1. INTERNATIONAL JOURNAL OF ELECTRICAL ENGINEERING
International Journal of Electrical Engineering and Technology (IJEET), ISSN 0976 –
6545(Print), ISSN 0976 – 6553(Online) Volume 4, Issue 1, January- February (2013), © IAEME
& TECHNOLOGY (IJEET)
ISSN 0976 – 6545(Print)
ISSN 0976 – 6553(Online) IJEET
Volume 4, Issue 1, January- February (2013), pp. 153-161
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GRAPH THEORETIC APPROACH TO SOLVE MEASUREMENT
PLACEMENT PROBLEM FOR POWER SYSTEM STATE
ESTIMATION
R. J. Motiyani1, Dr. A. R. Chudasama2
1
Department of Electrical Engineering,
S N Patel Institute of Technology & Research, Bardoli, Surat, India
2
Department of Electrical Engineering,
The M S University of Baroda, Vadodara, India
ABSTRACT
In this paper a new method based on graph theoretic approach is proposed to solve
the measurement placement problem for power system state estimation. The developed
method allows measurement placement without iterative addition. The simulation study is
performed on IEEE 14 bus test system. The P-δ and Q-V observable concepts are used to
check network observability by triangular factorization of the gain matrix. The
measurement system configuration designed through the proposed method maintains
network observability and accomplishes accuracy and bad data processing requirements for
state estimator. The developed method can be used for measurement systems planning to
maintain overall system observable even under branch contingencies and loss of
measurements.
Keywords: Bad data processing, Measurement placement, Network observability, Network
topology processor, Static state estimator.
1. INTRODUCTION
Within the energy management system state estimation is a key function to derive a
real time network model by extracting information from a redundant data set consisting
telemetred static data items. The state of electrical power system is defined as the vector of
voltage magnitude and angle at all network buses [1]. Static state estimator is related to
conventional power flow calculations. However, the static state estimator is designed to
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2. International Journal of Electrical Engineering and Technology (IJEET), ISSN 0976 –
6545(Print), ISSN 0976 – 6553(Online) Volume 4, Issue 1, January- February (2013), © IAEME
handle the many uncertainties. Uncertainties arise because of meter and communication
errors, incomplete metering and unexpected system changes. These uncertainties make
difference between the usual power flow studies done in office and online state estimation
done as a part of energy management system [2].
The real-time modeling of a power network usually involves following procedure [3]:
• Data gathering
• Network topology processing
• Observability analysis
• State estimation (SE)
• Processing of bad data and
• Identification of network model
An observability test should be executed prior to performing the state estimation. A.
Monticelli et al. presented two algorithms; one for testing the observability of a network and
identifying the observable islands when the network is unobservable and the other for
selecting a minimal set of additional measurements to make the network observable [4]. The
network observability algorithm will check whether the currently available set of
measurement from the power system network provides sufficient information for
computational requirements of state estimation? If state of power system can be estimated
throughout by processing a given set of measurement sent by the measuring devices in the
system, then the network is said to be observable, otherwise it is said to be unobservable.
The post state estimation procedure involves identification and elimination of bad data from
the available set of measurement. Sum of squares of residuals method of bad data detection
and identification is presented in [5].
Selection of measurement system aims at attending to requirements such as
observability and reliability- taking in to account the associated monetary costs is discussed
in [6]. In the same paper best measurement system configuration for IEEE 30 bus system is
presented. An optimization algorithm suitable to choose the optimal number and positions of
the measurement devices for state estimation in modern electric distribution network is
discussed in [7].
2. STATE ESTIMATION
The state estimation is a mathematical procedure by which the state of electric power
system is extracted from a set of measurement. In standard SE, in order to relate
measurements and non linear equations, the following model is used:
z = h (x ) + e (1)
Where,
z = m×1 measurement vector
h(x) = m ×1 vector of non linear functions
x = 2n ×1 state vector
e = m×1 measurement error vector
n = Total number of buses in the system and
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3. International Journal of Electrical Engineering and Technology (IJEET), ISSN 0976 –
6545(Print), ISSN 0976 – 6553(Online) Volume 4, Issue 1, January- February (2013), © IAEME
m = Total number of measurements
The state estimator is a mathematical algorithm formulated to minimize the error between a
real time measurement and a calculated value of the measurement. The minimization
criterion often selected is the weighted sum of error squares of all the measurements. The
estimator favors accurate measurements over the less accurate ones by weighing the errors
with the measurement standard deviation (σj) [8].
