. An important procedure in
electricity market clearing is to sequentially solve the
security-constrained economic dispatch (SCED) problem. However,
the scale of the SCED problem with al l N-1 constraints is huge.
Directly optimizing such a problem is inefficient and not robust.
With the development of smart grids, the frequency of market
clearing is increasing, which presents new requirements for fast
calculation of SCED. To solve this problem, we propose an
intelligent pre-screening method to identify the active constraints
of SCED based on deep learning. We utilize stacked denoising auto
encoders (SDAE) to extract the nonlinear relationship between the
system operating condition and the active constraint set of SCED.
Especially, the input/output feature vectors and learning strategy
are designed to improve the training efficiency and guarantee the
learning accuracy of the deep neural network (DNN). Besides, a
fast tuning strategy of neural network parameters based on
transfer learning is proposed to handle new scenarios like
topology change. The computational efficiency of the SCED
problem is significantly improved while the accuracy is not
influenced. The IEEE 30-bus, IEEE 118-bus, and practical utility
661-bus systems are used to demonstrate the effectiveness of the
proposed method.
Index Terms—Security-constrained economic dispatch (SCED),
constraint screening, active constraint set, deep neural network.
NOMENCLATURE
A. Variables
H1, H2
Operationa l costs
J
The minimized objective function for
unsupervised pre-training stage
L
The minimized objective function for
supervised tuning stage
m
The number of training sam ples
n
The number of encoding functions in SDAE
Y. Yang, Z. Yang, J. Yu and K. Xie are with State Key Laboratory of Power
Transmission Equipment & System Security and New Technology, College of
Electrical Engineering, Chongqing University, Chongqing 400044, China. L.
Jin is with State Grid Chongqing Electric Power Company, Chongqing 400014,
China. Corresponding author is Z. Yang, yzf1992@cqu.edu.cn.
This work is supported by National Natural Science Foundation of China
(No. 51807014), State Grid Corporation of China (The key technology and
application of active and reactive power scheduling based on nonlinear
projection), and Fundamental Research Funds for the Central Universities
(No. 2019CDXYDQ0010).
PD
Load demand
,,
line
i j c
P
Power flow on branch (i, j) in the cth
contingency
G
P
,
G
P
Upper and lower limits of generator output
PG
Generator output
R
Activation function
,,i j c
S
Power transfer distribution factor ma trix in the
cth contingency
vmean, vstd
Mea n a nd standard deviation of vector V
Wl, bl
Encoding parameters in the lth DAE
l
W
,
l
b
Decoding parameters in the lth DAE
X
Input feature vector of SDAE
Yout
Output feature vector of SDAE
Yl-1, Yl, Zl
Input vector, middle layer vector and output
vector of the lth DAE
θ={W, b}
Encoding parameters in SDAE
θ´={W´, b´}
Decoding parameters in SDAE
B. Abbreviations
BP
Back propa gation
CNN
Convolutional neural network
DBN
Deep belief network
DNN
Deep neural network
DAE
Denoising auto-encoder
ELM
Extreme learning machine
HELM
Hierarchical ELM
ISO
Independent system operator
LP
Linear programming
ReLU
Rectified linear unit
RNN
Recurrent neural network
SCED
Security-constrained economic dispatch
SDAE
Stacked denoising auto encoder
I. INTRODUCTION
Sma rt grid is evolving to provide a reliable, sustainable and
economic energy supply. The ele ctricity market is one
important component in sma rt grids, which aims to guide users
to reasonably use the electricity [1]-[3]. Induced by the pricing
signals, it is expected that the peak load can be reduced and the
utilization efficiency of resources can be improved [3]. An
important procedure of the electricity ma rket clearing is to
sequentially solve the single-interva l security-constrained
economic dispatch (SCED) model. However, the transmission
constraints in N-1 contingencies grea tly increase the scale of
Fast Economic Dispatch in Smart Grids Using
Deep Learning: An Active Constraint Screening
Approach
Yan Yang, Student Member, IEEE, Zhifang Yang, Member, IEEE, Juan Yu, Senior Member, IEEE,
Kaigui Xie, Senior Member, IEEE, Liming Jin
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the SCED model. It brin gs scalability cha llenges, even for
linear programming (LP) so lvers. Directly optimizing such a
problem is not efficient or robust. With the development of
smart grids, the number of market participants and the
frequency of market clearing are rapidly increasing, which
results in more complex clearing models and shorter market
clearing windows. Therefore, there ha s been a recent push to
further accelerate the calculation of SCED in practica l
operations.
Although the number of constraints is large in the SCE D
model, the percentage of active constraints in the fina l solution
is relatively small. Hence, it is a common practice for system
operators in the U.S. and China to first solve the unconstrained
SCED (i.e. economic dispatch problem) and then iteratively
add active constraints. This method could achieve the optimal
solution and reduce the computational pressure. However, the
computational time rises rapidly if the number of iterations is
large. For some independent system operators (ISOs), to
accelerate the computational speed, the set of transmission
constraints that is most likely to bind will be added to the SCED
model in the first iteration. Currently, such constraint set is
determined in an ad-hoc ma nner. If there is a way to completely
identify the active constraints before optimization and add
these constraints into the SCED model, the iterations will not be
needed a nd the computational speed of the SCED calculation
can be significantly accelerated. This improvement can provide
a promising way to rea lize real-time electricity trading. Besides,
it also allows the system operators to formulate more detailed
operational constraints or directly solve the multi-period SCED,
which is currently difficult because of the limited market
clearing window and large computational burden.
