A multi agent based power system hybrid dynamic state estimator
MULTI-AGENTS
A Multi-AgentBased Power System
Hybrid Dynamic
State Estimator
Ankush Sharma, Suresh Chandra Srivastava, and Saikat Chakrabarti, Indian Institute
of Technology Kanpur
C
onventional power system state estimation (PSSE) is performed with the
help of static state estimation (SSE) tools such as the weighted least squares
(WLS) approach.1 These tools use the measurements received from remote
For fast estimation
terminal units (RTUs)—typically, at an interval of 1 to 5 seconds—through a
of power system
state, a multi-agentbased power system
hybrid dynamic state
estimator uses field
measurements from
remote terminal
units and phasor
measurement units.
52
supervisory control and data acquisition
(SCADA) system. With the advent of phasor
measurement units (PMUs), the measurement data reporting rate has increased up
to 50 frames per second for 50-Hz systems
and up to 60 frames per second for 60-Hz
systems. But SSE execution takes approximately 30 to 60 seconds for a large system, indicating the updated measurements
received from RTUs and PMUs aren’t being utilized, which means that SSE results
lag behind actual system states. The development of dynamic state estimation (DSE)
tools could solve state estimation problems,
enabling available measurements to be utilized when estimating system states.
Despite the introduction of agent-based
software engineering,2,3 few efforts have been
made to apply a multi-agent-based approach
to solving the power system SSE problem.4,5
To our knowledge, nothing in the literature
describes solving the power system dynamic
state estimation (PSDSE) problem using
multi-agent-based software engineering.
However, quite a few efforts have focused on solving this problem by using
Kalman filters. One approach6 used the
extended Kalman filter (EKF) to estimate
power system states. The EKF uses firstorder approximation of the Taylor series
to solve the measurement function, causing
its state estimation results to deviate from
actual values, thereby rendering the EKF
not quite suitable for dynamic state estimation of nonlinear systems such as the power
system. Other researchers7–9 used the unscented Kalman filter (UKF) to estimate the
power system state, but because the UKF’s
performance deteriorates with the increase
in the number of state variables,10 it also
isn’t suitable for estimating the state in
large power systems. Recently, the cubature
Kalman filter (CKF)10–12 has demonstrated
potential benefits, such as accuracy and stability for the large state vector, over other
Kalman filtering techniques.
The CKF approach was originally developed to track aircraft trajectories. 10
1541-1672/15/$31.00 © 2015 IEEE
Published by the IEEE Computer Society
IEEE INTELLIGENT SYSTEMS
To make it suitable for power system hybrid dynamic state estimation
(PSHDSE), we developed a modified
CKF approach. The CKF doesn’t require Taylor series approximation
of the nonlinear function and the
Jacobian during its execution, and
unlike with the UKF, its performance
doesn’t deteriorate with the increase
in the size of the state variable vector.10 We developed a software-based
multi-agent model to scan and process the PMU and the conventional
RTU measurements separately; this
model then combines the CKF results
in every Kalman filter cycle to estimate system states. To speed up the
CKF processing, we used a factorization approach that factorizes the
large measurement vector into subvectors. The CKF is processed in parallel using the subvectors to estimate
the complete power system’s various
states.
PSHDSE formulation, state vector xk
comprises the bus voltage (Vk) and
the angle (pk) state subvectors. The
measurement vector zk comprises the
subvectors of voltage magnitude measurements (Vmk), voltage angle measurements (pmk), real power injection
measurements (Prmk), reactive power
injection measurements (Qrmk), real
power flow measurements (PFmk), and
reactive power flow measurements
(QFmk), received from the RTUs and
PMUs at the instant k, which are
given by
xk = [Vk | pk]T
(5)
zk = [Vmk | pmk | Prmk | Qrmk | PFmk | QFmk]T,
xk+1 = f(xk) + wk
(1)
zk+1 = h(xk+1) + vk+1
(2)
wk = N(0, Qk)
(3)
f(xk−1) = ak−1 + bk−1
(7)
vk+1 = N(0, Rk+1),
(4)
ak−1 = α k−1x k−1 + (1 − α k−1)x k−−1
(8)
where, xk is the state vector at the kth
instant; zk is the measurement vector
at the kth instant; wk is Gaussian process noise with zero mean; Qk is the
process noise error covariance; and
vk is the Gaussian measurement noise
with zero mean. In addition, Rk is the
measurement noise error covariance,
and f(•): ℜn → ℜn and h(•): ℜn → ℜm
are the nonlinear functions for a state
space of size n and the measurement
space of size m, respectively. For the
MAY/JUNE 2015
Cubature Kalman Filter
Under the Bayesian estimation approach, using the known posterior
probability density value at the k − 1th
instant, the prior probability density
at the next instant k− can be calculated using the Chapman-Kolmogorov
equation, given by10
p(x k | z1:k−1) =
∫ p(xk | xk−1)p(xk−1 | z1:k−1)dxk−1
bk−1 = ak−1 (ak−1 − ak−2) + (1 − ak−1)bk−2, (9)
where ak−1 and ak−1 are the parameters at instant k − 1, containing
values between 0 and 1; x k−−1 is the
predicted state vector at the instant
k − 1; ak−1 and bk−1 are the vectors defined by Equations 8 and 9 at instant
k − 1. To define the measurement
function h(•) for the power system,
we use standard bus power injection
equations (real and reactive), and the
www.computer.org/intelligent
(10)
ℜ
p(x k−1| z1:k−1) = N (xˆ k−1|k−1, Px ,k−1|k−1),
(11)
(6)
where subscript m indicates the size
of the corresponding measurement
samples at the kth instant.
We use the state forecasting tool13 in
our proposed approach to model the
state transition function. This function
accommodates variations in state due
to changes in system parameters, such
as load variations from one time step
to another time step. Our proposed
approach uses Holt’s two-parameter
linear exponential smoothing technique13 to forecast states. Using this
technique, we can define the state
transition function f(•) as
Power System
Dynamic Model
We can represent a nonlinear dynamical power system for state estimation
in a discrete form with the help of the
following equations:
line power flow equations (real and
reactive).14
where N (xˆ k−1|k−1, Px ,k−1|k−1) is the standard normal distribution for the
Gaussian density, with xˆ k−1|k−1 as the
estimated value (mean) and Px,k−1|k−1
as the covariance at k − 1th instant.
We can calculate the value of the
probability density p(xk | xk−1) by using Equation 1. The minus (−) and
plus (+) signs in the superscript indicate the prior and posterior values, respectively. The CKF is implemented in
two steps, as follows.10
Time Update
In the first step, we calculate the estimated value of the mean xˆ k|k−1 and the
error covariance Px,k|k−1 of the states
by using Equations 1 and 10, which
are given by
∫
xˆ k|k−1 = f (x k−1)p(x k−1| z1:k−1)dx k−1 (12)
ℜ
Px ,k|k−1 =
∫ f (x k−1)f T (x k−1)p(x k−1 | z1:k−1)dx k−1
ℜ
− xˆ k|k−1xˆ kT|k−1 + Qk−1,
(13)
where p(xk−1 | z1:k−1) is the posterior probability density, provided by the measurement update step at the k − 1th instant.
53
MULTI-AGENTS
Dynamic state estimation coordinator
Multi-Agent System
Agent for conventional data
processing
Agent for PMU data
processing
Data transfer agent
Data transfer agent
PMU data scanner
Conventional data scanner
Figure 1. Proposed multi-agent model for the power system hybrid dynamic state
estimation (PSHDSE) formulation. The agents are intelligent software modules
designed to scan and process remote terminal units (RTU) and phasor measurement
units (PMU) in parallel, and then exchange the results and status in a time-bound
manner to coordinate final state estimates.
Measurement Update
In the second step, we calculate the
predicted measurement zˆ k|k−1, its associated innovation covariance Pzz,k|k−1,
and the cross covariance P xz,k|k−1 by
using the following equations:
zˆ k|k−1 =
∫ h(xk)p(xk | z1:k−1)dxk−1
by using the third-degree sphericalradical cubature rule,15 given as
IN (f ) =
≈
∫
ξi =
ℜ
− zˆ k|k−1zˆ kT|k−1 + R k
Pxz ,k|k−1 =
∫ xkh T (xk)p(xk | z1:k−1)dxk
ℜ
− xˆ k|k−1zˆ kT|k−1.
(16)
When the new measurements are received, the estimated mean of the state
and its associated error covariance
are updated at the k+th instant by using the standard Kalman filtering approach, given by
xˆ k|k = xˆ k|k−1 + Kk (z k − zˆ k|k−1)
Pk|k =
Pk|k−1 − K k Pzz ,k|k−1K kT
−1
K k = Pxz ,k|k−1Pzz
,k|k−1.
(17)
(18)
(19)
To estimate the system states, the
solution of the integral terms in Equations 10 through 16 is approximated
54
∑ ω if (
i =1
ℜ
h(x k)h T (x k)p(x k | z1:k−1)dx k (15)
(20)
2n
(14)
Pzz ,k|k−1 =
∫ f (x)N (x | µ, Σ)dx
ℜ
n [e ]i , ω i =
Σξi + µ)
1
,
2n
(21)
where S is the error covariance matrix, m is the mean value, e is the
unity matrix, and w is the weight.