2
 ej 
m
min J ( x ) = ∑   (2)
j =1  σ j 
The condition for optimality is obtained at a point when the gradient of J(x) is zero. From
weighted least square (WLS) method, the iterative equation can be obtained as follows:
(
∆x = H T R −1 H )
−1
( ( ))
H T R −1 z − h x k (3)
x k +1 = x k + ∆x (4)
Where,
 ∂ h1 (x ) ∂ h1 (x ) ∂ h1 (x ) 
 ∂x L
∂x2 ∂ x Ns 
 1

 ∂ h 2 (x ) ∂ h 2 (x ) ∂ h 2 (x ) 
∂ h (x ) L (5)
H = =  ∂ x1 ∂x2 ∂ x Ns 
∂x  
M M L M
 ∂ h m (x ) ∂ h m (x ) ∂ h m (x ) 
 L 
 ∂ x1 ∂x2 ∂ x Ns 
 1 
 σ 2 
 1 
 1 
W = R − 1
=  σ 2 
 2
 (6)
 L 
 1 
 σ 2 
 m 
Considering one of the bus as a reference, n-1 angles and n voltage magnitudes (for n bus
system) are to be calculated. The state estimation jacobian (H) always has 2n-1 columns and
large number of rows based on number of measurements made.
The gain matrix is defined as
G = H T R −1 H (7)
While the power system not only has Supervisory Control and Data Acquisition system
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4. International Journal of Electrical Engineering and Technology (IJEET), ISSN 0976 –
6545(Print), ISSN 0976 – 6553(Online) Volume 4, Issue 1, January- February (2013), © IAEME
(SCADA), but also has Phasor Measurement Units (PMUs) placement, the sub problem is
formed by PMU placement and SCADA measurements [9]. The state variables measured by
PMU are assumed true value and the known state variables are x1. The unknown state
variables are required to be estimated by reduced power system state estimation model.
Hence, equations (1), (3) ~ (5) and (7) can be rewritten as follows:
z = h (x 2 ) + e (8)
( T
∆x 2 = H 2 R −1 H 2 )−1 T
( ( ))
H 2 R −1 z − h x 2
k (9)
x 2 + 1 = x 2 + ∆x 2
k k
(10)
G 2 = H 2 R −1 H 2
T
(11)
∂h ( x 2 ) (12)
H2 =
∂x 2
In presence of conventional measurements, one important criterion for PMU placement is
the improvement in state estimator performance [10], [11]. For example frequently
encountered problem in state estimation is the large value of the condition number of gain
matrix. The PMU placement can be done in such a way that the condition number of the
gain matrix is reduced when PMU placements combined with the SCADA measurements.
3. METER PLACEMENT
State estimator uses a set of measurement consisting of bus injections, branch flows
and bus voltages collected through SCADA System [12]. If all the quantities are measured
as shown in the figure 1, then the possible measurements are 3n + 4b where, n is the total
number of network buses and b is the total number of network branches. The state
estimation jacobian will have 3n+4b number of rows.
Figure 1. The Per-Phase Representation of Transmission Line- Showing Possible
Measurements
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5. International Journal of Electrical Engineering and Technology (IJEET), ISSN 0976 –
6545(Print), ISSN 0976 – 6553(Online) Volume 4, Issue 1, January- February (2013), © IAEME
The meter placement problem involves selection of number, type and place of
measurement. The main objective in designing a metering scheme is always to satisfy cost,
reliability, accuracy and bad data processing requirements. Considering all these
requirements of metering scheme, a meter placement method for IEEE 14 bus system was
developed by Mesut E Baran et al. [13]. The metering scheme designed using hybrid genetic
algorithm and simulated annealing (GA/SA) for 10 and 14 bus system is presented in [14].
3.1. PROPOSED METHOD
For observability, the presence or absence of the flow is of importance and not the
numerical value of the flow. If a branch that neither has a flow measurement on it nor an
injection measurement at one of its terminal nodes, that branch does not come into the
matrix H and thus it does not play any role either in observability analysis or in state
estimation.
Proposed measurement placement method is based on network graph theory. The
metering scheme assures that each branch of power system network is incidental by power
injection measurements at either ends or a flow measurement and an injection measurement
at one of its end. Selection of meter locations also assures least requisite of remote terminal
units (RTUs).
The proposed meter placement method proceeds as follows:
• Read bus data, initialize measurement set of interest by injection measurements at all
the zero injection buses in the power system network.