There have been some methods proposed to pre-screen the
contingency constraints. Many methods are derived based on
the analytical properties of the dispatch model considering N-1
contingencies. For instance, feasible cuts are derived based on
Benders’ decomposition in [4]. References [5]-[7] use
bounding techniques to obtain a set of potentially binding
constraints. A linea r programming model is proposed in [8]-[11]
to construct representative constra ints or identify inactive
constraints. Reference [12] uses line outage distribution factors
to filter the a ctive N-1 congestion constraints. The mentioned
approaches pre-identify the active constra ints based on certain
assumptions or solving relatively some small-scale
optimization problems. Essentially, the difficulty of a ctive
constraint identification is quite similar to that of obtaining the
globa l optimal solution of the SCED problem. Therefore, it is
nea rly impossible to identify a ll the active constraints by
analytical methods before optimizing the SCED.
Some studies use statistical methods to identify active
constraints. Reference [13] uses the vio lation ranking to select
the candidate constra ints. Reference [14] counts the number of
times when ea ch constraint is binding, and selects the most
frequent ones. References [15] and [16] use sta tistical lea rning
and neural nets respectively to identify the active constra ints.
The output feature vector of references [15] and [16] is the
binding status of the constraint set. However, the dimension of
the feature vector quickly rises when considering the enormous
constraints in the SCED problem .
With the fa st development of information technology, deep
learning techniques provide a promising way to effectively
capture the complex nonlinear relationship between the active
constraint set of the SCED a nd system opera ting condition. It
has been demonstrated that deep models can extra ct more
complex features than shallow models like back propagation
(BP) and ra dial basis function networks [17]-[19]. Deep models
generally ca n be generally divided into convolutional neural
networks (CNNs), recurrent neural networks (RNNs) a nd
fully-connected networks. CNNs are the go-to methods for
different types of prediction problem involving image da ta as
input. RNNs a re des igned to solve sequence prediction
problems. Fully-connected networks are suita ble for regression
and classification problems. Moreover, fully-connected
networks have been proved to have the ability to a pproximate
any function with high a ccuracy in theory [20], [21].
Taking advantage of the numerous historical operation data,
this paper utilizes a fully-connected neural network to solve the
pre-screening problem of contingency constraints. As a
representative of fully-connected neural networks, stacked
denoising auto encoders (SDAE) is utilized. Compared with
other types of fully-connected neural networks, such a s deep
belief network (DBN) [22], [23] and multi-layer extreme
learning machine (ELM) [24], [25], SDAE can more
effectively extract high-dimensiona l complex nonlinear
features by its deep stacked structure and encoding/decoding
process [26], [27]. However, the application of the deep
learning methods in power system domain still needs further
investigations. For example, the feature vector and learning
strategy need to be carefully selected to effectively extract the
features of power system knowledge. Besides, solutions need to
be proposed to adapt the evolving features in power systems
caused by topology change.
Regarding the problem mentioned above, an intelligent
pre-identification method is proposed to identify the active
constraints of the SCED problem in this paper. The
contributions are as follows:
1. A deep-learning structure is proposed to reduce the sca le
and accelerate the computational speed of the SCED problem.
Based on a DNN, a data-driven method is proposed to identify
the active constra ints of the SCED problem. The proposed
method can greatly im prove the efficiency of SCED calculation
without compromising its accuracy.
2. Identification model for the SCED problem and
corresponding learning strategy a re designed to improve
training efficiency. The identification model is developed by
the feature vector construction and DNN selection. Besides, the
training st rategy considering data pre-processing, activation
function, and lea rning algorithm is designed to effectively
extract the features of SCED problem.
3. A fast tuning strategy of DNN parameters based on
transfer learning is proposed to consider the topology change.
Because the well-trained DNN has a lready extract ed key
features of the SCED problem, transfer learning is used to
utilize the prior learned knowledge and improve training
efficiency.
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This paper is o rganized a s follows. Fast calculation
framework based on deep neura l network for the SCED
problem is designed in Section II. Identification method of
active constraints is proposed for the SCED problem in Section
III. Fast ca lculation of SCED based on deep lea rning is
presented in Section IV. The numerical test results a re shown in
Section V, followed by the conclusions in Section VI.
II. FAST CALCULATION FRAMEWORK BASED ON DEEP NEURAL
NETWORK FOR THE SCED PROBLEM
A. SCED Model Formulation
In pra ctical opera tion, the preventive control strategy is used
in the SCE D model to guarantee the security. The objective
function of the SCED model is shown as follows:
12
min +
G
TT
G G GPP H P H P
(1)
where PG represents the generator output; H1 and H2 represent
the operational costs. According to the current industrial
practice, a DC power flow model is used to model the power
flow [28], [29]. The constra ints in the SCE D model are shown
as follows:
System load balance
G G D D
=e P e P
(2)
where eG a nd eD are the “all-one” vector, and PD is the load
demand.
Generator output limits
GGG
PPP
(3)
where
G
P
and
G
P
are the upper and lower limits of generator
outputs, respectively.
Transmission line capacity constraints
( )
, , , , , , , , ,
line line line
i j c i j c i j c i j K c C PPP
(4)
where
( )
, , , ,
=
line
i j c i j c
GD
−P S P P
(5)
In this model,
,,
line
i j c
P
is the power flow on the branch (i, j) in
condition c (c=0 is the normal condition); Si ,j,c is to the power
transfer distribution factor matrix [30], C is the anticipated
contingency set; and K is the set of branches considered in the
dispatch model. In (4), the branch flow constraints in N-1
contingencies are considered. In order to gua rantee the
operational security and relia bility, the power system
scheduling requires to consider the N-1 contingency security
criterion. This criterion guarantees that the operational
constraints are not violated under any single branch outage. It
can be observed from the Eq. (4) tha t there will be N×N×2
constraints if power grids have N branches. The optimization
model (1)-(5) is a quadratic program ming problem. There are
many mature algorithms to solve the problem. However, the
enormous scale of the SCED problem adds great computational
burden for practically-sized systems.
The para meter of this model is the load PD (renewable energy
sources are regarded as load). Different generator scheduling
results can be obtained by setting different PD. Hence, the
system operating condition is reflected by PD in this paper.
