optimal meter placement using genetic algorithm to maintain network observability

Optimal meter placement using genetic algorithm

to maintain network observability

Amany El-Zonkoly *

Department of Electrical and Computer Control Engineering, Arab Academy for Science & Technology, P.O. 1029, Miami-Alexandria, Egypt

Abstract

This paper presents a genetic algorithm based method by which measurement system can be optimally determined and upgraded to maintain

network observability. Accurate monitoring of power system operation has become one of the most important functions in today’s deregulated

power markets. State estimators are the essential tools of choice in the implementation of this function. Determination of the best possible

combination of meters for monitoring a given power system is referred to as the optimal meter placement problem. The proposed algorithm yields

a measurement configuration that withstands any single branch outage and/or loss of single measurement, without losing network observability.

The proposed algorithm is based on the measurement Jacobian and sparse triangular factorization in its numerical part and based on artificial

intelligence in the decision making part. Details of the algorithm are presented using two case studies.

q 2005 Published by Elsevier Ltd.

Keywords: Artificial intelligence; Network observability; Meter placement; State estimation

1. Introduction

Accurate monitoring of power system operation has become

one of the most important functions in today’s deregulated

power markets. State estimators are the essential tools of

choice in the implementation of this function. Whether a new

state estimator is put into service or an existing one is being

upgraded, placing new meters for improving or maintaining

reliability of measurement system; is of great concern.

Determination of the best possible combination of meters for

monitoring a given power system is referred to as the optimal

meter placement problem.

This problem has been addressed earlier in various studies.

Some of them study the problem in the transmission networks

(Fetzer & Anderson, 1975; Koglin, 1975; Aam, & Holten.,

et al, 1983; Park, Moon, Choo, & Kwon, 1988; Sarma, & Raju,

et al., 1994; Clements, & Krumpholz, et al., 1983; Momticelli

& Wu, 1985; Korres & Contaxis, 1994; Baran, & Zhu et al.,

1995; Gouvea & SimoesCosta, 1996; Celik & Liu, 1995; Abur

& Magnago, 1999; Magnago & Abur, 2000; Milosevic &

Begovic, 2003) and others study the distribution systems

(Wang & Schulz, 2004). However, distribution systems have

0957-4174/$ - see front matter q 2005 Published by Elsevier Ltd.

doi:10.1016/j.eswa.2005.09.016

* Tel.: C20 35622586; fax: C20 35622366.

E-mail address: [email protected]

many features that are different from transmission systems

such as (i) radial topology, (ii) three-phase unbalanced

systems, (iii) high resistance to reactance ratio and (iv) very

limited number of real-time measurements. In this paper, we

will concentrate our research on transmission networks only.

While the majority of studies are concerned about designing

an observable system with minimum variance of errors in the

state estimates, others consider loss of lines and/or measure-

ments. They design measurement systems, which can keep the

systems observable during such unexpected disturbances.

As the power markets become more competitive, having

reliable measurement systems that can withstand branch

outages or loss of meters, will become more important and

the costs associated with metering upgrades will be justified.

Abur and Magnago (Abur & Magnago, 1999; Magnago &

Abur, 2000) presented a systematic method which provide

optimal measurement configuration for a given power system

that incorporated meter installation costs into the problem

formulation. This method made the selection not only

technically but also financially sound. In Abur and Magnago

(1999), the authors developed a topological method that

accounts for single branch outages only and solved the optimal

meter placement problem using linear programming. In

Magnago and Abur (2000), the authors presented a method

that generalizes the meter placement problem formulation in

such a way that considerations of both types of contingencies,

namely loss of a branch or a measurement, could be

simultaneously taken into account. The problem was then

Expert Systems with Applications 31 (2006) 193–198

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A. El-Zonkoly / Expert Systems with Applications 31 (2006) 193–198194

solved using integer programming. In Milosevic and Begovic

(2003), the authors consider the phasor measurement unit

(PMU) placement problem in which they try to minimize the

number of PMU’s used, so that the system is topologically

observable during its normal operation and following any

single-line contingency only. They tried also to maximize the

measurement redundancy so that instead of a unique optimal

solution, there was a set of best tradeoffs between competing

objectives, the so-called Pareto-optimal solutions (POS). A

specially tailored nondominated sorting genetic algorithm

(NSGA) for the PMU placement to find the POS was used.