• For n bus power system network, read line data and prepare n x n adjacency matrix
A= [aij] where; aij =1, if ith bus is incident to jth bus and aij =0, if otherwise. Modify
adjacency matrix by making all aii =0, as these elements of matrix represent the bus
itself.
• Compute total ones of each row of modified adjacency matrix. Identify buses of
maximum (p) and minimum (q) adjacency. Place RTUs and measure power
injections at the buses of adjacency p, p-1, p-2 ……., till p, p-1, p-2 ……., = q+2.
• Identify branches contain no power injection measurement at one of its end, place
RTU and measure injection at any end. Add power injection measurements at the
buses of q+1 adjacency and voltage measurements at all RTU locations till
redundancy becomes ≥ 1.
• Update line data file by removing all the lines comprising of injection measurements
at both ends. Measure power flows through the remnant lines such that no
requisite of additional RTUs.
In presence of PMU, the proposed method of measurement placement can be applied to
modified power system network by removing all the buses of PMU locations and branches
connected to them. The unknown state variables are to be estimated by reduced power
system state estimation model using the metering scheme obtained through the proposed
method.
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6. International Journal of Electrical Engineering and Technology (IJEET), ISSN 0976 –
6545(Print), ISSN 0976 – 6553(Online) Volume 4, Issue 1, January- February (2013), © IAEME
4. TESTS AND RESULTS
The simulation study is performed on IEEE 14 bus test system. Figure 2 shows
designed metering scheme obtained using proposed method of measurement placement. The
metering scheme consist power injection measurements at buses 2, 4, 5, 6, 7, 9, 10, 12, 13,
14; voltage magnitudes at buses 2, 4, 5, 6, 9, 10, 12, 13, 14 and power flows on branches
connecting buses 2-1, 5-1, 2-3, 4-3, 6-11, 10-11.
Figure 2. Metering Scheme Designed for IEEE14 Bus System Using Proposed Method
For the measurement set power system network was observable. Power flow through the line
(4-3) was detected as bad measurement. After removing bad data from the available set of
measurement, still network was found observable and redundancy becomes1.44.
Table 1 shows the actual state and estimated state obtained by WLS method using metering
schemes of Mesut E Baran et al., hybrid GA/SA and proposed method of measurement
placement. Graph 1 & 2 shows voltage magnitude and bus angle errors at all the buses of
IEEE 14 bus system.
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7. International Journal of Electrical Engineering and Technology (IJEET), ISSN 0976 –
6545(Print), ISSN 0976 – 6553(Online) Volume 4, Issue 1, January- February (2013), © IAEME
Table 1. Actual and Estimated State
0.4
0.3
Voltage Error (pu)
0.2
0.1
0
1 2 3 4 5 6 7 8 9 10 11 12 13 14
-0.1
Bus Number
-0.2 Proposed Method
Mesut E Baran et al.
Hybrid GA/SA
Graph 1. Voltage error
9
7
Bus Angle Error
5
(Degree)
3
1
-1 1 2 3 4 5 6 7 8 9 10 11 12 13 14
Bus Number
Proposed Method
Mesut E Baran et al.
Hybrid GA/SA
Graph 2. Bus angle error
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8. International Journal of Electrical Engineering and Technology (IJEET), ISSN 0976 –
6545(Print), ISSN 0976 – 6553(Online) Volume 4, Issue 1, January- February (2013), © IAEME
5. CONCLUSION
The developed method avoids iterative addition of measurements and instead allows
simultaneous placement of measurement set. The proposed method accomplishes bad data
processing and observability requirements and yields much accurate state of power system.
The developed method of measurement placement can be implemented in existing state
estimators as an off- line measurement system planning tool.
6. REFERENCES
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[13] Mesut, E. B., Jinxiang, Z., Hongbo, Z. and Kenneth, E. G (1995). “A Meter Placement
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AUTHORS’ INFORMATION
R. J. MOTIYANI has received the M.E degree in Electrical Power
Engineering in 2005 from The M. S. University of Baroda,
Vadodara, India. Currently he is working with S N Patel Institute of
Technology & Research Centre as Associate Professor and Head of
Electrical Engineering Department.
Prof. (Dr.) A. R. CHUDASAMA was born on May 9, 1956. He
received the M.E degree in Electrical Power Engineering from The
M. S. University of Baroda, India in 1986. He received the Doctoral
degree from The M. S. University, Baroda, India in 2003. He
published & presented more than 55 research papers.
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