B. Intelligent Fast Framework of the SCED Calculation
The ca lculation fram ework of the proposed method is
illustrated in Fig. 1. The proposed method uses the DNN to
identify the active constra ints (i.e., binding transmission
constraints in normal a nd contingency conditions) before
optimization. After the optimization, the N-1 analysis is
implemented to check whether the scheduling result meets the
N-1 security criterion. If there are no new active constraints in
the N-1 analysis, the optimal solution of the SCED model can
be obtained; otherwise, the new identified a ctive constra ints
will be added to the SCED model. The purpose of the proposed
method is to identify the entire active constraints before
optimization in most ca ses so that the iteration (the ora nge
cha in in Fig. 1) can be avoided. The proposed method leaves
the certain computational burden of the SCED calculation to
offline training of DNN tha t identifies active constraints.
Besides, the proposed framework would not deteriorate the
accuracy even though the DNN is a black box. The optimality
of the SCED solution is guaranteed by the N-1 analysis after
optimization.
The algorithm flow in Fig.1 is also used in current power
industries for solving the SCED problem, except that the
intelligent identification of active constra ints (the blue cha in in
Fig. 1) is not included [31], [32]. In our case studies, we found
that the SCED calculation normally takes 3~6 iterations to
converge. To make the SCED converge with one iteration, the
key is how accurate the DNN can identify all the active
constraints. Severa l techniques have been proposed in this
paper from the following aspects: 1) the model construction for
active constraints identification, 2) the effective lea rning
strategy, and 3) a fast tuning strategy of DNN for the situation
tha t the DNN is fa cing new scenarios like topology cha nge.
DNN
New active constraints?
System operating condition
No
The optimal solution
Yes
N-1 analysis
k=1
k=k+1
Active constraint sets
A(k)=A(k-1)∪Anew
SCED model
Active constraint
sets A(1)
s.t.
No
12
min G
TT
G G G
+
PP H P H P
G G D D
=e P e P
( )
( )
, , , , , , , , ,
i j c i j c i j c
line line line k
i j c A PPP
GG
P
Fig. 1 Calculation framework of the proposed method
III. IDENTIFICATION METHOD OF ACTIVE CONSTRAINTS
A. Proposed Identification Model of SCED Problem
To effectively extract the features of SCED problem, input
and output feature vectors are constructed. The DNN structure
for the a ctive constra int identification is designed based on
SDAE.
i)Feature vector construction
The set of a ctive constra ints A of the model (1)-(5) can be
regarded a s a nonlinear function of the system operating
condition PD. Fortunately, DNN has the a bility to approximate
any function with an arbitrary degree of accura cy according to
the universal approximation theorem [20], [21]. Essentially,
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DNN utilizes the sensitivity of output to the input to mine the
non-linear relationship between them. Therefore, the system
operating condition PD in Eq. 2 is chosen as the input feature
vector X.
For the output vector, the binding status of constraints at ea ch
contingency can be directly constructed as the output feature
vector Y. However, the dimension of the output vector equals
the number of contingencies times the number of branches. The
size of the DNN increa ses exponentia lly with the scale of the
test system. Moreover, the number of active constraints is much
less than that of inactive constraints in the matrix. The
unbalance will deteriorate the identification accura cy of active
constraints. Hence, we propose to use the decision variable PG
in the SCED model a s the output feature vector, and the
dimension of which is similar to the scale of the test system.
The proposed paradigm for active constraint identification is
illustrated in Fig. 2. The designed DNN is expected to map the
system operating condition PD to the decision variable PG in the
SCED model. Then the set of a ctive constraints will be
calculated via N-1 analysis.
PGN-1 analysis
0 0 0
1 0 0
0 0 1
nl
...
...
...
...
...
...
1
2
0 1 N
...
...
line
Contingency
...
Output feature
vector: generation
power (PG)
Input feature vector:
system operating
condition (PD)
PD1
PD2
PDi
DNN
N-1 analysis
Fig. 2 Proposed perspective for active constraint identification
ii)Selection of DNN for the SCED Problem
As illustrated in Fig.2, the re lationship between the input
feature vector and the output feature vector is inexplic it and
highly nonlinea r. The neural networks can be categorized to
shallow neural networks a nd deep neural networks. Compared
with shallow neural networks (BP neural networks , ELMs, etc.),
DNNs are a ble to extract complex features more effectively
than shallow neural networks [18]. Hence, DNNs are a dopted
in this paper.
Some of existing DNNs are designed for classification
problems. However, the SCED problem discussed in our study
is a regression problem. For the SCED problem, the set of
active constraints va ries with the system operating condition.
According to the multi-parameter programming theory, the
relationship between the input feature vector PD and the output
feature vector PG is a piecewise function (a set of active
constraints corresponds to a function segment) [2], [33]. DNN
with a good generalization ability is required to represent the
complex relationship. SDAE uses an unsupervised pre-training
approach, which effectively improve the generalization ability
compared with directly optimizing the labeled objective
function [26]. Therefore, this paper employs SDAE to extract
the complex nonlinear features of the SCED problem. In our
case studies, SDAE is compared with other neural networks
and its superiority is verified.
The structure of SDAE is illustrated in Fig. 3, SDAE has a
stacked structure with m ultiple denoising auto-encoders
(DAEs). For the lth DAE, the input layer is Yl-1 , the middle layer
is Yl, and the output la yer is Zl. Yl is determined by Yl-1 using the
encoding function f in (6). Output layer Zl is calculated by the
decoding function g shown in (7).
( ) ( )
11
l
l l l l l−−
= = +
θ
Y f Y R W Y b
(6)
( ) ( )
l
l l l l l
= = +
θ
Z g Y R W Y b
(7)
In (6) a nd (7), θ={W, b} and θ´={W´, b´} are the encoding
parameters and decoding pa rameters in the SDAE, respectively,
where W is the weight between the layers and b is the bias; R is
an activation function, which will be introduced in the next
section.