However, they did not consider the meter installation costs into

the problem formulation.

In this paper, the optimal meter placement problem against

both types of contingencies at the same time and taking into

consideration the meter installation costs will be solved using an

artificial intelligence method, namely the ‘Genetic Algorithm

(GA)’. The proposed algorithm provides global solution of the

problem that may be missed by other heuristic, sequential

selection schemes. It can also be extended to solve the problem

in case of loss of multi-branches and/or multi-meters.

The paper is organized in such a way that a review of a

direct method for network observability analysis is given first.

Then, the proposed algorithm for optimal meter placement is

introduced. Finally, simulation results are presented at the end

of the paper to demonstrate method’s application to typical

power system.

2. Network observability analysis using directnumerical method

The meter placement algorithm presented in this paper is

based on the observability analysis method introduced earlier in

Gou and Abur (2000). This method will be briefly reviewed first.

The static state estimation is a mathematical procedure to

compute the best estimate of the node voltage magnitude and

angle for each node from a given set of measurements.

Network observability must be checked prior to state

estimation. If there were any unobservable parts of the

network, then a meter placement procedure will be followed,

in order to make the entire network observable.

Consider the real power versus phase angle part of the

linearized and decoupled measurement equation. This is

obtained by using the first order approximation of the

decoupled nonlinear measurement equation around an operat-

ing point:

zZHqCe (1)

where

z mismatch between the measured and calculated real

power measurements;

H decoupled Jacobian of the real power measurements

versus all bus phase angles;

q incremental change in the bus phase angle at all

buses including the slack bus;

e measurement error vector.

The decoupled gain matrix for the real power measurements

can be formed as:

GZHTH (2)

where, measurement error covariance matrix is assumed to be

the identity matrix without loss of generality. Note that, since

the slack bus is also included in the formulation, the rank of H

(and G) will be at most (nK1) (n being the number of buses),

even for a fully observable system. This leads to the triangular

factorization of a singular and symmetric gain matrix.

Consider the step where the first zero pivot is encountered

during the factorization of the singular gain matrix, as

illustrated below:

ð3Þ

where

G0 Z LK1I LK1

iK1.LK11 GLKT

1 .LKTiK1L

KTi (4)

And Li’s are elementary factors given by:

ð5Þ

lTi Z ½LiC1;i; LiC2;i;.; Lni� (6)

Lij is the ijth entry of Li.

Setting LiC1ZInxn, the triangular factorization of G’ in Eq.

(3) can proceed with the (iC2)-nd column. This procedure can

be repeated each time a zero pivot is detected until completion

of the entire factorization. The following expression can then

be written:

DZ LK1n LK1

nK1.LK11 GLKT

1 .LKTnK1L

KTn Z LK1GLKT (7)

where D is a singular and diagonal matrix with zeros in rows

corresponding to zero pivots encountered during the factoriz-

ation of G, and L is a nonsingular lower triangular matrix. If the

matrix D have more than one zero on its diagonal then the

system is unobservable.

The above method is simple and non-iterative, which makes

it computationally very efficient. Thus, it will be used to

initialize the optimal meter placement algorithm developed in

this paper.

3. Proposed algorithm for optimal meter placement

This algorithm will be applied in two steps. In the first

step, an optimal measurement scheme is determined such that

A. El-Zonkoly / Expert Systems with Applications 31 (2006) 193–198 195

the network will be fully observable. The optimal scheme will

represent the essential measurement set. In the second step of

the algorithm, the needed set of measurement that is to be

added such that the network regains its observability during

both types of contingencies mentioned before is optimally

determined. The optimization problem in the two steps will be

solved using genetic algorithm.

