Observability of power system is a necessary condition for state estimation.
State estimation provides estimation for all metered and unmetered electrical
quantities of the power system. Its output is used for online operation and
management of power system. Power system observability analysis reveals if the
present set of measurements and their distribution are sufficient to solve the
current state of the power system. Traditional state estimators use Weighted
Least Squares (WLS) method to solve system state estimation problem with conventional
measurements such as voltage magnitudes, bus real and reactive power injections
and line real and reactive power flows. This is a nonlinear-iterative method
and its solution time is considerably more than linear methods. With development
of digital signal processing and global positioning systems, Phasor measurement
system was introduced in last 1980s. Because of the ability of PMUs to gather
synchronized phasors of voltages and currents from all over the system, they
have been used in various fields of modern power systems studies such as state
estimation, transient stability analysis, fault location studies, wide area
protection and small signal stability (Baldwin et al.,
1993; Phadke and Thorp, 2006; Kai
et al., 2006; Zhao et al., 2005).
Since, PMU measures the phase and amplitude of installed bus voltages and all connected branches currents, there is no need to install PMU in all buses, thus a suitable approach is needed to determine the optimum set and locations of PMUs in the electric network.
Lots of interests have been shown toward optimal PMU placement in recent years.
For first time optimal PMU placement problem is formulated using graph theoretical
observability analysis (Baldwin et al., 1993).
Zhao et al. (2005) dealt with the placement of
a minimal set of PMUs to make system completely observable and collected more
valuable dynamic data of power system at the same time using simulated annealing
optimization method. PMU placement considering complete and incomplete observability
is done using tree search method (Nuqui and Phadke, 2005).
Gou (2008a, b) had presented
a generalized integer linear programming formulation and solution approach for
PMU placement. (Chakrabarti and Kyriakides, 2008) used
a exhaustive binary search method to find the minimum number of PMUs for power
system complete observability.
Phasor Measurement Unit (PMU) placement for some other purposes such as voltage
stability and transient stability studies were also carried out by Mili
et al. (1990). The minimal PMU placement was carried out to make
observable the transmission network fault location (Kai
et al., 2006).
The PMU placement problem is formulated and solved using several optimization
methods and algorithms such as graph theoretic procedure and Dual Search (Baldwin
et al., 1993) Tabu search (Jiangnan et al.,
2006), integer programming (Bei and Abur, 2004),
Nondominated sorting genetic algorithm (Milosevic and Begovic,
2003), integer linear programming (Gou, 2008a,
b; Dua et al., 2008). The main drawback of
integer programming is that it may be trapped in local minima as mentioned by
Chakrabarti and Kyriakides (2008).
MATERIALS AND METHODS
Observability analysis is a fundamental component of real-time state estimation. There are two major algorithms for power network observability analysis: topology based algorithms and numerical methods. Topology methods use the decoupled measurement model and graph theory. In these methods decision is based on logical operations. Thus, they require only information about network connectivity, measurement types and their locations. If a full rank spanning tree can be constructed with current measurement set, the system will be observable.
Numerical methods, on the other hand, use either fully coupled or decoupled measurement models. These methods are based on numerical factorization of the measurement Jacobian or measurement information gain matrix. If any of these matrices is full rank, the system will be observable and there will be a unique solution for state estimation problem. Numerical methods are not suitable for large systems because they are involved with huge matrix manipulation and have their own computational complexity. Therefore, topology based method is used in this paper. It should be mentioned that OPP problem is formulated with numerical method based genetic algorithm, hitherto.
TOPOLOGY BASED FORMULATION
In this study, the optimal PMU placement problem is described as finding a
scheme with minimal PMUs and their installation locations such that the entire
system becomes observable. The used observability rules are as follows:
||For PMU installed buses, voltage phasor and current phasor
of all its incident branches are known. These are called as direct measurements
||If voltage and current phasors at one end of a branch are known then voltage
phasor at the other end of the branch can be obtained. These are called
||If voltage phasors of both ends of a branch are known then the current
phasor of this branch can be obtained directly. These measurements are also
called pseudo measurements
||For a zero-injection bus i in a N-bus system we have:
where, Yij is the ij-th element of admittance matrix of the system
and Vj is the voltage phasor of j-th buse.
Therefore, if there is a zero-injection bus without PMU whose incident branches
current phasors are all known but one, then the current phasor of the unknown
one could be obtainable using KCL equations.
||If there is a zero-injection bus with unknown voltage phasor
and voltage phasors of its adjacent buses are all known, then the voltage
phasor of the zero-injection node can be found by node equations
||If there exists a group of adjacent zero-injection buses whose voltage
phasors are unknown but the voltage phasors of all adjacent buses to the
group are known, then the voltage phasors of zero-injection buses can be
obtained through node equations
The measurements obtained from rules 4-6 are called extended measurements.