In the designed structure of DNN, the input vector X is PD,
which represents the system operating condition; the output
feature vector is genera tor output PG. According to the structure
of the SDAE, the relationship between PD and PG can be
described by formulation (8):
( )
( )
1n
GD
=θ θ
P f f P
(8)
In (8), n is the number of encoding functions in SDAE. Note
that the output layer Z of DAE is needed only in the pre-training
process. In consequence, the identification model of active
constraints for SCED problem is constructed based on the
designed feature vectors and the structure of SDAE.
2
θ
f
-1l
θ
f
l
θ
f
-1n
θ
f
PG1PG2PGi
n
Y
l
Y
1n−
Y
1D
P
Di
P
2D
P
1
θ
g
l
θ
g
1n−
θ
g
1n−
Z
DAEn-1
1
Z
l
Z
DAE1
DAEl
1
Y
( )
0
XY
1
θ
f
n
θ
f
Fig. 3 Designed structure of DNN.
B. Learning Strategy for the Identification Model
The learning ta rget is to obtain the optima l encoding
parameters θ to ca pture the nonlinear features. The learning
strategy of SDAE consists of four parts: training framework,
data pre-processing, a ctive function, and lea rning algorithm.
SDAE uses a two-stage training framework, including the
unsupervised pre-training and supervised tuning. In
unsupervised pre-training stage, the training target is to find a
set of θ and θ´ that minimizes the following J for each DAE:
( )
( )
2
11
1000 ll
ll
m
−−
=−J Y g f Y
(9)
where m is the number of training samples.
The main target of the unsupervised sta ge is to initialize θ.
Unsupervised pre-training sta ge has shown the ability to enable
the DNN to reach better loca l minimum and a chieve better
generalization than traditional ra ndom initialization methods
[34], [35]. Note tha t the output layer Z of DAE (marked by grey
in Fig. 3) is not included when calculating the output of SDAE.
In the supervised stage, the in itialized encoding parameters θ
are tuned to m inimize the following L:
( )
( )
2
1
1000 n
GD
m
=−L P f f P
. (10)
Data pre-processing e liminates the adverse influence of
singular samples on the training process and ma ke different
samples comparable. The common data processing methods
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5
include min-ma x normaliza tion method a nd z-score method. In
this paper, we use z-score method shown in (11) to normalize
the samples because it can effectively handle outliers:
0
=0
mean std
std
mean std
vif v
v
v if v
−
=
−
V
V
V
(11)
where vmean and vstd are the mean and the standard deviation of
vector V, respectively. Also, only the mean a nd standard
variance of historical statistics are required, which is suitable
for the numerical characteristics of the SCED model.
Activation function has a significant influence on the
training process. Although the sigmoid function R(x) = 1/(1+e-x)
is used in SDAE, saturation phenomenon of sigmoid-like
function may cause the parameters update come to a standstill.
Therefore, Rectified Linear Unit (ReLU) [36] is selected a s the
activation function:
0
() 00
x if x
Rx if x
=
(12)
Due to its piecewise linear formulation, ReLU avoids the
gradient vanishing effect, which is suita ble for the deep
learning problem in this paper. The norma lized output can be
less than zero, but the ReLU cannot reach a value les s than zero.
Therefore, the a ctivation function of the last la yer is designed to
be a linea r function in this paper.
( )
n
f x x=
(13)
We a dopt root mean square propagation (RMSProp) as the
learning a lgorithm, which is a common approach in existing
methods [37]. Parameters are updated by the RMSProp
algorithm a s follows:
( )
11
11
=1
tt
o
o
t t t t
oo
−−
−−
= − + −
+，
θ
θ
Or r O O
r
(14)
where
1t
o−
θO
is the pa rtial derivative of the objective function
O to the variables θo in the tth updating; ⊙is a Hadama rd
product. In this pa per, we set
=0.99, η=0.001,
=1×10-8.
Based on the proposed learning strategy above, the detailed
training process is illustrated in Fig. 4.
Trained SDAE for active constraint
identification
l=n-1?
Data preprocessing using (11), l=1 Supervised tuning using (10)-(12),
where O=L, θo=θ
l=l+1
Unsupervised pre-training of the lth
DAE using (9), (11), (12), where
O=J, θo={θ, θ'}
Meet the stop criteria?
YesNo
Yes
No
Fig. 4 Flowchart of the training process for the SCED problem
C. Fast Tuning Strategy based on Transfer Learning
The above lea rning strategy assumes that the training and test
data are in the same feature space and follow the same
distribution. However, the topology change is an important
issue in practical systems, which may generate a different spa ce
and influence the a ccuracy performance of the trained DNN.
Hence, a DNN needs to be rebuilt using the newly collected
training data when the topology changes. In practice, it is
time-consuming to recollect the needed training data and
rebuild the DNN model from scratch. Transfer lea rning
techniques try to transfer the “knowledge” from some previous
tasks to a target ta sk when the latter ha s fewer high-qua lity
training data [38]. As one of the transfer learning techniques,
parameter transfer technique improves the learning efficiency
by providing a better initialization (than a random initialization)
of DNN parameters, which has been successfully applied to
many tasks [39], [40]. Therefore, considering that the
well-trained DNN has a lready extracted useful complex
features of SCED, a tuning strategy for the DNN p a rameters
based on parameter transfer technique is proposed to reduce the
effort for recollecting the training data and rebuilding the DNN.
We use the parameter of the well-trained DNN a s the initial
parameters of the new DNN for topology changes, and then
updating the new DNN by the learning algorithm RMSProp.
The related pseudo code can be seen as follo ws. The approach
is simple yet effective for making use of the information of the
trained DNN for the SCED problem.
Algorithm 1
1: Preprocess the raw data as training samples, validation and test data by (11).
2: Initialize the new DNN using the parameters of the well-trained DNN.