The {x, w} set forms the cubature
points. The details for the CKF appear elsewhere.10
Power System Hybrid
Dynamic State Estimation
In the power system, RTU data is
typically reported every 1 to 5 seconds, whereas PMU data is reported
every 20 to 200 milliseconds for
50-Hz systems and 17 to 167 milliseconds for the 60-Hz system. The
role of the multi-agents in our proposed approach is to process these
measurements separately in a collaborative manner for PSHDSE and
then integrate PSHDSE results to finally estimate the overall power system’s states.
www.computer.org/intelligent
The multi-agent system comprises multiple interacting autonomous agents
that help manage the complex system
by distributing the system execution
tasks among themselves and sharing
information interactively. The agents
in the proposed system are intelligent
software modules designed to scan and
process RTU and PMU measurements
in parallel, and then exchange the results and status in a time-bound manner to coordinate final state estimates
(see Figure 1).
The PMU (RTU) data scanner
agent periodically checks the arrival
of new data from the field PMUs
(RTUs). The data transfer agent
(DTA) does a sanity check on newly
arrived data and acts as a first-level
filter to identify and remove outliers from the measurement set. The
other two agents—the PMU data
processing agent (PDPA) and the
conventional data processing agent
(CDPA)—form part of the dynamic
state estimation coordinator (DSEC)
and play an important role in executing our PSHDSE approach.
Specifically, the PDPA receives
PMU data from the DTA at regular intervals and runs the CKF. The
CKF uses the latest available state estimates and the latest available PMU
measurements to execute its measurement update step. In our proposed
approach, only the voltage magnitude
and the angle measurements from the
PMUs are considered for dynamic
state estimation. This keeps the measurement equation for the CKF linear
in nature, as given by
Vmk
θ mk
V
= e 0 k + vk , (22)
0 e θk
where e is the unity subvector and
0 is the zero subvector. It’s assumed
that the measurements provided by
various measurement devices are
IEEE INTELLIGENT SYSTEMS
Scan data receipt
Scan for data receipt
request
State forecasting
data
response
Scan new data
New data
received
PMU measurment data
PMU agent state
estimates
State forecasting
data
State
forecasting
data generated
Data received
Send forecasted state
request to conventional
data processing agent
Time update
Run state forecasting
Measurement update
CKF
Scan for data
measurement
transfer request from
update Execution
conventional data
State
estimate
data
processing agent
State estimate data transfer
Time update
Send request to
PMU data
processing agent
CKF time update
Execution
State initialization
State estimates
from PMU data
processing
agent received
State Initialization
CKF time update
Execution
Measurement update
CKF
measurement
update execution
New data
received
Scan new data
Final state estimate data
Conventional measurment data
Updated state estimate data transfer request
(a)
(b)
Figure 2. State transition diagram: (a) the PMU data processing agent (PDPA) and (b) the conventional data processing
agent (CDPA). CKF stands for cubature Kalman filter. The PDPA and the CDPA agents process the respective PMU and RTU
measurements separately in various stages and regularly exchange the state forecasting data in a time-bound manner to
coordinate the final state estimates.
uncorrelated. Hence, Equation 22
can be divided into two independent
equations for parallel and fast processing. Figure 2a shows the PDPA’s
state transition diagram.
The CDPA execution process is
divided into two parts. In the first,
on arrival of the new RTU measurements, the CKF is executed to estimate overall system states, utilizing
the latest RTU measurements and
PDPA state estimates. In the second
part, the CDPA forecasts the states
by utilizing the previous state estimation results. The second part of
the execution process repeats for two
consecutive RTU measurement refreshes—ultimately, the step is executed after receiving the request from
the PDPA to provide the state forecasting data. Figure 2b shows the
CDPA’s state transition diagram.
Kalman Filtering Approach
for PSHDSE
The existing CKF process10 is modified
in our work specifically for PSHDSE
execution. In our proposed approach,
to estimate the 2m − 1 states of an
m-bus power system using the CKF,
4m − 2 cubature points are required to
MAY/JUNE 2015
be evaluated per system state per iteration. For a bigger power system, due
to the large size of the cubature point
matrix (2m − 1 × 4m − 2), running
the CKF without using a factorization approach is a time-consuming job.
Hence, the state and measurement vectors are partitioned into subvectors, as
in Equations 5 and 6, and the CKF is
executed for the subvectors in parallel
to save time. The CKF-based PSHDSE
execution process assumes a flat start
with initial voltage magnitudes and
angles at all the power system’s buses
assumed to be at 1 per unit (p.u.) and
zero radians, respectively.
CDPA Time Update
The CDPA execution steps are as
follows:
• Calculate the cubature points xi for
the state vectors,
ξi =
n [ e ]i , i
−
n [ e ]i − n , i
= 1 n
(23)
= n + 1 2 n ,
where n is the size of the state vector, e is a unity matrix of size n ×
n, and xi is the ith cubature point
vector.
www.computer.org/intelligent
• Evaluate the cubature points at the
instant k−, given as
x
q
i ,k− |k−1
=
Pk−1|k−1 ξ i + xˆ k−1|k−1,
(24)
where x qi ,k− |k−1 is the evaluated cubature point vector for the ith
cubature point input. The field
measurements and, hence their cubature points, are assumed to be
uncorrelated. Therefore, utilizing
the Cholesky factorization technique, the square root of Pk−1|k−1 can
be divided into four independent
submatrices, as shown in Equation
26. These submatrices are utilized
by dividing the state equation into
two independent equations containing voltage magnitude (V) and the
voltage angle (q) subvectors separately for fast processing, as shown
in Equations 27 and 28:
q
x Vi ,k− |k−1
q
x θi ,k− |k−1
=
A
B
C
D
×
ξ V ,i
ξ θ ,i
, (25)
xˆ V ,k−1|k−1
xˆ θ ,k−1|k−1
+
55
MULTI-AGENTS
where
A
Pk−1|k−1 =
B
C
D
.
(26)
Hence,
xq − = [A × ξ
V ,i
Vi ,k |k−1
+ B × ξ θ ,i
]
(27)
+ xˆ V ,k−1|k−1
and
xq − = [C × ξ
V ,i
θi ,k |k−1
+ D × ξ θ ,i
]
(28)
+ xˆ θ ,k−1|k−1 .
• Propagate the cubature points
through the state function f(•), defined in Equations 7 through 9,
against each state subvector j (V, q)
at instant k−. The propagated cubature point vector x*ji ,k− |k−1 for the
ith cubature point input and the jth
state subvector are given by
x*ji ,k− |k−1 = f (x
q
).
ji ,k− |k−1
1
2n
2n
∑x
*
,
i ,k− |k−1
(30)
i =1
where
x*i ,k− |k−1 = x*Vi ,k− |k−1 x*θi ,k− |k−1 .
(31)
• Calculate the predicted state error
covariance, given as
Pk− |k−1 =
1
2n
−
x*i ,k− |k−1x *T −
i ,k |k−1
i =1
xˆ k− |k−1xˆ kT− |k−1
(32)
+ Q k−1.
On getting the state forecasting request from the PDPA, the state vector
and the state error covariance estimated in this step are sent to the PDPA’s measurement update step.
PDPA Measurement Update
The PDPA execution steps are as
follows:
56
qP
i ,k− |k−1
Pk− |k−1 ξ i + xˆ k− |k−1,
=
(33)
where x qP
is the evaluated cubai ,k− |k−1
ture point vector for the ith cubature point input, and the superscript
P indicates the evaluated cubature
point vector under the PDPA measurement update step. The cubature
points in Equation 33 are evaluated by dividing the equation into
subvectors, in the same manner as
Equations 25 through 28.
• Propagate the cubature points
through the measurement function
against each measurement subvector l (Vm, pm) at instant k−,
γ liP,k− |k−1 = h lP (x qP
),
i ,k− |k−1
(34)
P
where, γ li ,k− |k−1 is the propagated
cubature point subvector for the ith
cubature point input, and the lth
measurement subvector h lP is the
PDPA measurement function, as
defined in Equation 22.
• Calculate the estimated value of
the measurement vector at the k− th
instant,
zˆ
2n
∑
x
(29)
• Calculate the estimated value of the
state vector, given as
xˆ k− |k−1 =
• Evaluate the cubature points utilizing the updated state estimates, the
state error covariance values provided by the time update step, and
the cubature points calculated in
Equation 23 as
k− |k−1
=
1
2n
1
PP −
=
xz ,k |k−1 2n
2n
∑x
T
*
γP
i ,k− |k−1 i ,k− |k−1
i =1
− xˆ k− |k−1zˆ T
(38)
.
k− |k−1
On arrival of fresh PMU measurements, the estimated state value and
its associated error covariance are
updated using Equations 17 through
19 at the k+th instant. These updated
values are then sent to the PDPA time
update step for further processing.
