Enhancing observability in MILP-based optimal joint allocation of PMU channels and conventional measurements with new security concepts

  • Masoud Esmaili
  • Mohammad Ghamsari-Yazdel
  • Reza Sharifi
Original Paper


Since buses in power systems have a different number of connected branches, they require Phasor Measurement Units (PMUs) with different number of measuring channels. In this paper, a multi-objective channel-based method is proposed for optimal allocation of PMU base units, PMU measuring channels, and Flow Measurements (FMs) for power system observability. The main objective function encompasses cost of PMU base units and their measuring channels as well as FMs. Unlike previous methods, PMU channels are modeled as binary optimization variables, the optimal assignment of which is decided by the optimization problem. Under this framework, a PMU assigns measuring channels to observe its adjacent buses only if they are economically justified. The second objective function maximizes measurement redundancy at selected buses in a more controlled way. For a more economical solution, existing FMs are employed as network resources along with placing new PMUs and FMs. Moreover, PMU failures and branch outages are modeled using a novel approach that is more cost-effective than existing methods. Furthermore, channel failure is also introduced and modeled as a new type of contingency. Results obtained from the proposed method clarify its efficiency from economic perspective.


Phasor measurement unit PMU measuring channel Flow measurement Measurement redundancy Channel failure 

List of symbols


\( SB \)

Set of buses

\( SFM \)

Set of existing flow measurements

\( SB_{out}^{br} \)

Set of branches considered for outage


\( C_{PMU} \)

Unit cost for a base PMU

\( C_{ch} \)

Unit cost for a PMU measuring channel

\( C_{FM} \)

Unit cost for an FM

\( a_{ij} \)

Connectivity matrix binary entries: 1 if buses \( i \) and \( j \) are connected or if \( i = j \); otherwise 0

\( z_{i} \)

1 if bus \( i \) is a zero-injection bus, otherwise 0

\( \rho_{i} \)

1 if bus \( i \) has a conventional voltage measurement, otherwise 0

\( b_{ij} \)

1 if bus \( j \) is one of terminal buses in existing FM set \( i \), otherwise 0

\( y_{ij}^{0} \)

1 if there is an existing FM on branch \( ij \), otherwise 0

\( s_{i} \)

Significance of bus \( i \) for redundant observability

\( p_{i} \)

1 if bus \( i \) is connected to more than one ZIB, otherwise 0

\( \omega_{i} \)

Weighting factor of objective function \( i \)

\( F_{i}^{nadir} \), \( F_{i}^{ideal} \)

Nadir and ideal values for objective function \( i \)


\( f_{i} \)

Observability function of bus \( i \) provided from PMU channels or ZIB properties

\( g_{i} \)

Augmented observability function of bus \( i \) including FM observability

\( x_{i} \)

1 if a PMU is installed at bus \( i \), otherwise 0

\( x_{i}^{'} \)

1 if a secondary PMU is installed at bus \( i \), otherwise 0

\( ch_{ij} \)

1 if PMU at bus \( i \) assigns a voltage channel to observe its host bus (if \( i = j \)) or a current channel to observe its adjacent bus \( j \) across branch \( ij \), otherwise 0

\( ch_{ij}^{'} \)

Binary variable denoting measuring channels of secondary PMUs

\( y_{ij} \)

1 if a new FM is to be placed on branch \( ij \) to observe bus \( i \) through already observed bus \( j \), otherwise 0

\( u_{ij} \)

Auxiliary binary variable to handle ZIB property

\( q_{i} \)

Binary variable denoting buses that can have single observation in branch outages due to being connected to multiple zero injection buses

\( F_{i} \)

Objective function \( i \)

\( \mu_{i} \)

Fuzzy membership of objective function \( i \)


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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Electrical EngineeringWest Tehran Branch, Islamic Azad UniversityTehranIran

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