3: do
4: update the new DNN by RMSProp using input data X and output data Yn
=-
+
L
r
;
( )
1
= + − r r L L
;
( )
2
1
1
=−
=
L f f
nn
n
lYX
where Y0=X,
=0.99, η=0.001,
=1×10-8.
5: end for
The iterative number of epochs reach to the thres hold or the DNN meets the
condition of early s topping method[41].
IV. FAST CALCULATION OF SCED BASED ON DEEP LEARNING
In this section, the proposed fast calculation of SCED based
on deep learning is summa rized. The flowchart of the algorithm
is illustrated in Fig. 5. The procedure is described in detail as
follows：
Step 1): Data acquisition. Sa mples can be obtained by the
following two ways: i) opera tional data from practical system s;
ii) simulation data (using actual topology/generator information
and sim ulated loads). The former way can reflect the rea l
operating state of the system, but the number of historical
operation data may not meet the requirement of DNN training.
The latter way can effectively simulate various system
operating conditions, which can be regarded a s a
supplementation to the practical data. This paper combines two
methods to construct samples.
Step 2): Training of the intelligent pre-identifica tion model.
Based on the proposed lea rning strategy in Section IV, a ll the
training data should be normalized by (11). The training
process contains unsupervised pre-tra ining a nd supervised
tuning. The DNN needs to be trained only once offline and then
it ca n handle many new operating conditions [42].
For unsupervised pre-training stage, construct the objective
function (9) for the first DAE a ccording to (6), (7), (12) and
input data X. Substitute the objective function (9) into (14) to
update param eters. Then, the encoding a nd decoding
parameters of the first DAE can be obtained. Afterward, the
output of the middle layer of DAE obta ined from (6) is re garded
as the input of the next DAE. Apply the same methods to
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construct objective function and update parameters for each
DAE from the first to the last.
For the supervised tuning stage, use the encoding para meters
obtained from unsupervised pre-training stage to initialize
SDAE. According to (6), (12), input da ta X, and output data Y,
construct the objective function (10) for SDAE. Substitute the
objective function (10) into (14) to update parameters. All the
optimal parameters of encoders θ={W, b} in SDAE can be
determined. It is worth mentioning tha t Step 1) and Step 2) are
only needed to execute once offline.
Step 3): Identifying the active constraints by DNN. Feed the
system operating condition into the well-trained DNN from
Step 2). Determine the normalized optimal generation power
output by (8). Denormalize the optima l generation power to
obtain PG according to (11). Afterward, ca lculate the set of
active constra ints via N-1 ana lysis by (4), (5).
Step 4): Fast ca lculation of SCED problem. Apply the
system opera tion condition and identified active constraints
from Step 3) to construct the SCED model (1)-(5). Use LP
solver to obtain the new optimal genera tor output and judge if
there are active constraints. If not, the optimal solution of the
SCED model can be obtained; otherwise, the ne w a ctive
constraints will be added to update the SCED model (1)-(5) and
iterates until no new a ctive constraints are identified.
V.NUMERICAL TEST RESULTS
To verify the effectiveness of the proposed m ethod,
simulations are implemented in the IEEE 30-bus, IEEE 118-bus,
and practical 661-bus utility system .
A. Test Information and Methods for Comparison
The system data of the IEEE 30-bus and IEEE 118-bus
system s are given in [43]. The power demand is sampled
randomly by normal distribution to generate samples with a
standard devia tion (10% of the expected value). The load
dema nd in [43] is chosen as the expected value. Following
methods are compa red:
M0: SDAE with the proposed learning algorithm and the
output fea ture vector is the set of a ctive constraints.
M1: SDAE with lea rning strategy in [26], and the network
structure is the sam e as the proposed method.
M2: SDAE with the proposed lea rning a lgorithm, but the
activation function is only ReLU.
M3: BP neural network, there are 900 neurons in the hidden
layer. The activation function is ReLU.
M4: DBN in [23], there are 3 hidden layers, and each hidden
layer has 300 neurons.
M5: Hierarchical ELM (HELM) with unsupervised tra ining
in [25]. There are 900 hidden neurons in the la st ELM.
M6: SCED model with all possible contingencies
considered.
M7: Practical iterative approach similar to the algorithm
flow in Fig. 1 (without active constraint identification).
M8: Active constraints identification method [14].
M9: Proposed method. There are 3 DAEs in the DNN, ea ch
DAE has 300 neurons in the middle layer, respectively.
M10: Tra nsfer learning with the proposed method to
consider topology cha nge. The middle layers are the sa me as
M9.
The intentions for these comparison methods are listed in
Table I.
Step 1):Data acquisition
Step 2): Training intelligent pre-identification model
Supervised training
and fine-tuning of
neural network
Unsupervised pre-
training of neural
network layer by layer
Z-score data
preprocessing
ReLU and linear
function
RMSProp
learning algorithm
Step 3): Calculate the active constraints by DNN
Step 4): Fast calculation of SCED problem
PGActive constraintes
N-1 analysis
System operating
condition
New active constraints?
System operating condition and
Pre-identified active constraints
No
The optimal solution
Yes
N-1 analysis
k=k+1
Active constraint sets
A(k)=A(k-1)∪Anew
SCED model
s.t.
No
12
min G
TT
G G G
+
PP H P H P
G G D D
=e P e P
( )
( )
, , , , , , , , ,
i j c i j c i j c
line line line k
i j c A PPP
GG
P
Operation data Simulation data
Offline
Online
Fig. 5 Flowchart to determine the optimal solution based on SDAE.
TABLE I COMPARISON ME THODS AND THE INTENTIONS
Methods
Corres ponding intention
M0 and M9
Verify the effectiveness of the designed feature vector.
M1, M2 and M9
Verify the effectiveness of the proposed learning
strategy.
M3, M4, M5 and
M9
Verify the effectiveness of the chosen deep neural
network, SDAE.