PDPA Time Update
The PDPA time update step is executed by using Equations 23 through
32 and the results of the PDPA measurement update step. The time and
measurement update steps are executed sequentially for every PMU
measurement set received between two
consecutive RTU measurements. The
execution of the proposed PDPA time
and measurement update steps is fast
enough so that one cycle of the process completes before the arrival of the
next PMU measurement set. On the
arrival of the next RTU measurement
or upon receiving a request from the
CDPA, the updated state and the state
error covariance values are sent to the
CDPA measurement update step.
CDPA Measurement Update
The CDPA execution steps are as
follows:
2n
∑ γ Pi,k |k−1,
(35)
−
i =1
where
P
γ Pi ,k− |k−1 = γ V i ,k− |k−1
m
γP
T
θmi ,k− |k−1
. (36)
• Calculate the estimated value of the
innovation covariance,
1
=
PP
z z ,k− |k−1
2n
2n
∑γ
T
P
γP
i ,k− |k−1 i ,k− |k−1
i =1
− zˆ k− |k−1zˆ T
k− |k−1
+R
(37)
• Evaluate the cubature points using
the same process as followed in the
PDPA measurement update step.
• Propagate the cubature points
through the measurement function
against each measurement subvector
l, as defined in Equation 6, at instant
k−. The propagated cubature points
are given by
qC
γC
= hC
l (x i ,k− |k−1),
li ,k− |k−1
(39)
k
• Calculate the estimated value of the
cross covariance,
www.computer.org/intelligent
C
where γ li ,k− |k−1 is the propagated
cubature point subvector for the
IEEE INTELLIGENT SYSTEMS
Table 1. Error comparison for PSHDSE with respect to actual values.*
Method used
ith cubature point input and the lth
CDPA measurement subvector, and
hC
l is the CDPA measurement function14 against the lth measurement
subvector. The superscript C indicates the values corresponding to
the CDPA measurement update step.
The propagated cubature point subvector against the respective measurement subvector can be processed
individually because all the measurements and their cubature points are
assumed to be uncorrelated.
• Calculate the estimated value of the
measurement vector,
zˆ
k− |k−1
=
1
2n
2n
∑ γ Ci,
k− |k−1
,
(40)
i =1
where
γC
=
−
i ,k |k−1
γC
Vmi ,k− |k−1
γ CPr
m i ,k
−
|k−1
γC
Qrm i ,k− |k−1
γ CPF
m i ,k
γC
QF
−
m i ,k
|k−1
−
|k−1
.
(41)
• Calculate the estimated value of the
innovation covariance,
1
=
PC −
zz ,k |k−1 2n
2n
CT
,k− |k−1
∑ γ Ci,k |k−1γ i
−
i =1
zˆ T
− zˆ k− |k−1
k− |k−1
(42)
+ Rk .
• Calculate the estimated value of the
cross covariance,
1
PC −
=
xz ,k |k−1 2n
2n
∑x
T
*
γC
i ,k− |k−1 i ,k− |k−1
i =1
zˆ T
− xˆ k− |k−1
k− |k−1
(43)
.
On arrival of the fresh RTU measurement set, the estimated state and
its associated error covariance are
updated again using Equations 17
through 19 at the k+th instant. After
this step, one cycle of the PSHDSE
MAY/JUNE 2015
Maximum voltage
error (-p.u.)
Maximum angle
error (radian)
Average sum of
square error;
voltage (p.u.2)
Average sum of
square error;
angle (radian 2)
30-bus IEEE system
EKF
3.5 × 10 −02
4.2 × 10 −02
1.63 × 10 −05
2.41 × 10 −05
UKF
6.6 ×
7.4 ×
1.03 ×
10 −05
9.40 × 10 −05
CKF
3.0 × 10 −02
3.7 × 10 −02
1.07 × 10 −05
1.09 × 10 −05
10 −02
10 −02
246-bus Indian NRPG system
EKF
7.9 × 10 −02
8.8 × 10 −02
9.05 × 10 −05
10.24 × 10 −05
UKF
4.1 × 10 −02
4.2 × 10 −02
7.91 × 10 −05
8.14 × 10 −05
CKF
2.3 ×
2.8 ×
7.20 ×
6.08 × 10 −05
10 −02
10 −02
10 −05
* CKF= cubature Kalman filter; EKF = extended Kalman filter; NRPG = Northern Regional Power Grid; and UKF = unscented Kalman
filter.
execution completes, utilizing the
available PMU and RTU measurements. Executing the proposed
PSHDSE is fast enough that the whole
process completes before the arrival
of the next RTU measurement set.
To compare EKF- and UKF-based
methods with the CKF, we used a
similar procedure, except the time
and measurement update equations
were utilized corresponding to the
implemented KF. Details about the
EKF6 and UKF7,8 time and measurement update equations are well established in the literature.
Simulation Results
The methodology proposed in this
work has been demonstrated on the
30-bus IEEE system16 and the 246bus Indian Northern Regional Power
Grid (NRPG; http://docslide.us/documents/nrpg-datapdf.html). We developed a Matlab-, Simulink-, and
SimEvents-based application (www.
mathworks. in/products/simevents),
including software agents, to test
and verify our proposed CKF-based
PSHDSE approach. For comparison, we also executed the EKF- and
UKF-based PSHDSE approaches on
the 30-bus IEEE and 246-bus NRPG
systems. The actual values of the system states at various time instants are
obtained by running the load flow
repeatedly for the various operating
conditions by varying the loads ranwww.computer.org/intelligent
domly between ±5 percent and ±30
percent. Using different standard
deviation quantities for the various
types of measurements, the Gaussian
noise is added into the actual values
of the load-flow results to generate
the RTU and PMU measurement sets.
Because we tested the PSHDSE
methodology using simulated measurement data, data scanner modules aren’t
implemented in the present approach.
For the PSHDSE execution, we assume that RTU measurements refresh
every one second and PMU measurements refresh every 40 milliseconds.
The simulations are carried out on an
Intel Core-i7 3.4-GHz processor-based
computer with 4 Gbytes RAM.
30-Bus IEEE System
The PSHDSE was implemented on
the 30-bus IEEE system using the
RTU measurement sets for 100 simulation time-steps under various
loading conditions. Between the two
consecutive RTU measurement simulation time-steps, 25 simulation timesteps for the PMU measurements
were also processed.
For the 30-bus IEEE system, the
PMUs are considered at buses 6, 9,
and 12. The values of a and b, used
in the load-forecasting process, are
estimated as 0.778 and 0.52, respectively, using 200 Monte Carlo (MC)
simulations. Table 1 compares the estimation errors of the CKF-, UKF-,
57
MULTI-AGENTS
Table 2. Execution time comparison for PSHDSE with respect to actual values.
One PMU measurement set
processing time (milliseconds)
One PSHDSE cycle execution
time (seconds)
30-bus IEEE
system
246-bus Indian
NRPG system
30-bus IEEE
system
246-bus Indian
NRPG system
EKF
7.9
17.7
0.971
1.480
UKF
6.7
9.2
0.968
0.993
CKF
6.1
8.9
0.965
0.989
Method
used
THE AUTHORS
Ankush Sharma is currently working in the Power System Centre of Excellence (CoE) di-
vision of Tata Consultancy Services (TCS), Pune, India. His research interests are state
estimation, power system deregulation, smart grid technology, and IT applications in
power system. Sharma has a PhD in electrical engineering from the Indian Institute of
Technology Kanpur, India. Contact him at ankushsharma@ieee.org.
Suresh Chandra Srivastava is a professor in the Department of Electrical Engineering at
the Indian Institute of Technology Kanpur. His research interests include energy management systems, synchrophasor applications, power system security, stability, and technical issues in electricity markets. Srivastava has a PhD in electrical engineering from the
Indian Institute of Technology Delhi, India. He’s a fellow of Indian National Academy of
Engineering (INAE) India, Institution of Engineers (IE) India, and the Institution of Electronics and Telecommunication Engineers (IETE) India and a senior member of IEEE.
Contact him at scs@iitk.ac.in.
Saikat Chakrabarti is an associate professor in the Department of Electrical Engineer-
ing at the Indian Institute of Technology Kanpur, India. His research interests include
power system dynamics and stability, state estimation, and synchrophasor applications.
Chakrabarti has a PhD in electrical engineering from Memorial University of Newfoundland, Canada. Contact him at saikatc@iitk.ac.in.
and EKF-based state estimation results, as compared to the actual values of the states for the 30-bus IEEE
system. From Table 1, we can see that
PSHDSE using the CKF is more accurate compared to that with the EKF
and the UKF.
Table 2 compares the execution
time in processing one PMU measurement set and one cycle of PSHDSE
execution using the CKF, UKF, and
EKF approaches for the 30-bus IEEE
system. From the table, we can conclude that the three KF approaches
will finish processing one PMU measurement set well before the arrival of
the next set. Furthermore, the execution of one PSHDSE cycle completes
before the arrival of the next RTU
measurement set. Out of the three KF
approaches, the CKF is the fastest:
its cubature points and weights are
58
independent of the nonlinear measurement function, hence they can be
calculated and stored offline.