3.1. Solution using genetic algorithm

The genetic algorithm will proceed as follows:

1. Initialization: in this step a population of possible

solutions of the problem is initiated. Each individual

consists of a number of bits corresponding to the number

of possible meters to be added in the system. Each

individual will be assigned a fitness value which equal to

the reciprocal of the cost of the measurement set

suggested by the individual.

2. Selection: selection of individuals with highest fitness

value is performed to generate the next generation of

individuals.

3. Crossover and mutation: these are the main two operators

of genetic algorithm. Through them the information

contained by each individual are exchanged and

manipulated in order to have new individuals with better

and better fitness values.

4. The previous steps are repeated until the optimal solution

is found.

3.2. Step 1: selection of optimal measurement scheme

The novel algorithm, which will identify all the necessary

measurements to be used in order to have an observable system

will proceed as follows:

1. For each set of measurement suggested by an individual the

Jacobian (H) of the system is formed.

2. Calculate the gain matrix (G) and perform triangular

factorization to get the diagonal matrix (D).

3. If the matrix (D) have more than one zero in its diagonal

then the system is classified to be unobservable. Hence, this

individual is refused and another one is generated instead

and tested in the same manner.

4. Every individual in any population must yield an

observable system.

5. The GA search is performed with testing all of the

individuals through the generations until an optimal

solution is reached.

3.3. Step 2: optimal meter placement against measurement loss

and branch outage

With the set of measurements determined in the first step

of the solution, the proposed algorithm is to be applied

again. The algorithm will be applied in case of loss of any

of the predetermined measurements and in case of outage of

any single branch that will affect the observability of the

system to find the optimal additional set of measurements to

maintain the system observability. The measurement

Jacobian is to be formed considering the set of measure-

ments determined in step 1. During any of the two types

contingencies under consideration, the Jacobian must be

modified in accordance.

3.3.1. Loss of single measurement

In case of loss of one of the measurements, the Jacobian (H),

will be modified (Hmod) by removing one row that corresponds

to the lost measurement. Calculating the gain matrix of the

modified Jacobian and decomposing it to its factors will show

that the system is now unobservable.

The GA is applied to determine the additional set of

measurement to be used to regain the system observability. To

carry on with the observability analysis as described in Section

2, the Jacobian (Hmod) is modified again by adding rows

corresponding to the suggested set of measurements.

3.3.2. Loss of a single branch

It is known that network observability will be drastically

affected by topology changes. In general, it is not necessary to

check network observability following the outage of every

single branch in the system. It is sufficient to make the

measurement system robust against the outage of every single

tree branch only. The tree should correspond to the chosen set

of essential measurements. Thus, the loss of a co-tree branch

(or link) will not have to be considered since it will have no

effect on observability. Therefore, for a system of n buses, it is

sufficient to check the outage of nK1 tree branches. Assuming

that the tree branch kKj is outaged, the measurement Jacobian

will be modified as Hmod, where: Hmodik ZHmod

ij Z0, if

measurement i is a line flow. Hmodij Z0, Hmod

ik ZHikCHij, if

measurement I is an injection at bus k. Hmodik Z0,

Hmodij ZHikCHij, if measurement I is an injection at bus j

(Magnago & Abur, 2000).

3.3.3. Optimal selection

The selection of optimal meter placement must satisfy the

condition that the system is observable under any case of losing

either a single measurement or a single branch. The optimal

solution will yield the minimum number of added measure-

ments with minimum cost that will make the system observable

again. The selection mechanism will proceed as follows:

1. Form the Jacobian (H) corresponding to the set of

measurements determined in step1.

2. Apply the GA search as described in Section 3.1 such that

each solution is tested for observability under all possible

contingencies as described in Section 2.

3. The optimal solution is determined in the end of the

search.

Fig. 2. Cost value through the generation.


4. Simulation results

4.1. Case study 1

The simple 6 bus system shown in Fig. 1 is considered to

illustrate the proposed method. All the branch impedances are

set equal to j1 pu.