M6, M7, M8, and
M9
Verify the effectiveness of the proposed active
constraint identification method.
M7 and M10
Verify the effectiveness of the proposed fast tuning
algorithm for topology change.
The training process stops if the designed neural network
meets the condition of early stop ping method [41] or the
number of iterative epochs reaches the threshold. The ea rly
stopping method is utilized to alleviate the over-fitting issue.
The maximum number of epochs in unsupervised and
supervised stages are 300 a nd 500, respectively. The number of
batches is 300. The number of training samples, validation data,
and test sa mples are 30000, 1000 and 2000, respectively. The
training samples are used for both pre-training and fine-tuning
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stage. All simulations are performed on a PC equipped with
Intel(R) Core(TM) i7-7500U CPU @ 2.70GHz 32GB RAM.
Gurobi is used as the optimization solver. Matlab is used as the
training platform.
B. Comparison and Analysis of Different Feature Vectors
Following two indexes are used to compare the performance
of different feature vectors: 1) ACC, which refers to the
proportion between the correctly classified constraints and the
total constraints; 2) PRE, which refers to the proportion
between the correctly classified active constraints and the total
active constraints. The expression of ACC and PRE are shown
as follows:
TP TN
TP TN FP FN
ACC +
=+ + +
(15)
TP
TP FP
PRE =+
(16)
where TP/T N is the number of correctly identified
active/ina ctive constraints; FP/FN is the number of
active/ina ctive constra ints that the proposed method fails to
identify.
Table II shows the pre-identification results of transmission
constraints by M0 and M9 for the 2000 test samples. Fo r Ca se
30, the identification PREs of M9 and M0 are 99.7% and 96.3%
respectively. The active constraints are more correctly
identified by the proposed output feature vector in M9 tha n M0.
From the respective of ACC, M0 gets a higher value than M9.
Therefore, M0 identifies more inactive constraints. The reason
is that the small number of a ctive constraints are easily
subm erged by a la rge number of inactive constraints by M 0
method.
For Case 118, the output vector of M0 has 186*186=34596
columns. Hence, 30000 sa mples have overwhelmed the
memory of our computing device in the training process. The
situation of Case 661 is the same as that of Case 118 by M0.
Therefore, it is not a good choice to directly choose the binding
status of constraints a s the output fea ture vector. If generation
power PG is used as the output feature vector, the required
memory is only linearly a ssociated with the scale of power
system s. Moreover, the proposed fea ture vector gets h igh
precision. The higher the PRE, the probability of convergence
within one itera tion is higher. The ACC in Ca se 118 and Case
661 are both higher than 99.6%. It can be inferred that only a
few inactive constra ints a re identified as active constraints. In
conclusion, M9 not only achieves higher identify accura cy of
active constraints, but also occupies less memory than M0. The
designed output feature vector for the SCED problem is
reasona ble and effective.
TABLE II
IDEN TIFICATION RESU LTS OF TRANSMISSION CONSTRAINTS WITH DIFFERENT
OUTPUT FEATURE VECTORS IN DIFFERENT CASES (2000 TEST SAMPLES).
Cases
Meth.
PRE
ACC
TP
FP
TN
FN
Case
30
M0
96.3%
99.3%
265740
10210
3072191
13859
M9
99.7%
94.7%
275020
930
2907810
178240
Case
118
M0
Out of memory in Matlab platform
M9
99.8%
99.6%
112906
229
68794233
284632
Case
661
M0
Out of memory in Matlab platform
M9
100.0%
99.9%
4650690
0
2186902272
865038
C. Training Efficiency Comparison with Different Training
Strategies
Fig. 6 Supervised learning process by M1, M2 and M9 in Case 118.
Fig. 7 Supervised learning process by M1, M2, and M9 in Case 661.
To validate the effectiveness of the proposed learning
strategy, the SDAE with traditional learning strategy M1 as in
[26] and a common learning strategy M2 a re compared with the
proposed method M9 in Case 118 and Case 661. For the
traditional learning strategy, the a ctivation function is sigmoid,
and the learning algorithm is the stocha stic gradient descent
algorithm. For the common learning strategy, the activa tion
function uses ReLU. The other pa rameters of M1 and M2 in the
training process are the same as the proposed learning strategy.
The relationship between the loss reduction and the number
of epochs in Case 118 and Case 661 are shown in Fig. 6 and Fig.
7, respective ly. The average running time of each epoch in Case
118 is 0.8642, 0.8739 and 0.8728 seconds for M1, M2 and M9,
respectively. The a verage running time of each epoch in Case
661 is a bout 1.8415, 1.8110 and 1.8089 seconds for M1, M2
and M9, respectively. The running time of each epoch for M1,
M2 and M9 is comparable. It can be seen from Fig. 6 and Fig. 7
that M1 has the largest value of loss function. The satura tion
phenomenon using M1 comes early. The main rea son is that the
traditional lea rning strategy M1 uses sigmoid a ctivation
function. The parameters update come to a standstill in the
saturation region of sigmoid function. Compared with M2, the
proposed method M9 can achieve lower convergence error
because the linear output layer is capable of capturing wider
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output. Therefore, the proposed learning strategy for the SCED
problem is more effective than the traditional lea rning strategy
of SDAE and the comm on learning strategy.
D.Accuracy Comparison with Different Neural Networks
The effectiveness of DNN selection is verified by comparing
the accuracy of predicted optimal generation power PG with
different neural networks. Parameter pa represents the
probability that the a bsolute error of generation output exceeds
5 MW; pr denotes the probability that the relative error of
generation output exceeds 5%. The values of pr a nd pa
calculated by different neural networks in Case 118 and Case
661 a re compared in Table III. For BP neural network, the
activation function is ReLU function and the learning algorithm
is stochastic gradient descent algorithm. For DBN neural
network, the code is referred to [44], the min-max
normalization is used a s the data processing method because its
learning algorithm [23] require s the data between 0 and 1.