246-Bus Indian NRPG System
Similar to the 30-bus IEEE system,
the 246-bus Indian NRPG system
also processed two consecutive RTU
measurement simulation time steps
and 25 PMU measurement simulation time steps. For the 246-bus Indian NRPG system, the PMUs are
considered at 30 buses.
Similar to the previous case, the
values of a and b, used for state forecasting, are estimated as 0.83 and
0.64, respectively, via 200 MC simulations. Table 1 compares the EKF,
UKF, and CKF PSHDSE results with
actual values; similar to the 30-bus
IEEE system, PSHDSE using the CKF
is the most accurate.
www.computer.org/intelligent
Table 2 compares the execution
time for processing one PMU measurement set and one cycle of the
PSHDSE execution using the CKF,
UKF, and EKF approaches for the
246-bus NRPG system. In comparing
the execution of one cycle, we can see
that the EKF-based PSHDSE doesn’t
complete its execution before the arrival of the next RTU measurement
set—this is because the EKF needs an
extra step to compute the derivatives
of the measurement functions while
processing the KF. The UKF and CKF
approaches execute one PSHDSE cycle within a second. Because it’s the
fastest of the three KF approaches,
the CKF is a better choice for multiagent-based PSHDSE.
T
he proposed method is generic
and can be applied to larger
power system networks. Hence, it
could be adopted in various control
centers to gain the faster situational
awareness required in smart grid implementations. The proposed method
can also be extended to estimate the
dynamic states of the power system
network under certain disturbances,
such as power swing and voltage collapse.
Acknowledgments
The Department of Science and Technology,
New Delhi, India, provided partial financial
support under project DST/EE/20100258.
References
1. F.C. Schweppe and J. Wildes, “Power
System Static State Estimation, Part I:
Exact Model,” IEEE Trans. Power Apparatus and System, vol. PAS-89, no. 1,
1970, pp. 120–125.
2. M. Wooldridge, “Agent-Based Software
Engineering,” IEE Proc. Software Eng.,
vol. 144, no. 1, 1997, pp. 26–37.
3. N.R. Jennings, “On Agent-Based
Software Engineering,” Artificial
IEEE INTELLIGENT SYSTEMS
Intelligence, vol. 117, no. 2, 2000,
pp. 277–296.
4. M.M. Nordman and M. Lehtonen,
“Distributed Agent-Based State Estimation for Electrical Distribution Networks,” IEEE Trans. Power Systems,
vol. 20, no. 2, 2005, pp. 652–658.
5. P.H. Nguyen and W.L. Kling, “Distributed State Estimation for Multi-Agent
Based Active Distribution Networks,”
IEEE Power and Energy Soc. General
Meeting, 2010, pp. 1–7.
6. J.K. Mandal, A.K. Sinha, and L. Roy,
“Incorporating Nonlinearities of
Measurement Function in Power System
Dynamic State Estimation,” IEE Generation, Transmission & Distribution,
vol. 142, no. 3, 1995, pp. 289–296.
7. G. Valverde and V. Terzija, “Unscented
Kalman Filter for Power System Dynamic State Estimation,” IET Generation, Transmission & Distribution, vol.
5, no. 1, 2011, pp. 29–37.
8. W. Shaobu, G. Wenzhong, and A.P.S.
Meliopoulos, “An Alternative Method
for Power System Dynamic State Estimation Based on Unscented Transform,” IEEE Trans. Power Systems,
vol. 27, no. 2, 2012, pp. 942–950.
9. G. Valverde et al., “A Constrained Formulation for Hybrid State Estimation,”
IEEE Trans. Power Systems, vol. 26,
no. 3, 2011, pp. 1102–1109.
10. I. Arasaratnam and S. Haykin, “Cubature Kalman Filters,” IEEE Trans.
Automatic Control, vol. 54, no. 6,
2009, pp. 1254–1269.
11. B. Jia, M. Xin, and Y. Cheng, “High-Degree Cubature Kalman Filter,” Automatica, vol. 49, no. 2, 2013, pp. 510–518.
12. I. Arasaratnam, S. Haykin, and T.R.
Hurd, “Cubature Kalman Filtering for
Continuous-Discrete Systems: Theory
and Simulations,” IEEE Trans. Signal
Processing, vol. 58, no. 10, 2010,
pp. 4977–4993.
13. A.M. Leite da Silva, M.B. Do Coutto
Filho, and J.F. De Queiroz, “State Forecasting in Electric Power Systems,” IEE
Proc. Generation, Transmission and
Distribution C, vol. 130, no. 5, 1983,
pp. 237–244.
14. A. Abur and A.G. Exposito, Power
System State Estimation: Theory and
Implementation, Mercel Dekker, 2004,
ch. 2, pp. 9–36.
15. A.H. Stroud, Approximate Calculation
of Multiple Integrals, Prentice Hall,
1971.
16. S. Chakrabarti, and E. Kyriakides, “Optimal Placement of Phasor Measurement
Units for Power System Observability,”
IEEE Trans. Power Systems, vol. 23,
no. 3, 2008, pp. 1433–1440.
Selected CS articles and columns
are also available for free at
http://ComputingNow.computer.org.
EXECUTIVE STAFF
PURPOSE: The IEEE Computer Society is the world’s largest association of computing
professionals and is the leading provider of technical information in the field.
MEMBERSHIP: Members receive the monthly magazine Computer, discounts, and
opportunities to serve (all activities are led by volunteer members). Membership is open to
Executive Director: Angela R. Burgess; Director, Governance & Associate Executive
Director: Anne Marie Kelly; Director, Finance & Accounting: John G. Miller; Director,
Information Technology Services: Ray Kahn; Director, Membership: Eric Berkowitz;
Director, Products & Services: Evan M. Butterfield; Director, Sales & Marketing: Chris
Jensen
all IEEE members, affiliate society members, and others interested in the computer field.
COMPUTER SOCIETY OFFICES
COMPUTER SOCIETY WEBSITE: www.computer.org
Washington, D.C.: 2001 L St., Ste. 700, Washington, D.C. 20036-4928
Phone: +1 202 371 0101 • Fax: +1 202 728 9614 • Email: hq.ofc@computer.org
Los Alamitos: 10662 Los Vaqueros Circle, Los Alamitos, CA 90720
Phone: +1 714 821 8380 • Email: help@computer.org
Next Board Meeting: 1–5 June 2015, Atlanta, GA, USA
EXECUTIVE COMMITTEE
President: Thomas M. Conte
President-Elect: Roger U. Fujii; Past President: Dejan S. Milojicic; Secretary: Cecilia Metra;
Treasurer, 2nd VP: David S. Ebert; 1st VP, Member & Geographic Activities: Elizabeth
L. Burd; VP, Publications: Jean-Luc Gaudiot; VP, Professional & Educational Activities:
Charlene (Chuck) Walrad; VP, Standards Activities: Don Wright; VP, Technical &
Conference Activities: Phillip A. Laplante; 2015–2016 IEEE Director & Delegate Division
VIII: John W. Walz; 2014–2015 IEEE Director & Delegate Division V: Susan K. (Kathy) Land;
2015 IEEE Director-Elect & Delegate Division V: Harold Javid
BOARD OF GOVERNORS
Term Expiring 2015: Ann DeMarle, Cecilia Metra, Nita Patel, Diomidis Spinellis, Phillip
A. Laplante, Jean-Luc Gaudiot, Stefano Zanero
Term Expriring 2016: David A. Bader, Pierre Bourque, Dennis J. Frailey, Jill I. Gostin,
Atsuhiro Goto, Rob Reilly, Christina M. Schober
MEMBERSHIP & PUBLICATION ORDERS
Phone: +1 800 272 6657 • Fax: +1 714 821 4641 • Email: help@computer.org
Asia/Pacific: Watanabe Building, 1-4-2 Minami-Aoyama, Minato-ku, Tokyo 107-0062,
Japan • Phone: +81 3 3408 3118 • Fax: +81 3 3408 3553 • Email: tokyo.ofc@computer.org
IEEE BOARD OF DIRECTORS
President & CEO: Howard E. Michel; President-Elect: Barry L. Shoop; Past President:
J. Roberto de Marca; Director & Secretary: Parviz Famouri; Director & Treasurer:
Jerry Hudgins; Director & President, IEEE-USA: James A. Jefferies; Director & President,
Standards Association: Bruce P. Kraemer; Director & VP, Educational Activities: Saurabh
Sinha; Director & VP, Membership and Geographic Activities: Wai-Choong Wong;
Director & VP, Publication Services and Products: Sheila Hemami; Director & VP,
Technical Activities: Vincenzo Piuri; Director & Delegate Division V: Susan K. (Kathy)
Land; Director & Delegate Division VIII: John W. Walz
Term Expiring 2017: David Lomet, Ming C. Lin, Gregory T. Byrd, Alfredo Benso,
Forrest Shull, Fabrizio Lombardi, Hausi A. Muller
MAY/JUNE 2015
revised 27 Jan. 2015
www.computer.org/intelligent
59
A Multi-AgentBased Power System
Hybrid Dynamic
State Estimator
Ankush Sharma, Suresh Chandra Srivastava, and Saikat Chakrabarti, Indian Institute
of Technology Kanpur
C
onventional power system state estimation (PSSE) is performed with the
help of static state estimation (SSE) tools such as the weighted least squares
(WLS) approach.1 These tools use the measurements received from remote
For fast estimation
terminal units (RTUs)—typically, at an interval of 1 to 5 seconds—through a
of power system
state, a multi-agentbased power system
hybrid dynamic state
estimator uses field
measurements from
remote terminal
units and phasor
measurement units.