The resulting Jacobian for all possible measurements is

given below (Magnago & Abur, 2000):

q1 q2 q3 q4 q5 q6

Inj:1

Inj:2

Inj:3

Inj:4

Inj:5

Inj:6

Fl:1

Fl:2

Fl:3

2 K1 K1

2 K1 K1

K1 2 K1

K1 K1 3 K1

K1 2 K1

K1 K1 K1 3

1 K1

1 K1

1 K1

26666666666666666664

37777777777777777775

where, Inj.k and Fl.j represent the net injection at bus k and the

power flow through branch j, respectively.

4.1.1. Step 1

The genetic algorithm parameters used in the first step to run

the search for the optimal set of measurements to make the

system observable are set as follows:

Maximum generationZ100

Population sizeZ100

Crossover probabilityZ0.8

Mutation probabilityZ0.01

The cost of the best solution through out the generations is

shown in Fig. 2. Where measurements are assigned relative

cost to each other, which can be replaced by true values.

Fig. 1. 6 bus system example.

The optimal and essential set of measurements were found

to be:

½Inj:1; Inj:2; Inj:4; Inj:5; Fl:4�

4.1.2. Step 2

The genetic algorithm parameters used in the second step

are the same as in the first one. In this step, the search is

run to find the optimal set of added measurements to

maintain the system observability under all possible loss of a

measurement or a branch. The contingencies considered are

the loss of any of the measurements determined in step 1

and the outage of branches 2–3, 2–5 and 5–6. The outage of

these branches only, one at a time, will make the system

unobservable.


shown in Fig. 3.

The optimal measurement to be added to maintain system

observability under any single branch outage or loss of any

single measurement was found to be Inj.6 (the power injection

at bus 6).

4.2. Case study 2

The IEEE 30 bus system shown in Fig. 4 is considered also

to illustrate the proposed method. The system data is given in

Saadat (1994).


Fig. 4. IEEE 30 bus system.


A. El-Zonkoly / Expert Systems with Applications 31 (2006) 193–198 197

4.2.1. Step 1

The genetic algorithm parameters used in the first step to run

the search for the optimal set of measurements to make the

system observable are set as follows:

Maximum generationZ100

Population sizeZ100

Crossover probabilityZ0.8

Mutation probabilityZ0.01


shown in Fig. 5. Where measurements are assigned relative

cost to each other which can be replaced by true values.

The optimal and essential set of measurements were found

to be:


½Inj:8; Inj:9; Inj:10; Inj:12; Inj:17; Inj:18; Inj:21; Inj:27;

Fl:1K2; Fl:1K3; Fl:2K5; Fl:2K6; Fl:3K4;

Fl:4K12; Fl:6K7; Fl:8K28; Fl:9K11;

Fl:10K21; Fl:12K13; Fl:14K15; Fl:15K18;

Fl:15K23; Fl:16K17; Fl:22K24; Fl:24K25;

Fl:25K26; Fl:27K30; Fl:29K30�

4.2.2. Step 2

The genetic algorithm parameters used in the second step

are the same as in the first one. In this step, the search is run to

find the optimal set of added measurements to maintain the

system observability under all possible loss of a measurement

and/or a branch. The contingencies considered are the loss of

any of the measurements determined in step 1 and the outage of

any branch of the network one at a time.


shown in Fig. 6.

The optimal set of measurements to be added to maintain

system observability under any single branch outage and/or

loss of any single measurement was found to be as follows:

½Inj:20; Inj:28; Inj:23; Inj:25; Inj:7; Fl:5K7�

5. Conclusion

This paper presents a novel and unified algorithm to account

for contingencies when designing or upgrading measurement

systems for state estimation. Loss of a single branch and a

single measurement are considered as the two possible

contingencies, however the type and number of contingencies

can be enlarged without affecting the formulation of the

proposed algorithm. The developed algorithm avoids iterative

addition of measurements and instead allows simultaneous

placement of a minimal and optimal set of measurements that

will maintain the system observability. It is based on a

previously developed method for observability analysis and


makes use of genetic algorithm in deciding on the placement of

measurements that account for all possible contingencies

considered. The proposed algorithm is computationally very

attractive, yet simple to implement in existing state estimators

as an off-line planning tool.

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optimal meter placement using genetic algorithm to maintain network observability

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