HELM uses an unsupervised stage to reconstruct input and
feeds the reconstructed input into an ELM. Therefore, HELM
can be regarded a s a n ELM when calculating PG. There are two
hidden layers in the unsupervised stage of HELM. The first and
second hidden layers ha ve 700 and 500 neurons, respectively.
It can be concluded from the results of M4 that DBN is not
suitable for the regression problem a lthough it has shown good
performance for many classification problems. The pa
calculated by BP neural network M3 and HELM M5 are both
higher than 13% in two cases. The value of pr is larger than 7%
by M3 and reaches 5% by M5 in Case 118. However, the values
of pr by M3 and M5 are increased sharply with the scale of
power systems. Consequently, those results of BP neural
network and HELM are lim ited by the shallow network
structure.
For the proposed method M9, it can be observed from Ta ble
III that the values of pa are 0.9% and 3.5% respectively in the
two cases. The values of pr are 1.8% a nd 1.2% respectively.
The absolute error and relative error calculated by the proposed
method M9 are both sign ificantly reduced compared with M3,
M4, and M5. In conclusion, SDAE is a desired fully-connected
neural network for the SCED problem because of its deep
stacked structure and unsupervised criterion. The well-trained
SDAE ca n effectively predict generation output from system
operating conditions.
TABLE III
ACCURACY OF GENERATION OUTPU T WITH DIFFE RENT METHODS IN CA SE
118 AND CASE 661.
Cases
Method
pa
pr
Case 118
M3
13.7%
7.6%
M4
42.6%
42.1%
M5
14.0%
5.0%
M9
0.9%
1.8%
Case 661
M3
26.2%
15.5%
M4
58.5%
58.9%
M5
35.3%
25.3%
M9
3.5%
1.2%
E. Performance of the Proposed Identification Method
The effectiveness of the proposed approach for the SCED
problems is shown in Table IV a nd Table V. Benefiting from
the high prec ision of active constraint identification by SDAE,
M9 takes one iteration to converge in most ca ses. There are no
more than three iterations to converge by the proposed method
M9. The computational time with M6-M9 is compared in Table
V. It shows tha t M6 takes the most computational time in Case
118. For Ca se 118, compared with pra ctica l approach M7, the
average computa tional speed of the SCED problem is improved
by 1.8 times by M9. The computational speed in the most
time-consum ing ca se of M7 can be improved by 2.7 times by
the proposed approach M9. Compared with method M8, it can
be observed from Table IV that our proposed approach (M9)
generally has a higher percentage to converge within one
iteration. Besides, it can be observed from Table V that M9 has
a faster calculation speed for each sample. The reason is that
our proposed method adds less redundant constraints than M8.
For example, M8 identifies 1892 active constraints for Case
118. It can be calculated from Table II that the number of
minimal a ctive constra ints in Case 118 is
(112906+229)/2000=57 for each sample on a verage. Our
proposed method M9 identifies (112906+229+284632)/2000
=199 active constraints on average.
TABLE IV
PERCEN TAGE OF THE NUMBE R OF ITERATIO NS WITH M7- M9 IN CA SE 118 AND
CASE 661.
Cases
The number of iterations
1
2
3
4
5
6
M7
Case 118
-
2.4%
83.5%
14.1%
-
-
Case 661
-
-
-
34.0%
65.6%
0.4%
M8
Case 118
94.4%
5.6%
-
-
-
-
Case 661
100.0%
0.0%
-
-
-
-
M9
Case 118
99.5%
0.5%
-
-
-
-
Case 661
100.0%
-
-
-
-
-
TABLE V
COMPUTATIONAL TIME WITH M6-M9 IN CA SE 118 AND CASE 661.
Cases
Computational time of 2000 tes t
samples on average (s)
Computational time in the
mos t time-consuming case
for M7 (s)
M6
M7
M8
M9
M7
M8
M9
Case
118
10.6
1.4
1.0
0.8
1.9
0.8
0.7
Case
661
Numerical
failure
112.1
77.0
31.9
155.1
75.7
30.9
In the practical 661-bus system, M6 fa ces numerical
problems. This fact illustrates the necessity of iteratively
solving the SCED in practical operations. From Table V, the
practical iterative a pproach M7 takes 4~6 iterations to converge.
For a ll of the 2000 test samples, the SCED problem is solved
within one iteration by the proposed method M9. M7 takes
112.1 seconds for ea ch sample on average, and 155.1 seconds
in the most time-consuming case. Compared with M7, it can be
observed from Table V that the computational speed of M9 is
improved by 3.5 times on average, and 5.0 times for the worst
case of M7. Compared with method M8, it can be observed
from Table IV that our proposed a pproach M9 and M8 can both
make the test samples converge within one iteration. Similar
with the IEEE 118-bus system, it can be observed from Table V
that the proposed method is more computationally efficient
than M8 because less redundant constraints are a dded. For a
certain sample in Ca se 661, there a re 52602 a nd 2758
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constraints are identified to be a ctive by M8 and M9,
respectively. The number of minimal active constra ints in Case
661 is 2325.
Above all, the proposed method can effectively identify the
active constraints a nd improve the calculation efficiency of
SCED problem.
F. Fast Tuning Strategy for Topology Change
To dem onstrate the effectiveness of the proposed fast tuning
strategy of DNN parameters for topology change, two modified
cases from IEEE 118-bus system are utilized. The deta iled
num erical tests are a s follows.
Case 0: The original IEEE 118-bus system.
Case 1: A new branch from bus 81 to bus 80 is added. The
branch from bus 81 to bus 80 is the most frequently binding line
in the original IEEE 118-bus system.
Case 2: There is a load bus a nd two bra nches (from bus 119
to bus 118; from bus 119 to bus 40) added in the original IEEE
118-bus system.