52
supervisory control and data acquisition
(SCADA) system. With the advent of phasor
measurement units (PMUs), the measurement data reporting rate has increased up
to 50 frames per second for 50-Hz systems
and up to 60 frames per second for 60-Hz
systems. But SSE execution takes approximately 30 to 60 seconds for a large system, indicating the updated measurements
received from RTUs and PMUs aren’t being utilized, which means that SSE results
lag behind actual system states. The development of dynamic state estimation (DSE)
tools could solve state estimation problems,
enabling available measurements to be utilized when estimating system states.
Despite the introduction of agent-based
software engineering,2,3 few efforts have been
made to apply a multi-agent-based approach
to solving the power system SSE problem.4,5
To our knowledge, nothing in the literature
describes solving the power system dynamic
state estimation (PSDSE) problem using
multi-agent-based software engineering.
However, quite a few efforts have focused on solving this problem by using
Kalman filters. One approach6 used the
extended Kalman filter (EKF) to estimate
power system states. The EKF uses firstorder approximation of the Taylor series
to solve the measurement function, causing
its state estimation results to deviate from
actual values, thereby rendering the EKF
not quite suitable for dynamic state estimation of nonlinear systems such as the power
system. Other researchers7–9 used the unscented Kalman filter (UKF) to estimate the
power system state, but because the UKF’s
performance deteriorates with the increase
in the number of state variables,10 it also
isn’t suitable for estimating the state in
large power systems. Recently, the cubature
Kalman filter (CKF)10–12 has demonstrated
potential benefits, such as accuracy and stability for the large state vector, over other
Kalman filtering techniques.
The CKF approach was originally developed to track aircraft trajectories. 10
1541-1672/15/$31.00 © 2015 IEEE
Published by the IEEE Computer Society
IEEE INTELLIGENT SYSTEMS
To make it suitable for power system hybrid dynamic state estimation
(PSHDSE), we developed a modified
CKF approach. The CKF doesn’t require Taylor series approximation
of the nonlinear function and the
Jacobian during its execution, and
unlike with the UKF, its performance
doesn’t deteriorate with the increase
in the size of the state variable vector.10 We developed a software-based
multi-agent model to scan and process the PMU and the conventional
RTU measurements separately; this
model then combines the CKF results
in every Kalman filter cycle to estimate system states. To speed up the
CKF processing, we used a factorization approach that factorizes the
large measurement vector into subvectors. The CKF is processed in parallel using the subvectors to estimate
the complete power system’s various
states.
PSHDSE formulation, state vector xk
comprises the bus voltage (Vk) and
the angle (pk) state subvectors. The
measurement vector zk comprises the
subvectors of voltage magnitude measurements (Vmk), voltage angle measurements (pmk), real power injection
measurements (Prmk), reactive power
injection measurements (Qrmk), real
power flow measurements (PFmk), and
reactive power flow measurements
(QFmk), received from the RTUs and
PMUs at the instant k, which are
given by
xk = [Vk | pk]T
(5)
zk = [Vmk | pmk | Prmk | Qrmk | PFmk | QFmk]T,
xk+1 = f(xk) + wk
(1)
zk+1 = h(xk+1) + vk+1
(2)
wk = N(0, Qk)
(3)
f(xk−1) = ak−1 + bk−1
(7)
vk+1 = N(0, Rk+1),
(4)
ak−1 = α k−1x k−1 + (1 − α k−1)x k−−1
(8)
where, xk is the state vector at the kth
instant; zk is the measurement vector
at the kth instant; wk is Gaussian process noise with zero mean; Qk is the
process noise error covariance; and
vk is the Gaussian measurement noise
with zero mean. In addition, Rk is the
measurement noise error covariance,
and f(•): ℜn → ℜn and h(•): ℜn → ℜm
are the nonlinear functions for a state
space of size n and the measurement
space of size m, respectively. For the
MAY/JUNE 2015
Cubature Kalman Filter
Under the Bayesian estimation approach, using the known posterior
probability density value at the k − 1th
instant, the prior probability density
at the next instant k− can be calculated using the Chapman-Kolmogorov
equation, given by10
p(x k | z1:k−1) =
∫ p(xk | xk−1)p(xk−1 | z1:k−1)dxk−1
bk−1 = ak−1 (ak−1 − ak−2) + (1 − ak−1)bk−2, (9)
where ak−1 and ak−1 are the parameters at instant k − 1, containing
values between 0 and 1; x k−−1 is the
predicted state vector at the instant
k − 1; ak−1 and bk−1 are the vectors defined by Equations 8 and 9 at instant
k − 1. To define the measurement
function h(•) for the power system,
we use standard bus power injection
equations (real and reactive), and the
www.computer.org/intelligent
(10)
ℜ
p(x k−1| z1:k−1) = N (xˆ k−1|k−1, Px ,k−1|k−1),
(11)
(6)
where subscript m indicates the size
of the corresponding measurement
samples at the kth instant.
We use the state forecasting tool13 in
our proposed approach to model the
state transition function. This function
accommodates variations in state due
to changes in system parameters, such
as load variations from one time step
to another time step. Our proposed
approach uses Holt’s two-parameter
linear exponential smoothing technique13 to forecast states. Using this
technique, we can define the state
transition function f(•) as
Power System
Dynamic Model
We can represent a nonlinear dynamical power system for state estimation
in a discrete form with the help of the
following equations:
line power flow equations (real and
reactive).14
where N (xˆ k−1|k−1, Px ,k−1|k−1) is the standard normal distribution for the
Gaussian density, with xˆ k−1|k−1 as the
estimated value (mean) and Px,k−1|k−1
as the covariance at k − 1th instant.
We can calculate the value of the
probability density p(xk | xk−1) by using Equation 1. The minus (−) and
plus (+) signs in the superscript indicate the prior and posterior values, respectively. The CKF is implemented in
two steps, as follows.10
Time Update
In the first step, we calculate the estimated value of the mean xˆ k|k−1 and the
error covariance Px,k|k−1 of the states
by using Equations 1 and 10, which
are given by
∫
xˆ k|k−1 = f (x k−1)p(x k−1| z1:k−1)dx k−1 (12)
ℜ
Px ,k|k−1 =
∫ f (x k−1)f T (x k−1)p(x k−1 | z1:k−1)dx k−1
ℜ
− xˆ k|k−1xˆ kT|k−1 + Qk−1,
(13)
where p(xk−1 | z1:k−1) is the posterior probability density, provided by the measurement update step at the k − 1th instant.
53
MULTI-AGENTS
Dynamic state estimation coordinator
Multi-Agent System
Agent for conventional data
processing
Agent for PMU data
processing
Data transfer agent
Data transfer agent
PMU data scanner
Conventional data scanner
Figure 1. Proposed multi-agent model for the power system hybrid dynamic state
estimation (PSHDSE) formulation. The agents are intelligent software modules
designed to scan and process remote terminal units (RTU) and phasor measurement
units (PMU) in parallel, and then exchange the results and status in a time-bound
manner to coordinate final state estimates.
Measurement Update
In the second step, we calculate the
predicted measurement zˆ k|k−1, its associated innovation covariance Pzz,k|k−1,
and the cross covariance P xz,k|k−1 by
using the following equations:
zˆ k|k−1 =
∫ h(xk)p(xk | z1:k−1)dxk−1
by using the third-degree sphericalradical cubature rule,15 given as
IN (f ) =
≈
∫
ξi =
ℜ
− zˆ k|k−1zˆ kT|k−1 + R k
Pxz ,k|k−1 =
∫ xkh T (xk)p(xk | z1:k−1)dxk
ℜ
− xˆ k|k−1zˆ kT|k−1.
(16)
When the new measurements are received, the estimated mean of the state
and its associated error covariance
are updated at the k+th instant by using the standard Kalman filtering approach, given by
xˆ k|k = xˆ k|k−1 + Kk (z k − zˆ k|k−1)
Pk|k =
Pk|k−1 − K k Pzz ,k|k−1K kT
−1
K k = Pxz ,k|k−1Pzz
,k|k−1.
(17)
(18)
(19)
To estimate the system states, the
solution of the integral terms in Equations 10 through 16 is approximated
54
∑ ω if (
i =1
ℜ
h(x k)h T (x k)p(x k | z1:k−1)dx k (15)
(20)
2n
(14)
Pzz ,k|k−1 =
∫ f (x)N (x | µ, Σ)dx
ℜ
n [e ]i , ω i =
Σξi + µ)
1
,
2n
(21)
where S is the error covariance matrix, m is the mean value, e is the
unity matrix, and w is the weight.
