Journal of Power and Energy Engineering, 2014, 2, 162-169
Published Online September 2014 in SciRes. http://www.scirp.org/journal/jpee
How to cite this paper: Xia, C.J., Zhou, Y., Men, K. and Xie, Y.G. (2014) Study on Identification of Inductive-Motors Load Par-
tition Based on Coherence. Journal of Power and Energy Engineering, 2, 162-169.
Study on Identification of Inductive-Motors
Load Partition Based on Coherence
Chengjun Xia1, Yun Zhou1, Kun Men2, Yinggeng Xie1
1School of Electric Power, South China University of Technology, Guangzhou, China
2Electric Power Research Institute of CSG, China Southern Power Grid, Guangzhou, China
Received June 2014
A new inductive motors load equivalence algorithm based on coherence is proposed in this paper.
In order to partite motors load rapidly and accurately, fuzzy c-means clustering along with parti-
cle swarm optimization (PSO-FCM) algorithm is proposed to identify coherent motors base on its
physical essence of fuzziness. The merits of PSO algorithm are independent to initial value and
convergent to optimum value rapidly, and the validity function is constructed to assess clustering
validity. The test on IEEE 39-Bus System is presented to evaluate the effectiveness of the new algo-
rithm, the membership matrix definite not only coherence group of motors but also correlation
value of coherence between motors. The algorithm can be used to partite motor load based on co-
herency in dynamic equivalence with power system operating on different modes.
Load Model, Inductive-Motor Load, PSO, FCM, Clustering Validity, Dynamic Equivalent
To simplify the analysis of the large scale interconnected power system, a series of power system equivalent
methods had been proposed in recent years. Numerous investigations had shown that the load model has an im-
portant and complex role on the dynamic simulation and response of power system after fault. Because most
power system loads involve many inductive motors, it makes sense to develop an effective identification theory
for industrial loads which comprised of different induction motors. In order to obtain more accurate results in
transient analysis and simulation, it is necessary to identify inductive motors based on coherence -.
For actual power system, there is no coherence load in the strict sense because of the uncertainty of load
model. The similar dynamic characteristics exist in coherence load due to its ambiguity. In , clustering me-
thod is used to partite motor load, and criteria of coherence were proposed in dynamic control and analysis in
power system. In , dynamic performance of power system load representation is analyzed and the importance
of load modeling is emphasized. In , the identifiability of the equivalent motor parameters is investigated in
the composite load model, case studies verify the validity of the proposed method.
This paper proposes a method for the identification of inductive motors load partition based on coherence. In
order to partite motors load rapidly and accurately, fuzzy c-means clustering along with particle swarm optimi-
C. J. Xia et al.
zation (PSO-FCM) algorithm is applied to identify coherent motors base on its physical essence of fuzziness.
Then fuzzy partition entropy function is used as a measurement index to measure the performance of algorithm
and evaluate the effectiveness of fuzzy c-mean clustering algorithm based on particle swarm optimization. The
computational results show that control strategy works effectively and can be applicable to the identification
2. Load Models in Power System
2.1. Static Load Models 
The load models are traditionally classified into two broad categories: static models and dynamic models. A
static load model expresses the characteristics of the load at any instant of time as algebraic functions of the bus
voltage magnitude and frequency at that instant, including ZIP model, exponential model and polynomial model.
An alternative model which has been widely used to represent the voltage dependency of loads is the ZIP model,
as it is composed of constant impedance (Z), constant current (I), and constant power (P) components.
2.2. Dynamic Load Models
There are, howev er, many cases where it is necessary to account for the dynamic of load components. Studied of
inter-area oscillations, voltage stability, and long-term stability often require load dynamics to be modeled.
Study of system with large concentrations of motors also requires representation of load dynamics.
The dynamic characteristics of induction machine should be taken in account in terms of influence of induc-
tion machine on power system. The dynamic equation of an induction machine may be written as:
=-[-(X -X )]-(-1)E
=-[-(X -X )]-(-1)E
are transient electric potential,
is rotor angular velocity,
; TJ is motor inertia time constant, S is slip speed of rotor.
T Ei Ei= +
is electromagnetic torque;
XX X= +
is rotor open circuit reactance.
2.3. Synthesis Load Model
For actual power system, the modeling of loads is complicated because a typical bus represented in stability
studies is composed of a large number of devices. The exact composition of load is difficult to estimate. The
characteristics of load can be represented by the ZIP models and induction machine. The synthesis load model
consists of third-order induction motor in paralleled with ZIP. The schematic structure of this synthesis load
model is shown in Figure 1.
In Figure 1, Rs is stator equivalence resistance, Xs is stator equivalence leak reactance, Rr is rotor equivalence
resistance, Xr is rotor equivalence leak reactance, Xm is value of mutual inductance between stator and rotor
Model order reduction techniques are often used to reduce the model complexity. When analyzing the tran-
sients of circuits and electrical machines, the detailed model is often built first to describe the plant as accurately
as possible. Then the model order can be reduced according to the specific problems under investigation.