Fig. 8 shows the probability density function of the 11th
generation output power in Case 0, Case1 and Case 2. The deep
learning approa ch requires that the distribution of test data is
the sa me as that of the tra ining data for DNN. For Case 1, it can
be observed that there a re slight differences of the 11 th
generation output distribution between Case 0 and Case 1.
Benefiting from the strong generaliza tion ability of the DNN of
Case 0, the a ccuracy of predicted optimal generation power PG
for Case 1 is a cceptable when directly utilizing the DNN of
Case 0. The pa is 5.1% and the pr is just 2.4%. Bes ides, 98.2%
of the 2000 test samples are solved within one iteration, and the
rest sam ples are solved within two iterations. For Case 2, the
number of bus nodes increases. It ca n be observed from Fig. 8
that the distribution of the 11th genera tion output power in Case
2 is quite different from that in Case 0. Therefore, the DNN for
Case 0 cannot be applied to Ca se 2. A new DNN needs to be
trained for Case 2.
Fig. 8 Probabilistic density function of the 11th generation output power in
different cases.
Table VI shows the results by the proposed fa st tuning
strategy for topology change. When applying the proposed
tuning strategy (M10) to Case 1, it can be observed from Table
VI that the pa and pr can be reduced to 4.5% and 2.2%
respectively only using 10000 samples and 100 epochs (i.e., 40
seconds in tota l). For Case 2, only half of the number of training
samples in Ca se 0 is needed. After 50 epochs (i.e., 48 seconds
in total), the pa and pr of Case 2 can be both less than 5%.
Besides, it can be observed from Table VII that for 99% out of
the 2000 test samples, the SCED problem is solved within one
iteration for both Case 1 and Case 2. The proposed tuning
strategy based on transfer learning can effectively build a new
DNN with fewer training sa mples.
TABLE VI
TEST CONDITIONS AND CORRESPONDING RESULTS BY THE PROPOSED
STRATEGY FOR TOPOLOGY CHANGE.
Training
time (s)
pa
pr
Nepoch
Ntrain
Nvalid
Ntest
Case 1
40
4.5%
2.2%
100
10000
2000
2000
Case 2
48
3.9%
1.7%
50
15000
2000
2000
TABLE VII
PERFORMA NCE OF TH E PROPOSED STRATEGY AND M7 FOR TOPOLOGY
CHANGE.
Cases
Method
Percentage of different iteration number
1
2
3
4
5
6
Case 1
M7
-
1.3%
91.4%
7.2%
0.1%
-
M10
99.4%
0.6%
-
-
-
-
Case 2
M7
-
0.1%
73.2%
26.3%
0.1%
-
M10
99.6%
0.4%
-
-
-
-
VI. CONCLUSIONS
In this paper, the traditiona l SCED problem is embedded
with deep learning techniques to improve the computational
efficiency of SCED without any accuracy loss. SDAE is
utilized to extract the nonlinear relationship between the system
operating condition and the set of active constraints. The input
and output feature vector and learning strategy a re designed to
improve training efficiency so tha t the learning accuracy of the
SDAE ca n be guaranteed. In our case studies, the SCED
calculation normally takes 3~6 iterations to converge while the
proposed method does not need iterations in most cases.
Besides, a fast tuning strategy ba sed on transfer learning for
DNN parameters is proposed to handle the situation of topology
change. The computational efficiency of the SCED problem is
significantly improved. Because the proposed method does not
affect the computational a ccuracy and convergence
performance of the SCED calculation, it shows excellent
potentials for practical applications in the rea l-time ma rket
clearing.
As shown in this pa per, although DNN cannot guarantee the
correctness of all the predicted values, the useful reference
information ca n be selected to accelerate the computational
speed of the SCED without compromising the accuracy. Hence,
the idea of improving the efficiency of power system opera tion
analysis by deep lea rning techniques is worthy of further study.
This paper focuses on the single -interval SCED model, which
is commonly used in the United Sta tes, China , and other
countries to clear the ma rket. Our study shows tha t deep
learning techniques are capable of mining the deep complex
relationship between the set of active constraints and system
operating condition. Therefore, for more complicated problems
such as unit commitment, multi-interval economic dispatch,
and market clearing considering bid price, DNN is also a
promising tool for digging the information from generated or
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10
recorded dataset. However, how to effectively obtain training
samples and properly utilize the deep learning techniques
regarding the specific properties these problems should be
further investigated. Besides, the physical models of the power
system operation a re known. Therefore, to improve the learning
performance combining with power doma in expertise is worthy
of further exploration. Besides, the tra nsfer learning method
proposed in this pa per also has certain limitations. If the
topology substantially changes, the features of the learning
target will notably change, a nd then, more powerful transfer
learning techniques are needed.
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Yan Ya ng (S’2017) received the B. S. degree
from the School of Electrical Engineering,
Chongqing University, China, in 2016, where
she is currently pursuing the Ph.D. degree. Her
research interests include big data, deep
learning, a nd their applications in power and
energy systems.
Zhifang Yang (S’2013-M’2018) received his
Ph.D. degree in electrical engineering from
Tsinghua University in 2018. He currently
works a s an assistant professor at Chongqing
University. His research interests include
power system ana lysis a nd electricity market.
Juan Y u (M’2007, SM’2015) received the
Ph.D. degree in electrical engineering from
Chongqing University, China, in 2007.
Currently, she is a full professor at Chongqing
University. Her research interests include big
data analytics a nd power system ana lysis.
Kaigui X ie (M’10–SM’13) is a Fu ll Professor
with the School of Electrical Engineering,
Chongqing Unive rsity, China. His ma in
research interests focus on a reas of power
system relia bility, planning, a nd analysis. He
is an Editor of the IEEE Transactions on Power
Systems.
Liming Jin received the M. S. degree in
electrical engineerin g from North China
Electric Power Unive rsity, China, in 2007.
Currently, he is a n engineer a t Sta te Grid
Chongqing Electric Power Company, China.
His research interests include power system
analysis and optimization.