The {x, w} set forms the cubature
points. The details for the CKF appear elsewhere.10
Power System Hybrid
Dynamic State Estimation
In the power system, RTU data is
typically reported every 1 to 5 seconds, whereas PMU data is reported
every 20 to 200 milliseconds for
50-Hz systems and 17 to 167 milliseconds for the 60-Hz system. The
role of the multi-agents in our proposed approach is to process these
measurements separately in a collaborative manner for PSHDSE and
then integrate PSHDSE results to finally estimate the overall power system’s states.
www.computer.org/intelligent
The multi-agent system comprises multiple interacting autonomous agents
that help manage the complex system
by distributing the system execution
tasks among themselves and sharing
information interactively. The agents
in the proposed system are intelligent
software modules designed to scan and
process RTU and PMU measurements
in parallel, and then exchange the results and status in a time-bound manner to coordinate final state estimates
(see Figure 1).
The PMU (RTU) data scanner
agent periodically checks the arrival
of new data from the field PMUs
(RTUs). The data transfer agent
(DTA) does a sanity check on newly
arrived data and acts as a first-level
filter to identify and remove outliers from the measurement set. The
other two agents—the PMU data
processing agent (PDPA) and the
conventional data processing agent
(CDPA)—form part of the dynamic
state estimation coordinator (DSEC)
and play an important role in executing our PSHDSE approach.
Specifically, the PDPA receives
PMU data from the DTA at regular intervals and runs the CKF. The
CKF uses the latest available state estimates and the latest available PMU
measurements to execute its measurement update step. In our proposed
approach, only the voltage magnitude
and the angle measurements from the
PMUs are considered for dynamic
state estimation. This keeps the measurement equation for the CKF linear
in nature, as given by
Vmk
θ mk
V
= e 0 k + vk , (22)
0 e θk
where e is the unity subvector and
0 is the zero subvector. It’s assumed
that the measurements provided by
various measurement devices are
IEEE INTELLIGENT SYSTEMS
Scan data receipt
Scan for data receipt
request
State forecasting
data
response
Scan new data
New data
received
PMU measurment data
PMU agent state
estimates
State forecasting
data
State
forecasting
data generated
Data received
Send forecasted state
request to conventional
data processing agent
Time update
Run state forecasting
Measurement update
CKF
Scan for data
measurement
transfer request from
update Execution
conventional data
State
estimate
data
processing agent
State estimate data transfer
Time update
Send request to
PMU data
processing agent
CKF time update
Execution
State initialization
State estimates
from PMU data
processing
agent received
State Initialization
CKF time update
Execution
Measurement update
CKF
measurement
update execution
New data
received
Scan new data
Final state estimate data
Conventional measurment data
Updated state estimate data transfer request
(a)
(b)
Figure 2. State transition diagram: (a) the PMU data processing agent (PDPA) and (b) the conventional data processing
agent (CDPA). CKF stands for cubature Kalman filter. The PDPA and the CDPA agents process the respective PMU and RTU
measurements separately in various stages and regularly exchange the state forecasting data in a time-bound manner to
coordinate the final state estimates.
uncorrelated. Hence, Equation 22
can be divided into two independent
equations for parallel and fast processing. Figure 2a shows the PDPA’s
state transition diagram.
The CDPA execution process is
divided into two parts. In the first,
on arrival of the new RTU measurements, the CKF is executed to estimate overall system states, utilizing
the latest RTU measurements and
PDPA state estimates. In the second
part, the CDPA forecasts the states
by utilizing the previous state estimation results. The second part of
the execution process repeats for two
consecutive RTU measurement refreshes—ultimately, the step is executed after receiving the request from
the PDPA to provide the state forecasting data. Figure 2b shows the
CDPA’s state transition diagram.
Kalman Filtering Approach
for PSHDSE
The existing CKF process10 is modified
in our work specifically for PSHDSE
execution. In our proposed approach,
to estimate the 2m − 1 states of an
m-bus power system using the CKF,
4m − 2 cubature points are required to
MAY/JUNE 2015
be evaluated per system state per iteration. For a bigger power system, due
to the large size of the cubature point
matrix (2m − 1 × 4m − 2), running
the CKF without using a factorization approach is a time-consuming job.
Hence, the state and measurement vectors are partitioned into subvectors, as
in Equations 5 and 6, and the CKF is
executed for the subvectors in parallel
to save time. The CKF-based PSHDSE
execution process assumes a flat start
with initial voltage magnitudes and
angles at all the power system’s buses
assumed to be at 1 per unit (p.u.) and
zero radians, respectively.
CDPA Time Update
The CDPA execution steps are as
follows:
• Calculate the cubature points xi for
the state vectors,
ξi =
n [ e ]i , i
−
n [ e ]i − n , i
= 1 n
(23)
= n + 1 2 n ,
where n is the size of the state vector, e is a unity matrix of size n ×
n, and xi is the ith cubature point
vector.
www.computer.org/intelligent
• Evaluate the cubature points at the
instant k−, given as
x
q
i ,k− |k−1
=
Pk−1|k−1 ξ i + xˆ k−1|k−1,
(24)
where x qi ,k− |k−1 is the evaluated cubature point vector for the ith
cubature point input. The field
measurements and, hence their cubature points, are assumed to be
uncorrelated. Therefore, utilizing
the Cholesky factorization technique, the square root of Pk−1|k−1 can
be divided into four independent
submatrices, as shown in Equation
26. These submatrices are utilized
by dividing the state equation into
two independent equations containing voltage magnitude (V) and the
voltage angle (q) subvectors separately for fast processing, as shown
in Equations 27 and 28:
q
x Vi ,k− |k−1
q
x θi ,k− |k−1
=
A
B
C
D
×
ξ V ,i
ξ θ ,i
, (25)
xˆ V ,k−1|k−1
xˆ θ ,k−1|k−1
+
55
MULTI-AGENTS
where
A
Pk−1|k−1 =
B
C
D
.
(26)
Hence,
xq − = [A × ξ
V ,i
Vi ,k |k−1
+ B × ξ θ ,i
]
(27)
+ xˆ V ,k−1|k−1
and
xq − = [C × ξ
V ,i
θi ,k |k−1
+ D × ξ θ ,i
]
(28)
+ xˆ θ ,k−1|k−1 .
• Propagate the cubature points
through the state function f(•), defined in Equations 7 through 9,
against each state subvector j (V, q)
at instant k−. The propagated cubature point vector x*ji ,k− |k−1 for the
ith cubature point input and the jth
state subvector are given by
x*ji ,k− |k−1 = f (x
q
).
ji ,k− |k−1
1
2n
2n
∑x
*
,
i ,k− |k−1
(30)
i =1
where
x*i ,k− |k−1 = x*Vi ,k− |k−1 x*θi ,k− |k−1 .
(31)
• Calculate the predicted state error
covariance, given as
Pk− |k−1 =
1
2n
−
x*i ,k− |k−1x *T −
i ,k |k−1
i =1
xˆ k− |k−1xˆ kT− |k−1
(32)
+ Q k−1.
On getting the state forecasting request from the PDPA, the state vector
and the state error covariance estimated in this step are sent to the PDPA’s measurement update step.
PDPA Measurement Update
The PDPA execution steps are as
follows:
56
qP
i ,k− |k−1
Pk− |k−1 ξ i + xˆ k− |k−1,
=
(33)
where x qP
is the evaluated cubai ,k− |k−1
ture point vector for the ith cubature point input, and the superscript
P indicates the evaluated cubature
point vector under the PDPA measurement update step. The cubature
points in Equation 33 are evaluated by dividing the equation into
subvectors, in the same manner as
Equations 25 through 28.
• Propagate the cubature points
through the measurement function
against each measurement subvector l (Vm, pm) at instant k−,
γ liP,k− |k−1 = h lP (x qP
),
i ,k− |k−1
(34)
P
where, γ li ,k− |k−1 is the propagated
cubature point subvector for the ith
cubature point input, and the lth
measurement subvector h lP is the
PDPA measurement function, as
defined in Equation 22.
• Calculate the estimated value of
the measurement vector at the k− th
instant,
zˆ
2n
∑
x
(29)
• Calculate the estimated value of the
state vector, given as
xˆ k− |k−1 =
• Evaluate the cubature points utilizing the updated state estimates, the
state error covariance values provided by the time update step, and
the cubature points calculated in
Equation 23 as
k− |k−1
=
1
2n
1
PP −
=
xz ,k |k−1 2n
2n
∑x
T
*
γP
i ,k− |k−1 i ,k− |k−1
i =1
− xˆ k− |k−1zˆ T
(38)
.
k− |k−1
On arrival of fresh PMU measurements, the estimated state value and
its associated error covariance are
updated using Equations 17 through
19 at the k+th instant. These updated
values are then sent to the PDPA time
update step for further processing.