C. J. Xia et al.
Figure 1. Equivalent circuit of the synthesis
3. The Characteristics for Load Partition Identification
The dynamic characteristics of equivalence load will be poor in the circumstance because there are amount of
discrepant load in equivalence load. To guarantee the accuracy of dynamic simulation of power syste m, it’s
necessary to partite load in appropriate way .
The essence of partition equivalence load is to observe the dynamic characteristics of aggregation load. For
induction machine, the best tool to study the performance is slip. The relationship between slip and rotor speed
is s = (n0−n)/n0, so s is corresponding to speed completely. If Tm is constant, the formula of Te can be expressed
where, Scr = Rr/Xl is critical slip, Xl is sum of rotor leak reactance and stator leak reactance. Therefore, the lin-
earized forms of Equation (4) is
Generally, the value of SN is quite small, so compare
to Scr, it can be ignored. Then,
Therefore, the linearized forms of Equation (7) is
From the given Equation (8), the corresponding solution is
is determined by TJRr. If values of TJRr of the induction machine are similar, then they own similar
dynamic characteristics. Finally, to evaluate the state of induction machine, (TJRr, KL) is chosen to be a charac-
teristic quantity which is used to identify coherence inductive motors in power system. So, coherent inductive
motors own similar characteristic quantity.
4. Improved Fuzzy c-Means Clustering Algorithm
4.1. Fuzzy c- Means Clustering
Fuzzy c-means clustering (FCM) approach is widely implemented to tackle practical problems with uncertainty,
C. J. Xia et al.
imprecision and vague peculiarity. Initial categories and parameters are given before the program began to iter-
ate. Satisfying solution for project can be obtained though modification, as well as iteration. Robust results for
categories can be achieved based on fuzzy clustering. The formula of FCM can be expressed as following:
= == =
where, xi is sample, cj is a set of class center, U = (uij) is a membership matrix of samples.
To solve Equation (10) is impossible. Formula for soft clusters was given by mathematician Bezdek  :
where, m is constant of vague degree, dij represents the distance between sample and class center.
4.2. Particle Swarm Optimization
In PSO, the solution of the objective function can be imaged as a point in d-dimension, just as a bird fly in the
sky. The process of finding the solution is based on the analogy of swarm of bird flocking and fish schooling. A
swam of particles represent a candidate solution to the optimization problem  .
Each particle adjusts its position according to its own experience, and the experience of its neighbouring par-
ticles. The position and velocity of the ith particle in the d-dimensional search space are represented in (14) and
ididid idgd id
where, Xik is position of individual i at iteration k, Vik is velocity of individual i at iteration k,
is weight pa-
rameter, c1 is cognitive factor, c2 is social factor, pid is the best position of individual i until iteration k, Pgd is the
best position of the group until iteration k, rand1, rand2 are random numbers between 0 and 1.
To evaluate the merit rating of the particles, we can set a function to measure which called the fitness function.
According to the fitness function, the best particle is selected. And initial positions and velocities are vector.
Particles can move to a better position by sensing its own position and the particles nearby. In every step each
agent’s position is updated by the best fitness value called as pibest. Also each agent knows the best value in the
group called as pgbest among the pibest. After a specified number of iterations, the agents searching points gradu-
ally reach the global optimal point using pibest and pgbest.
An inertia weight in PSO algorithm was introduced in 1998, in order to provide better control exploration .
The inertia weight is used to control the impact of the previous history of velocity, and thus to influence the
tradeoff between global and local exploration abilities of the moving agents. In PSO, the balances between the
global and local exploration abilities are mainly controlled by inertia weights. Both convergence speed and ac-
curacy of solution will be affected. The inertia weight is suggested to decrease linearly from 0.9 to 0.4 during
the run. The inertia weight is formulated as in (16).
max max minmin
C. J. Xia et al.
is the attenuation index of inertia weight,
is the initial value of the inertia weight,
final value of the inertia weight, Tmax is the maximum iteration number; t is the current iteration number.
4.3. Particle Swarm Optimization Fuzzy c-Means Clustering Algorithm
As fuzzy c-means algorithm is easy to convergent to partial optimum, PSO is introduced to program instead of
the process based on gradient descent. The performance of the algorithm is fantastic and effective in the process
of iteration, and it is insensitive to initial values. Finding the optimal particle center is main task in the process
of iteration. Hence, class center ci is encoded as a particle. The particle can be expressed as following:
where, cij is the center ith with clustering method jth.
To evaluate the performance of particle, objective function is recommended. In the paper, JH is used to be ob-
jective function to compare the performance among particles. The objective function can be expressed as fol-
Solving Equation (18), we can obtain the fitness for every particle. Particle optimum point can be obtained
after comparing the global optimal point and particle optimal point. If f(xid) is greater than f(pgd), then fitness of
global optimal point will be updat ed, vice versa.