BRANCH AND BOUND (B AND B) APPROACH
The OPP formulation based on topological observability method that used in
B and B approach is as follows:
where, N is total No. of system buses, wi is weight factor accounting
to the cost of installed PMU at bus i, X is a binary variable vector whose entries
are defined as Eq. 3 and F(X) is a vector function that its
entries are non-zero if the corresponding bus voltage is observable using the
given measurement set and according to observability rules mentioned in earlier;
otherwise its entries are zero.
Two different methods have been proposed to take into account the zero-injection
buses. These are topology transformation and nonlinear constraint functions
methods. In this study, constraint functions considering adjacent zero-injection
buses are constructed using a novel hybrid topology transformation-nonlinear
constraint method. This method is illustrated below using an example.
In the New England 39-bus test system (Fig. 1), buses 5 and
6 are zero-injection buses. First, these two buses are combined together to
make the new bus 5 with the new constraint function (topology transformation):
Next, the new bus constraint function is eliminated from the formulation using
nonlinear constraint method. Thus, the neighboring buses constraint functions
are corrected as follows:
Branch and bound is an optimization method suitable for non-convex problems
and works very well for solving OPP. In this study, the problem is spitted into
sub-problems (equal to the number of nodes) and for each node the upper and
lower bounds are calculated using relaxation methods. Whenever the lower bound
of a node becomes larger than its upper bound, that node is removed from the
tree. In this case, in each step, each sub-problem is divided into two sub-trees
according to fixing xi to either zero or one and a sub-tree which
has the lowest lower bound is selected to continue optimization.
||New England 39-bus test system
In the next step, a variable having the value which is closest to either 0
or 1 will be selected to be fixed.
GENETIC ALGORITHM APPROACH
Genetic algorithm is one of the effective meta-heuristic methods developed
in order to solve nonlinear and non-convex optimization problems. This algorithm
is not dependent to objective function gradient. Besides, it has the capability
of globally convergence. In order to formulate the OPP in this approach, topology
of the system is shown by bus connection matrix (A). A is a N_busxN_bus symmetric
binary matrix that its arrays defined as Eq. 6.
In this method, the OPP problem is formulated as follows:
where, yij is observability index and defined as:
From observability rules mentioned earlier, it can be conducted that a zero
injection bus is observable if all its adjacent buses be observable. Thus, to
consider zero injection buses effect, a modified A matrix is developed by merging
zero injection buses with one of the adjacent buses. In Fig. 2,
bus-4 is zero injection bus and it merged with third bus and made bus numbered
Figure 3 shows the flowchart of OPP problem solution using
||Zero injection bus merging method
||Flowchart of OPP solution using GA
CASE STUDY RESULTS
The IEEE 14-bus, 30 -bus, 57-bus and 118-bus and the New England 39-bus test systems are used for observability analysis studies using Branch and Bound (B and B) and Genetic Algorithm (GA) approaches. Table 1 shows the data for these systems.
Optimum number of PMUs and their location using B and B approach and without considering zero injection buses information are shown in Table 2.
In Table 3, optimum PMU placement is done using GA approach and without considering zero injection buses information.
Optimum number of PMUs and their location using B and B approach and with considering
zero injection buses information are shown in Table 4.
||Test systems data
||Optimum number and location of PMUs using B and B approach
without considering ZI buses information
||Optimum number and location of PMUs using GA approach without
considering ZI buses information
||Optimum number and location of PMUs using B and B approach
with considering ZI buses information
||Optimum number and location of PMUs using GA approach with
considering ZI buses information
||Comparison of results of different algorithms
|N/A: Not available
In Table 5, optimum PMU placement is done using GA approach
and with considering zero injection buses information.
The results obtained using proposed approaches are compared with some other
algorithms such as Tabu search (Jiangnan et al.,
2006), Integer linear programming (Gou, 2008a),
nondominated sorting genetic algorithm (Milosevic and Begovic,
2003), Graph theoretic procedure and Dual Search (Baldwin
et al., 1993). Table 6 shows the comparison of
results. In Table 6, N/A shows that the result was not available
for that case.
In this study, the problem of optimum PMU placement for power system observability was investigated. The OPP problem was formulated using topology based algorithms and solved using branch and bound algorithm and genetic algorithm, proposed algorithms applied to some standard test systems.
Simulation results on the IEEE 14, 30, 57 and 118-bus test systems and the New England 39-bus test system indicate that the proposed placement methods satisfactorily provides observable system measurements with minimum number of PMUs.
The results show good improvement in decreasing the number of installed PMUs comparing with earlier applied methods.