PDPA Time Update
The PDPA time update step is executed by using Equations 23 through
32 and the results of the PDPA measurement update step. The time and
measurement update steps are executed sequentially for every PMU
measurement set received between two
consecutive RTU measurements. The
execution of the proposed PDPA time
and measurement update steps is fast
enough so that one cycle of the process completes before the arrival of the
next PMU measurement set. On the
arrival of the next RTU measurement
or upon receiving a request from the
CDPA, the updated state and the state
error covariance values are sent to the
CDPA measurement update step.
CDPA Measurement Update
The CDPA execution steps are as
follows:
2n
∑ γ Pi,k |k−1,
(35)
−
i =1
where
P
γ Pi ,k− |k−1 = γ V i ,k− |k−1
m
γP
T
θmi ,k− |k−1
. (36)
• Calculate the estimated value of the
innovation covariance,
1
=
PP
z z ,k− |k−1
2n
2n
∑γ
T
P
γP
i ,k− |k−1 i ,k− |k−1
i =1
− zˆ k− |k−1zˆ T
k− |k−1
+R
(37)
• Evaluate the cubature points using
the same process as followed in the
PDPA measurement update step.
• Propagate the cubature points
through the measurement function
against each measurement subvector
l, as defined in Equation 6, at instant
k−. The propagated cubature points
are given by
qC
γC
= hC
l (x i ,k− |k−1),
li ,k− |k−1
(39)
k
• Calculate the estimated value of the
cross covariance,
www.computer.org/intelligent
C
where γ li ,k− |k−1 is the propagated
cubature point subvector for the
IEEE INTELLIGENT SYSTEMS
Table 1. Error comparison for PSHDSE with respect to actual values.*
Method used
ith cubature point input and the lth
CDPA measurement subvector, and
hC
l is the CDPA measurement function14 against the lth measurement
subvector. The superscript C indicates the values corresponding to
the CDPA measurement update step.
The propagated cubature point subvector against the respective measurement subvector can be processed
individually because all the measurements and their cubature points are
assumed to be uncorrelated.
• Calculate the estimated value of the
measurement vector,
zˆ
k− |k−1
=
1
2n
2n
∑ γ Ci,
k− |k−1
,
(40)
i =1
where
γC
=
−
i ,k |k−1
γC
Vmi ,k− |k−1
γ CPr
m i ,k
−
|k−1
γC
Qrm i ,k− |k−1
γ CPF
m i ,k
γC
QF
−
m i ,k
|k−1
−
|k−1
.
(41)
• Calculate the estimated value of the
innovation covariance,
1
=
PC −
zz ,k |k−1 2n
2n
CT
,k− |k−1
∑ γ Ci,k |k−1γ i
−
i =1
zˆ T
− zˆ k− |k−1
k− |k−1
(42)
+ Rk .
• Calculate the estimated value of the
cross covariance,
1
PC −
=
xz ,k |k−1 2n
2n
∑x
T
*
γC
i ,k− |k−1 i ,k− |k−1
i =1
zˆ T
− xˆ k− |k−1
k− |k−1
(43)
.
On arrival of the fresh RTU measurement set, the estimated state and
its associated error covariance are
updated again using Equations 17
through 19 at the k+th instant. After
this step, one cycle of the PSHDSE
MAY/JUNE 2015
Maximum voltage
error (-p.u.)
Maximum angle
error (radian)
Average sum of
square error;
voltage (p.u.2)
Average sum of
square error;
angle (radian 2)
30-bus IEEE system
EKF
3.5 × 10 −02
4.2 × 10 −02
1.63 × 10 −05
2.41 × 10 −05
UKF
6.6 ×
7.4 ×
1.03 ×
10 −05
9.40 × 10 −05
CKF
3.0 × 10 −02
3.7 × 10 −02
1.07 × 10 −05
1.09 × 10 −05
10 −02
10 −02
246-bus Indian NRPG system
EKF
7.9 × 10 −02
8.8 × 10 −02
9.05 × 10 −05
10.24 × 10 −05
UKF
4.1 × 10 −02
4.2 × 10 −02
7.91 × 10 −05
8.14 × 10 −05
CKF
2.3 ×
2.8 ×
7.20 ×
6.08 × 10 −05
10 −02
10 −02
10 −05
* CKF= cubature Kalman filter; EKF = extended Kalman filter; NRPG = Northern Regional Power Grid; and UKF = unscented Kalman
filter.
execution completes, utilizing the
available PMU and RTU measurements. Executing the proposed
PSHDSE is fast enough that the whole
process completes before the arrival
of the next RTU measurement set.
To compare EKF- and UKF-based
methods with the CKF, we used a
similar procedure, except the time
and measurement update equations
were utilized corresponding to the
implemented KF. Details about the
EKF6 and UKF7,8 time and measurement update equations are well established in the literature.
Simulation Results
The methodology proposed in this
work has been demonstrated on the
30-bus IEEE system16 and the 246bus Indian Northern Regional Power
Grid (NRPG; http://docslide.us/documents/nrpg-datapdf.html). We developed a Matlab-, Simulink-, and
SimEvents-based application (www.
mathworks. in/products/simevents),
including software agents, to test
and verify our proposed CKF-based
PSHDSE approach. For comparison, we also executed the EKF- and
UKF-based PSHDSE approaches on
the 30-bus IEEE and 246-bus NRPG
systems. The actual values of the system states at various time instants are
obtained by running the load flow
repeatedly for the various operating
conditions by varying the loads ranwww.computer.org/intelligent
domly between ±5 percent and ±30
percent. Using different standard
deviation quantities for the various
types of measurements, the Gaussian
noise is added into the actual values
of the load-flow results to generate
the RTU and PMU measurement sets.
Because we tested the PSHDSE
methodology using simulated measurement data, data scanner modules aren’t
implemented in the present approach.
For the PSHDSE execution, we assume that RTU measurements refresh
every one second and PMU measurements refresh every 40 milliseconds.
The simulations are carried out on an
Intel Core-i7 3.4-GHz processor-based
computer with 4 Gbytes RAM.
30-Bus IEEE System
The PSHDSE was implemented on
the 30-bus IEEE system using the
RTU measurement sets for 100 simulation time-steps under various
loading conditions. Between the two
consecutive RTU measurement simulation time-steps, 25 simulation timesteps for the PMU measurements
were also processed.
For the 30-bus IEEE system, the
PMUs are considered at buses 6, 9,
and 12. The values of a and b, used
in the load-forecasting process, are
estimated as 0.778 and 0.52, respectively, using 200 Monte Carlo (MC)
simulations. Table 1 compares the estimation errors of the CKF-, UKF-,
57
MULTI-AGENTS
Table 2. Execution time comparison for PSHDSE with respect to actual values.
One PMU measurement set
processing time (milliseconds)
One PSHDSE cycle execution
time (seconds)
30-bus IEEE
system
246-bus Indian
NRPG system
30-bus IEEE
system
246-bus Indian
NRPG system
EKF
7.9
17.7
0.971
1.480
UKF
6.7
9.2
0.968
0.993
CKF
6.1
8.9
0.965
0.989
Method
used
THE AUTHORS
Ankush Sharma is currently working in the Power System Centre of Excellence (CoE) di-
vision of Tata Consultancy Services (TCS), Pune, India. His research interests are state
estimation, power system deregulation, smart grid technology, and IT applications in
power system. Sharma has a PhD in electrical engineering from the Indian Institute of
Technology Kanpur, India. Contact him at ankushsharma@ieee.org.
Suresh Chandra Srivastava is a professor in the Department of Electrical Engineering at
the Indian Institute of Technology Kanpur. His research interests include energy management systems, synchrophasor applications, power system security, stability, and technical issues in electricity markets. Srivastava has a PhD in electrical engineering from the
Indian Institute of Technology Delhi, India. He’s a fellow of Indian National Academy of
Engineering (INAE) India, Institution of Engineers (IE) India, and the Institution of Electronics and Telecommunication Engineers (IETE) India and a senior member of IEEE.
Contact him at scs@iitk.ac.in.
Saikat Chakrabarti is an associate professor in the Department of Electrical Engineer-
ing at the Indian Institute of Technology Kanpur, India. His research interests include
power system dynamics and stability, state estimation, and synchrophasor applications.
Chakrabarti has a PhD in electrical engineering from Memorial University of Newfoundland, Canada. Contact him at saikatc@iitk.ac.in.
and EKF-based state estimation results, as compared to the actual values of the states for the 30-bus IEEE
system. From Table 1, we can see that
PSHDSE using the CKF is more accurate compared to that with the EKF
and the UKF.
Table 2 compares the execution
time in processing one PMU measurement set and one cycle of PSHDSE
execution using the CKF, UKF, and
EKF approaches for the 30-bus IEEE
system. From the table, we can conclude that the three KF approaches
will finish processing one PMU measurement set well before the arrival of
the next set. Furthermore, the execution of one PSHDSE cycle completes
before the arrival of the next RTU
measurement set. Out of the three KF
approaches, the CKF is the fastest:
its cubature points and weights are
58
independent of the nonlinear measurement function, hence they can be
calculated and stored offline.