( )( )
The iteration structure of particle swarm optimization fuzzy c-means clustering algorithm is presented in
4.4. Evaluating the Effectiveness of Clustering Algorithm
To evaluate the effectiveness of fuzzy c-mean clustering algorithm based on particle swarm optimization, fu z zy
Set the initial parameters
Calculate membership for
clustering center as in (11)
function as in (10)
Calculate the fitness:
fi as in (18)
Update Pibest and
Pgbest as in (19)
Update the new
Vi and Xi
Out optimal parameters
center as in (12)
Figure 2. Block diagram of PSO-FCM algorithm.
C. J. Xia et al.
partition entropy S(U,K) is used as a measurement index to measure the performance of algorithm. The small
S(U,K) is, the dispersion degree between particle is smaller, that is to say, the performance of clustering algo-
rithm is more perfect. S(U,K) can be expressed as:
The distance between clusters and the data points assigned to them should be minimized and the distance be-
tween clusters should be maximized .
5. Simulation and Numerical Analysis
To verify the effectiveness of the proposed algorithm, source program is developed for identification of induc-
tive-motors load partition based on coherence for simulations and is applied to the standard IEEE-39 bus
benchmark system. The parameters used for chaos particle swarm optimization algorithm are as following: num-
ber of particles is 30,
, number of iterations is itermax = 100,
Research system and external system are shown in Figure 3. Broken line passing through bus 39, bus 3 and
bus 27 is shown in Figure 3. Taking actual research into consideration, research system is one above the broken
line. The program is developed to partite the coherence motors in outside system including 13 motors.
It is clearly known that clustering results with the relationship among membership matrix. The results of iden-
tification of inductive-motors load partition based on coherence can be written as following.
2 Group: (3, 4, 7, 8, 12, 15, 18);
(16, 20, 21, 23, 24, 27)
3 Group: (3, 4, 7, 8, 12, 15, 18);
(16, 20, 21); (23, 24, 27)
4 Group: (3, 4, 7, 8, 12, 15, 18);
(16, 20, 21); (23, 27); (24)
The value of fuzzy partition entropy S(U,K) is shown in Table 1.
Therefore, it can be inferred that the cluster performance is best in group 2. Optimum membership matrix U
can be obtained after calculation and simulation.
0.0084 0.0006 0.0064 0.0033 0.0092 0.00060.9841 0.0033 0.9767 0.9609 0.9657 0.9870 0.9657
0.9916 0.9994 0.9936 0.9967 0.9908 0.99940.0159 0.9967 0.0232 0.0391 0.0343 0.0130 0.0343
For this improved PSO-FCM algorithm, the program runs after parameters initialized, and convergen ce re-
sults of iteration are shown in Figure 4.
Figure 3. Diagram of IEEE-39 with 10 generators.
C. J. Xia et al.
The progress of PS O-FCM algorithm can be illustrated easily in Figure 4. After 35 times iteration, the task of
iteration is over. And the optimum solution can be obtained. So identification of inductive motors load partition
based on coherence can be easily implemented.
To illustrate the stability of the proposed algorithm, source program is developed for identification of induc-
tive motors load partition, and runs for several times, which contain 2 groups of inductive motors. The results
are shown in Tab le 2. It is clearly know that the results are same nearly. The value of fuzzy partition entropy in
iteration 3rd, 5th, 6th, 8th, and 10th is 7.7845e−04. The value in iteration 2nd, 7th is 7.7853e−04. The error existed
between maximum and minimum is only 0.0034. It is sufficient to illustrate the stability of proposed algorithm.
Considering convergence rate, the PSO-F CM method can always generate precipitous convergence rate to-
ward an acceptable solution, wh i ch show that the PSO-FCM method has good convergence performance. As
shown in Figure 5, the convergence characteristic from 20 to 30 times is different. The program based on PSO-
FCM is easier to be convergent with certain reliability. And it can be applied to large-scale power systems.
Table 1. The validity function value.
Group 2 3 4
S(U,K) 7.7862e−004 0.0437 0.2261
Table 2. S(U,K) value of ten times simulation (e−004).
Times 1 2 3 4 5
S(U,K) 7.7862 7.7853 7.7845 7.7879 7.7845
Times 6 7 8 9 10
S(U,K) 7.7845 7.7853 7.7845 7.7845 7.7845
010 20 30 40 50 60 70 80 90100
Figure 4. Fitness function evolution curve.
010 20 30 40 50 60 70 80 90100
Figure 5. Fitness function evolution curve in ten times.
C. J. Xia et al.
To investigate the identification of inductive motors load partition based on coherence, PSO -FCM algorithm is
proposed considering the characteristic of inductive motor load. The proposed algorithm uses PSO to find opti-
mum instead of the process of gradient descent. The coherence motors can be identified according to member-
ship matrix, as well as intensity of coherence motors. For the sake of validating the coherence motor in dynamic
studies, a series of simulation tests are performed and compared. And the analysis results of measured simula-
tion curves show that the coherence motors identified by the proposed strategy possess good description ability
for load dynamics. What’s more, the proposed identification strategy can improve the stability of identification
results, by which promotes the practical application of the load model. Case study also shows the effectiveness
and feasibility of the proposed identification strategy.
This work was financially supported by National High Technology Research and Development Program of
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