246-Bus Indian NRPG System
Similar to the 30-bus IEEE system,
the 246-bus Indian NRPG system
also processed two consecutive RTU
measurement simulation time steps
and 25 PMU measurement simulation time steps. For the 246-bus Indian NRPG system, the PMUs are
considered at 30 buses.
Similar to the previous case, the
values of a and b, used for state forecasting, are estimated as 0.83 and
0.64, respectively, via 200 MC simulations. Table 1 compares the EKF,
UKF, and CKF PSHDSE results with
actual values; similar to the 30-bus
IEEE system, PSHDSE using the CKF
is the most accurate.
www.computer.org/intelligent
Table 2 compares the execution
time for processing one PMU measurement set and one cycle of the
PSHDSE execution using the CKF,
UKF, and EKF approaches for the
246-bus NRPG system. In comparing
the execution of one cycle, we can see
that the EKF-based PSHDSE doesn’t
complete its execution before the arrival of the next RTU measurement
set—this is because the EKF needs an
extra step to compute the derivatives
of the measurement functions while
processing the KF. The UKF and CKF
approaches execute one PSHDSE cycle within a second. Because it’s the
fastest of the three KF approaches,
the CKF is a better choice for multiagent-based PSHDSE.
T
he proposed method is generic
and can be applied to larger
power system networks. Hence, it
could be adopted in various control
centers to gain the faster situational
awareness required in smart grid implementations. The proposed method
can also be extended to estimate the
dynamic states of the power system
network under certain disturbances,
such as power swing and voltage collapse.
Acknowledgments
The Department of Science and Technology,
New Delhi, India, provided partial financial
support under project DST/EE/20100258.
References
1. F.C. Schweppe and J. Wildes, “Power
System Static State Estimation, Part I:
Exact Model,” IEEE Trans. Power Apparatus and System, vol. PAS-89, no. 1,
1970, pp. 120–125.
2. M. Wooldridge, “Agent-Based Software
Engineering,” IEE Proc. Software Eng.,
vol. 144, no. 1, 1997, pp. 26–37.
3. N.R. Jennings, “On Agent-Based
Software Engineering,” Artificial
IEEE INTELLIGENT SYSTEMS
Intelligence, vol. 117, no. 2, 2000,
pp. 277–296.
4. M.M. Nordman and M. Lehtonen,
“Distributed Agent-Based State Estimation for Electrical Distribution Networks,” IEEE Trans. Power Systems,
vol. 20, no. 2, 2005, pp. 652–658.
5. P.H. Nguyen and W.L. Kling, “Distributed State Estimation for Multi-Agent
Based Active Distribution Networks,”
IEEE Power and Energy Soc. General
Meeting, 2010, pp. 1–7.
6. J.K. Mandal, A.K. Sinha, and L. Roy,
“Incorporating Nonlinearities of
Measurement Function in Power System
Dynamic State Estimation,” IEE Generation, Transmission & Distribution,
vol. 142, no. 3, 1995, pp. 289–296.
7. G. Valverde and V. Terzija, “Unscented
Kalman Filter for Power System Dynamic State Estimation,” IET Generation, Transmission & Distribution, vol.
5, no. 1, 2011, pp. 29–37.
8. W. Shaobu, G. Wenzhong, and A.P.S.
Meliopoulos, “An Alternative Method
for Power System Dynamic State Estimation Based on Unscented Transform,” IEEE Trans. Power Systems,
vol. 27, no. 2, 2012, pp. 942–950.
9. G. Valverde et al., “A Constrained Formulation for Hybrid State Estimation,”
IEEE Trans. Power Systems, vol. 26,
no. 3, 2011, pp. 1102–1109.
10. I. Arasaratnam and S. Haykin, “Cubature Kalman Filters,” IEEE Trans.
Automatic Control, vol. 54, no. 6,
2009, pp. 1254–1269.
11. B. Jia, M. Xin, and Y. Cheng, “High-Degree Cubature Kalman Filter,” Automatica, vol. 49, no. 2, 2013, pp. 510–518.
12. I. Arasaratnam, S. Haykin, and T.R.
Hurd, “Cubature Kalman Filtering for
Continuous-Discrete Systems: Theory
and Simulations,” IEEE Trans. Signal
Processing, vol. 58, no. 10, 2010,
pp. 4977–4993.
13. A.M. Leite da Silva, M.B. Do Coutto
Filho, and J.F. De Queiroz, “State Forecasting in Electric Power Systems,” IEE
Proc. Generation, Transmission and
Distribution C, vol. 130, no. 5, 1983,
pp. 237–244.
14. A. Abur and A.G. Exposito, Power
System State Estimation: Theory and
Implementation, Mercel Dekker, 2004,
ch. 2, pp. 9–36.
15. A.H. Stroud, Approximate Calculation
of Multiple Integrals, Prentice Hall,
1971.
16. S. Chakrabarti, and E. Kyriakides, “Optimal Placement of Phasor Measurement
Units for Power System Observability,”
IEEE Trans. Power Systems, vol. 23,
no. 3, 2008, pp. 1433–1440.
Selected CS articles and columns
are also available for free at
http://ComputingNow.computer.org.
EXECUTIVE STAFF
PURPOSE: The IEEE Computer Society is the world’s largest association of computing
professionals and is the leading provider of technical information in the field.
MEMBERSHIP: Members receive the monthly magazine Computer, discounts, and
opportunities to serve (all activities are led by volunteer members). Membership is open to
Executive Director: Angela R. Burgess; Director, Governance & Associate Executive
Director: Anne Marie Kelly; Director, Finance & Accounting: John G. Miller; Director,
Information Technology Services: Ray Kahn; Director, Membership: Eric Berkowitz;
Director, Products & Services: Evan M. Butterfield; Director, Sales & Marketing: Chris
Jensen
all IEEE members, affiliate society members, and others interested in the computer field.
COMPUTER SOCIETY OFFICES
COMPUTER SOCIETY WEBSITE: www.computer.org
Washington, D.C.: 2001 L St., Ste. 700, Washington, D.C. 20036-4928
Phone: +1 202 371 0101 • Fax: +1 202 728 9614 • Email: hq.ofc@computer.org
Los Alamitos: 10662 Los Vaqueros Circle, Los Alamitos, CA 90720
Phone: +1 714 821 8380 • Email: help@computer.org
Next Board Meeting: 1–5 June 2015, Atlanta, GA, USA
EXECUTIVE COMMITTEE
President: Thomas M. Conte
President-Elect: Roger U. Fujii; Past President: Dejan S. Milojicic; Secretary: Cecilia Metra;
Treasurer, 2nd VP: David S. Ebert; 1st VP, Member & Geographic Activities: Elizabeth
L. Burd; VP, Publications: Jean-Luc Gaudiot; VP, Professional & Educational Activities:
Charlene (Chuck) Walrad; VP, Standards Activities: Don Wright; VP, Technical &
Conference Activities: Phillip A. Laplante; 2015–2016 IEEE Director & Delegate Division
VIII: John W. Walz; 2014–2015 IEEE Director & Delegate Division V: Susan K. (Kathy) Land;
2015 IEEE Director-Elect & Delegate Division V: Harold Javid
BOARD OF GOVERNORS
Term Expiring 2015: Ann DeMarle, Cecilia Metra, Nita Patel, Diomidis Spinellis, Phillip
A. Laplante, Jean-Luc Gaudiot, Stefano Zanero
Term Expriring 2016: David A. Bader, Pierre Bourque, Dennis J. Frailey, Jill I. Gostin,
Atsuhiro Goto, Rob Reilly, Christina M. Schober
MEMBERSHIP & PUBLICATION ORDERS
Phone: +1 800 272 6657 • Fax: +1 714 821 4641 • Email: help@computer.org
Asia/Pacific: Watanabe Building, 1-4-2 Minami-Aoyama, Minato-ku, Tokyo 107-0062,
Japan • Phone: +81 3 3408 3118 • Fax: +81 3 3408 3553 • Email: tokyo.ofc@computer.org
IEEE BOARD OF DIRECTORS
President & CEO: Howard E. Michel; President-Elect: Barry L. Shoop; Past President:
J. Roberto de Marca; Director & Secretary: Parviz Famouri; Director & Treasurer:
Jerry Hudgins; Director & President, IEEE-USA: James A. Jefferies; Director & President,
Standards Association: Bruce P. Kraemer; Director & VP, Educational Activities: Saurabh
Sinha; Director & VP, Membership and Geographic Activities: Wai-Choong Wong;
Director & VP, Publication Services and Products: Sheila Hemami; Director & VP,
Technical Activities: Vincenzo Piuri; Director & Delegate Division V: Susan K. (Kathy)
Land; Director & Delegate Division VIII: John W. Walz
Term Expiring 2017: David Lomet, Ming C. Lin, Gregory T. Byrd, Alfredo Benso,
Forrest Shull, Fabrizio Lombardi, Hausi A. Muller
MAY/JUNE 2015
revised 27 Jan. 2015
www.computer.org/intelligent
59